Research Article Proof of Some Conjectures of Melham Using...
7
Research Article Proof of Some Conjectures of Melham Using Ramanujan’s 1 1 Formula Bipul Kumar Sarmah Department of Mathematical Sciences, Tezpur University, Napaam, Sonitpur, Assam 784028, India Correspondence should be addressed to Bipul Kumar Sarmah; [email protected] Received 5 February 2014; Accepted 16 June 2014; Published 10 July 2014 Academic Editor: Wolfgang zu Castell Copyright © 2014 Bipul Kumar Sarmah. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We employ Ramanujan’s 1 1 formula to prove three conjectures of R. S. Melham on representation of an integer as sums of polygonal numbers. 1. Introduction Jacobi’s classical two-square theorem is as follows. eorem 1 (see [1]). Let {◻ + ◻}() denote the number of representations of as a sum of two squares, counting order and sign, and let , () denote the number of positive divisors of congruent to modulo . en {◻ + ◻} () = 4 ( 1,4 () − 3,4 ()). (1) e above theorem can also be recasted in terms of Lambert series as ∞ ∑ =0 {◻ + ◻} () =1+4 ∞ ∑ =0 ( 4+1 1− 4+1 − 4+3 1− 4+3 ). (2) Similar representation theorems involving squares and tri- angular numbers were found by Dirichlet [2], Lorenz [3], Legendre [4], and Berndt [5]. For example, the following two theorems are due to Lorenz and Ramanujan, respectively. eorem 2 (see [3]). Let {◻ + ◻}() denote the number of representations of as a sum of times a square and times a square. en {◻ + 3◻} () = 2 ( 1,3 () − 2,3 ()) + 4 ( 4,12 () − 8,12 ()). (3) eorem 3 (see [5]). Let {Δ + Δ}() denote the number of representations of as a sum of times a triangular number and times a triangular number. en {Δ + 3Δ} () = 1,3 (2 + 1) − 2,3 (2 + 1) . (4) Hirschhorn [6, 7] obtained forty-five similar identities (including those obtained by Legendre and Ramanujan) involving squares, triangular numbers, pentagonal numbers, and octagonal numbers employing dissection of the - series representations of the identities obtained by Jacobi, Dirichlet, and Lorenz. In [8], Baruah and the author obtained twenty-five more such identities involving squares, triangular numbers, pentagonal numbers, heptagonal numbers, octag- onal numbers, decagonal numbers, hendecagonal numbers, dodecagonal numbers, and octadecagonal numbers. More works on this topic have been done in [9–11]. In [11], Melham presented 21 conjectured analogues of Jacobi’s two-square theorem which are verified using computer algorithms. In [12], Toh offered a uniform approach to prove these conjec- tures using known formulae for {◻ + ◻}(). In this paper, we show that some of these conjectures can also be proved by using Ramanujan’s famous 1 1 formula. We prove three conjectures enlisted in the following theorem which have appeared as (6), (7), and (8), respectively, in [11]. Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2014, Article ID 738948, 6 pages http://dx.doi.org/10.1155/2014/738948
Transcript of Research Article Proof of Some Conjectures of Melham Using...
Research ArticleProof of Some Conjectures of Melham UsingRamanujanrsquos
11205951
Formula
Bipul Kumar Sarmah
Department of Mathematical Sciences Tezpur University Napaam Sonitpur Assam 784028 India
Correspondence should be addressed to Bipul Kumar Sarmah bipultezuernetin
Received 5 February 2014 Accepted 16 June 2014 Published 10 July 2014
Academic Editor Wolfgang zu Castell
Copyright copy 2014 Bipul Kumar SarmahThis is an open access article distributed under theCreativeCommonsAttributionLicensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We employ Ramanujanrsquos11205951formula to prove three conjectures of R S Melham on representation of an integer 119899 as sums of
polygonal numbers
1 Introduction
Jacobirsquos classical two-square theorem is as follows
Theorem 1 (see [1]) Let 119903◻ + ◻(119899) denote the number ofrepresentations of 119899 as a sum of two squares counting orderand sign and let 119889
119894119895(119899) denote the number of positive divisors
of 119899 congruent to 119894modulo 119895 Then
119903 ◻ + ◻ (119899) = 4 (11988914(119899) minus 119889
34(119899)) (1)
The above theorem can also be recasted in terms ofLambert series asinfin
sum
119899=0
119903 ◻ + ◻ (119899) 119902119899= 1 + 4
infin
sum
119899=0
(1199024119899+1
1 minus 1199024119899+1minus1199024119899+3
1 minus 1199024119899+3)
(2)
Similar representation theorems involving squares and tri-angular numbers were found by Dirichlet [2] Lorenz [3]Legendre [4] and Berndt [5] For example the following twotheorems are due to Lorenz and Ramanujan respectively
Theorem 2 (see [3]) Let 119903119897◻ + 119898◻(119899) denote the number ofrepresentations of 119899 as a sum of 119897 times a square and119898 times asquare Then
119903 ◻ + 3◻ (119899) = 2 (11988913(119899) minus 119889
23(119899))
+ 4 (119889412(119899) minus 119889
812(119899))
(3)
Theorem 3 (see [5]) Let 119903119897Δ +119898Δ(119899) denote the number ofrepresentations of 119899 as a sum of 119897 times a triangular numberand119898 times a triangular number Then
119903 Δ + 3Δ (119899) = 11988913(2119899 + 1) minus 119889
23(2119899 + 1) (4)
Hirschhorn [6 7] obtained forty-five similar identities(including those obtained by Legendre and Ramanujan)involving squares triangular numbers pentagonal numbersand octagonal numbers employing dissection of the 119902-series representations of the identities obtained by JacobiDirichlet and Lorenz In [8] Baruah and the author obtainedtwenty-fivemore such identities involving squares triangularnumbers pentagonal numbers heptagonal numbers octag-onal numbers decagonal numbers hendecagonal numbersdodecagonal numbers and octadecagonal numbers Moreworks on this topic have been done in [9ndash11] In [11] Melhampresented 21 conjectured analogues of Jacobirsquos two-squaretheorem which are verified using computer algorithms In[12] Toh offered a uniform approach to prove these conjec-tures using known formulae for 119903119897◻ + 119898◻(119899) In this paperwe show that some of these conjectures can also be provedby using Ramanujanrsquos famous
11205951formula We prove three
conjectures enlisted in the following theorem which haveappeared as (6) (7) and (8) respectively in [11]
Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2014 Article ID 738948 6 pageshttpdxdoiorg1011552014738948
2 International Journal of Mathematics and Mathematical Sciences
Theorem 4 Considerinfin
sum
119899=0
119903 Δ + 5Δ (119899) 119902119899=
infin
sum
119899=0
(1199023119899+ 1199027119899+1
1 minus 11990220119899+5minus11990213119899+9+ 11990217119899+12
1 minus 11990220119899+15
)
(5)infin
sum
119899=0
119903 Δ + 6Δ (119899) 119902119899=
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9
minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
(6)infin
sum
119899=0
119903 2Δ + 3Δ (119899) 119902119899=
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3+1199027119899+2
1 minus 11990224119899+9
minus11990217119899+10
1 minus 11990224119899+15minus11990219119899+16
1 minus 11990224119899+21)
(7)
The next section of this paper is devoted to notationsdefinitions and preliminary results
2 Notations and Preliminary Results
Ramanujanrsquos general theta-function 119891(119886 119887) is defined by [5page 34 (181)]
119891 (119886 119887) =
infin
sum
119896=minusinfin
119886119896(119896+1)2119887119896(119896minus1)2 |119886119887| lt 1 (8)
Jacobirsquos famous triple product identity takes the simple form[5 page 35 Entry 19]
119891 (119886 119887) = (minus119886 119886119887)infin(minus119887 119886119887)
infin(119886119887 119886119887)
infin (9)
where here and throughout the paper for |119902| lt 1
(119886 119902)119899=
119899
prod
119896=0
(1 minus 119886119902119896minus1) (119886 119902)
infin=
infin
prod
119896=0
(1 minus 119886119902119896)
(10)
We also use the following notation for the sake of brevityof expressions
[1198861 1198862 1198863 119886
119899 119902]infin
= (1198861 119902)infin(1198862 119902)infin(1198863 119902)infinsdot sdot sdot (119886119899 119902)infin
(11)
One important special case of 119891(119886 119887) is
120595 (119902) = 119891 (119902 1199023) =
infin
sum
119896=0
119902119896(119896+1)2=
(1199022 1199022)infin
(119902 1199022)infin
(12)
where the product representations in (12) arise from (9)From [5 page 46 Entry 30] we find that
119891 (119886 119887) = 119891 (1198863119887 1198861198873) + 119886119891 (119886
minus1119887 11988651198873) (13)
Also if 119886119887 = 119888119889 then [5 page 45 Entry 29]
119891 (119886 119887) (119888 119889) = 119891 (119886119888 119887119889) 119891 (119886119889 119887119888)
+ 119886119891 (119887119888minus1 1198861198882119889)119891 (119887119889
minus1 1198861198881198892)
(14)
We recall Ramanujanrsquos11205951formula in the form [5 page
34 (176)]infin
sum
119899=minusinfin
(119886 119902)119899
(119887 119902)119899
119911119899=
(119886119911 119902)infin(119902119886minus1119911minus1 119902)infin(119902 119902)infin(119887119886minus1 119902)infin
(119911 119902)infin(119902119886minus1119911minus1 119902)infin(119887 119902)infin(119902119886minus1 119902)infin
(15)
where10038161003816100381610038161003816119887119886minus110038161003816100381610038161003816lt |119911| lt 1 (16)
Finally we note thatinfin
sum
119899=0
119903 119886Δ + 119887Δ (119899) 119902119899= 120595 (119902
119886) 120595 (119902
119887) (17)
3 Proof of the Conjectures (5)ndash(7)Proof of Conjecture (5) In view of (17) we rewrite (5) as
120595 (119902) 120595 (1199025) =
infin
sum
119899=0
(1199023119899+ 1199027119899+1
1 minus 11990220119899+5minus11990213119899+9+ 11990217119899+12
1 minus 11990220119899+15
) (18)
Replacing 119902 by 11990220 and then setting 119911 = 1199023 119886 = 1199025 and119887 = 11990225 in (15) we have
infin
sum
119899=minusinfin
(1199025 11990220)119899
(11990225 11990220)119899
1199023119899
=
(1199028 11990220)infin(11990212 11990220)infin(11990220 11990220)2
infin
(1199023 11990220)infin(11990217 11990220)infin(11990225 11990220)infin(11990215 11990220)infin
(19)
Expanding the 119902-products on the left side of (19) and notingthat (1 minus 1199025)(11990225 11990220)
infin= (1199025 11990220)infin we obtain
infin
sum
119899=minusinfin
1199023119899
1 minus 11990220119899+5
=
(1199028 11990220)infin(11990212 11990220)infin(11990220 11990220)2
infin
(1199023 11990220)infin(11990217 11990220)infin(1199025 11990220)infin(11990215 11990220)infin
(20)
Alsoinfin
sum
119899=minusinfin
1199023119899
1 minus 11990220119899+5=
infin
sum
119899=0
1199023119899
1 minus 11990220119899+5+
minus1
sum
119899=minusinfin
1199023119899
1 minus 11990220119899+5
=
infin
sum
119899=0
1199023119899
1 minus 11990220119899+5+
infin
sum
119899=1
119902minus3119899
1 minus 119902minus20119899+5
=
infin
sum
119899=0
1199023119899
1 minus 11990220119899+5minus
infin
sum
119899=1
11990217119899minus5
1 minus 11990220119899minus5
=
infin
sum
119899=0
1199023119899
1 minus 11990220119899+5minus
infin
sum
119899=0
11990217119899+12
1 minus 11990220119899+15
(21)
International Journal of Mathematics and Mathematical Sciences 3
Thus from (20) and (21) we haveinfin
sum
119899=0
1199023119899
1 minus 11990220119899+5minus
infin
sum
119899=0
11990217119899+12
1 minus 11990220119899+15
=
(1199028 11990220)infin(11990212 11990220)infin(11990220 11990220)2
infin
(1199023 11990220)infin(11990217 11990220)infin(1199025 11990220)infin(11990215 11990220)infin
(22)
Replacing 119902 by 11990220 and then setting 119911 = 1199023 119886 = 1199025 and 119887 = 11990225in (15) and proceeding as in case of (22) we find thatinfin
sum
119899=0
1199027119899
1 minus 11990220119899+5minus
infin
sum
119899=0
11990213119899+8
1 minus 11990220119899+15
=
(11990212 11990220)infin(1199028 11990220)infin(11990220 11990220)2
infin
(1199027 11990220)infin(11990213 11990220)infin(1199025 11990220)infin(11990215 11990220)infin
(23)
Multiplying (23) by 119902 and adding to (22) we obtaininfin
sum
119899=0
(1199023119899+ 1199027119899+1
1 minus 11990220119899+5minus11990213119899+9+ 11990217119899+12
1 minus 11990220119899+15
)
=
(1199028 11990220)infin(11990212 11990220)infin(11990220 11990220)2
infin
(1199023 11990220)infin(11990217 11990220)infin(1199025 11990220)infin(11990215 11990220)infin
+ 119902
(11990212 11990220)infin(1199028 11990220)infin(11990220 11990220)2
infin
(1199027 11990220)infin(11990213 11990220)infin(1199025 11990220)infin(11990215 11990220)infin
=
[1199028 11990212 11990220 11990220]infin
[1199023 1199025 1199027 11990213 11990215 11990217 11990220]infin
times ((1199027 11990220)infin(11990213 11990220)infin(11990220 11990220)infin
+ 119902(1199023 11990220)infin(11990217 11990220)infin(11990220 11990220)infin)
=
[1199028 11990212 11990220 11990220]infin
[1199023 1199025 1199027 11990213 11990215 11990217 11990220]infin
times (119891 (minus1199027 minus11990213) + 119902119891 (minus119902
3 minus11990217))
(24)
Setting 119886 = 119902 and 119887 = minus1199024 in (13) we find that
119891 (minus1199027 minus11990213) + 119902119891 (minus119902
3 minus11990217) = 119891 (119902 minus119902
4) (25)
Thus (24) reduces toinfin
sum
119899=0
(1199023119899+ 1199027119899+1
1 minus 11990220119899+5minus11990213119899+9+ 11990217119899+12
1 minus 11990220119899+15
)
=
[1199028 11990212 11990220 11990220]infin
[1199023 1199025 1199027 11990213 11990215 11990217 11990220]infin
119891 (119902 minus1199024)
(26)
Now
119891 (119902 minus1199024) = (minus119902 minus119902
5)infin(1199024 minus1199025)infin(minus1199025 minus1199025)infin (27)
We have
(minus119902 minus1199025)infin= (minus119902 119902
10)infin(1199026 11990210)infin
=
(1199022 11990220)infin(1199026 11990210)infin
(119902 11990210)infin
=
[1199022 1199026 11990216 11990220]infin
[119902 11990211 11990220]infin
(28)
Similarly
(1199024 minus1199025)infin=
[1199024 11990214 11990218 11990220]infin
[1199029 11990219 11990220]infin
(minus1199025 minus1199025)infin=
[11990210 11990210 11990220 11990220]infin
[1199025 11990215 11990220]infin
(29)
Hence from (27) (28) and (29) we have
119891 (119902 minus1199024) =
[1199022 1199024 1199026 11990210 11990210 11990214 11990216 11990218 11990220 11990220]infin
[119902 1199025 1199029 11990211 11990215 11990219 11990220]infin
(30)
Using (30) in (26) we find that
infin
sum
119899=0
(1199023119899+ 1199027119899+1
1 minus 11990220119899+5minus11990213119899+9+ 11990217119899+12
1 minus 11990220119899+15
)
=
[1199022 1199024 1199026 1199028 11990210 11990210 11990212 11990214 11990216 11990218 11990220 11990220 11990220]infin
[119902 1199023 1199025 1199025 1199027 1199029 11990211 11990213 11990215 11990215 11990217 11990219 11990220]infin
= ([1199022 1199024 1199026 1199028 11990210 11990212 11990214 11990216 11990218 11990220 11990220]infin
times[11990210 11990220 11990220]infin)
times ([119902 1199023 1199025 1199027 1199029 11990211 11990213 11990215 11990217 11990219 11990220]infin
times[1199025 11990215 11990220]infin)minus1
=
(1199022 1199022)infin(11990210 11990210)infin
(119902 1199022)infin(1199025 11990210)infin
= 120595 (119902) 120595 (1199025)
(31)
This completes the proof of (18)
Proof of Conjecture (6) In view of (17) we rewrite (6) as
120595 (119902) 120595 (1199026) =
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9
minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
(32)
4 International Journal of Mathematics and Mathematical Sciences
Replacing 119902 by 11990224 and then setting 119911 = 1199027 119886 = 1199023 and 119887 = 11990227in (15) and proceeding as in case of (22) we find that
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3minus11990217119899+14
1 minus 11990224119899+21)
=
(11990210 11990224)infin(11990214 11990224)infin(11990224 11990224)2
infin
(1199027 11990224)infin(11990217 11990224)infin(1199023 11990224)infin(11990221 11990224)infin
(33)
Replacing 119902 by 11990224 and then setting 119911 = 1199025 119886 = 1199029 and119887 = 11990233 in (15) and proceeding in a way similar to obtaining
(22) we have
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+9minus11990219119899+10
1 minus 11990224119899+15)
=
(11990214 11990224)infin(11990210 11990224)infin(11990224 11990224)2
infin
(1199025 11990224)infin(11990219 11990224)infin(1199029 11990224)infin(11990215 11990224)infin
(34)
Multiplying (34) by 119902 and then adding to (33) we find that
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
(11990210 11990224)infin(11990214 11990224)infin(11990224 11990224)2
infin
(1199027 11990224)infin(11990217 11990224)infin(1199023 11990224)infin(11990221 11990224)infin
+ 119902
(11990214 11990224)infin(11990210 11990224)infin(11990224 11990224)2
infin
(1199025 11990224)infin(11990219 11990224)infin(1199029 11990224)infin(11990215 11990224)infin
=
[11990210 11990214 11990224]infin
[1199023 1199025 1199027 1199029 11990215 11990217 11990219 11990221 11990224]infin
times (119891 (minus1199025 minus11990219) 119891 (minus119902
9 minus11990215)
+ 119902119891 (minus1199027 minus11990217) 119891 (minus119902
3 minus11990221))
(35)
Setting 119886 = minus119902 119887 = minus11990211 119888 = minus1199024 and 119889 = minus1199028 in (14) wefind that
119891 (minus1199025 minus11990219) 119891 (minus119902
9 minus11990215) + 119902119891 (minus119902
7 minus11990217) 119891 (minus119902
3 minus11990221)
= 119891 (119902 11990211) 119891 (minus119902
4 minus1199028)
(36)
Using (36) in (35) we have
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
[11990210 11990214 11990224]infin
[1199023 1199025 1199027 1199029 11990215 11990217 11990219 11990221 11990224]infin
times 119891 (119902 11990211) 119891 (minus119902
4 minus1199028)
(37)
Now
119891 (119902 11990211) = (minus119902 119902
12)infin(minus11990211 11990212)infin(11990212 11990212)infin
=
(1199022 11990224)infin(11990222 11990224)infin(11990212 11990212)infin
(119902 11990212)infin(11990211 11990212)infin
=
[1199022 11990212 11990222 11990224 11990224]infin
[119902 11990211 11990213 11990223 11990224]infin
119891 (minus1199024 minus1199028) = (119902
4 11990212)infin(1199028 11990212)infin(11990212 11990212)infin
= [1199024 1199028 11990212 11990216 11990220 11990224 11990224]infin
(38)
Using (38) in (37) we have
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
[1199022 1199024 1199028 11990210 11990212 11990212 11990214 11990216 11990220 11990222 11990224 11990224 11990224]infin
[119902 1199023 1199025 1199027 1199029 11990211 11990213 11990215 11990217 11990219 11990221 11990223 11990224]infin
= ([1199022 1199024 1199026 1199028 11990210 11990212 11990214 11990216 11990218 11990220 11990222 11990224 11990224]infin
times [11990212 11990224 11990224]infin)
times ([119902 1199023 1199025 1199027 1199029 11990211 11990213 11990215 11990217 11990219 11990221 11990223 11990224]infin
times [1199026 11990212 11990224]infin)minus1
=
(1199022 1199022)infin(11990212 11990212)infin
(119902 1199022)infin(1199026 11990212)infin
= 120595 (119902) 120595 (1199026)
(39)
This completes the proof of (32)
Proof of Conjecture (7) In view of (17) we rewrite (7) as
120595 (1199022) 120595 (119902
3) =
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3+1199027119899+2
1 minus 11990224119899+9
minus11990217119899+10
1 minus 11990224119899+15minus11990219119899+16
1 minus 11990224119899+21)
(40)
International Journal of Mathematics and Mathematical Sciences 5
Replacing 119902 by 11990224 and then setting 119911 = 1199025 119886 = 1199023 and119887 = 11990227 in (15) and proceeding as in case of (22) we find that
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3minus11990219119899+16
1 minus 11990224119899+21)
=
(1199028 11990224)infin(11990216 11990224)infin(11990224 11990224)2
infin
(1199025 11990224)infin(11990219 11990224)infin(1199023 11990224)infin(11990221 11990224)infin
(41)
Also replacing 119902 by 11990224 and then setting 119911 = 1199027 119886 = 1199029and 119887 = 11990233 in (15) and proceeding similar to (22) we obtain
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+9minus11990217119899+8
1 minus 11990224119899+15)
=
(11990216 11990224)infin(1199028 11990224)infin(11990224 11990224)2
infin
(1199027 11990224)infin(11990217 11990224)infin(1199029 11990224)infin(11990215 11990224)infin
(42)
Multiplying (42) by 1199022 and then adding to (41) we find that
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3+1199027119899+2
1 minus 11990224119899+9minus11990217119899+10
1 minus 11990224119899+15minus11990219119899+16
1 minus 11990224119899+21)
=
(1199028 11990224)infin(11990216 11990224)infin(11990224 11990224)2
infin
(1199025 11990224)infin(11990219 11990224)infin(1199023 11990224)infin(11990221 11990224)infin
+ 1199022(11990216 11990224)infin(1199028 11990224)infin(11990224 11990224)2
infin
(1199027 11990224)infin(11990217 11990224)infin(1199029 11990224)infin(11990215 11990224)infin
=
[1199028 11990216 11990224]infin
[1199023 1199025 1199027 1199029 11990215 11990217 11990219 11990221 11990224]infin
times (119891 (minus1199027 minus11990217) 119891 (minus119902
9 minus11990215)
+ 1199022119891 (minus119902
5 minus11990219) 119891 (minus119902
3 minus11990221))
(43)
Setting 119886 = 1199022 119887 = minus11990210 119888 = minus1199025 and 119889 = minus1199027 in (14) we findthat
119891 (minus1199027 minus11990217) 119891 (minus119902
9 minus11990215) + 1199022119891 (minus119902
5 minus11990219) 119891 (minus119902
3 minus11990221)
= 119891 (1199022 11990210) 119891 (minus119902
5 minus1199027)
(44)
Hence (43) reduces toinfin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
[1199028 11990216 11990224]infin
[1199023 1199025 1199027 1199029 11990215 11990217 11990219 11990221 11990224]infin
times 119891 (1199022 11990210) 119891 (minus119902
5 minus1199027)
(45)
Now
119891 (1199022 11990210) = (minus119902
2 11990212)infin(minus11990210 11990212)infin(11990212 11990212)infin
=
(1199024 11990224)infin(11990220 11990224)infin(11990212 11990212)infin
(1199022 11990212)infin(11990210 11990212)infin
=
[1199024 11990212 11990220 11990224 11990224]infin
[1199022 11990210 11990214 11990222 11990224]infin
119891 (minus1199025 minus1199027) = (119902
5 11990212)infin(1199027 11990212)infin(11990212 11990212)infin
= [1199025 1199027 11990212 11990217 11990219 11990224 11990224]infin
(46)
Using (46) in (45) we have
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
[1199024 1199025 1199027 1199028 11990212 11990212 11990216 11990217 11990219 11990220 11990224 11990224 11990224]infin
[1199022 1199023 1199025 1199027 1199029 11990210 11990214 11990215 11990217 11990219 11990221 11990222 11990224]infin
=
[1199024 1199028 11990212 11990216 11990220 11990224 11990224]infin[1199026 11990212 11990218 11990224 11990224]infin
[1199022 1199026 11990210 11990214 11990218 11990222 11990224]infin[1199023 1199029 11990215 11990221 11990224]infin
=
(1199024 1199024)infin(1199026 1199026)infin
(1199022 1199024)infin(1199023 1199026)infin
= 120595 (1199022) 120595 (119902
3)
(47)
This completes the proof of (40)
4 Concluding Remarks
All conjectures in [11] can easily be reformulated as thetafunction identity using Ramanujanrsquos
11205951formula However
these identities might be too complicated to actually have aproof
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] C G J Jacobi Fundamenta Nova Theoriae Functionum Ellipti-carum vol 107 of Werke I 162-163 Letter to Legendre 991828Werke I 424 1829
[2] P G L Dirichlet J Math 21 (1840) 3 6 Werke 463 466[3] L Lorenz ldquoBidrag til tallenes theorirdquo Tidsskrift for Mathematik
vol 3 no 1 pp 97ndash114 1871[4] A M Legendre Traite des Fonctions Elliptiques et des Integrales
Euleriennes III Huzard-Courcier Paris France 1828[5] B C Berndt Ramanujanrsquos Notebooks Part III Springer New
York NY USA 1991
6 International Journal of Mathematics and Mathematical Sciences
[6] M D Hirschhorn ldquoThe number of representations of a numberby various formsrdquo Discrete Mathematics vol 298 no 1ndash3 pp205ndash211 2005
[7] M D Hirschhorn ldquoThe number of representation of a numberby various forms involving triangles squares pentagons andoctagonsrdquo in Ramanujan Rediscovered N D Baruah B CBerndt S Cooper T Huber and M Schlosser Eds RMS Lec-ture Note Series No 14 pp 113ndash124 Ramanujan MathematicalSociety 2010
[8] ND Baruah and B K Sarmah ldquoThenumber of representationsof a number as sums of various polygonal numbersrdquo Integersvol 12 2012
[9] H Y Lam ldquoThe number of representations by sums of squaresand triangular numbersrdquo Integers vol 7 article A28 2007
[10] R S Melham ldquoAnalogues of two classical theorems on therepresentations of a numberrdquo Integers vol 8 article A51 2008
[11] R S Melham ldquoAnalogues of Jacobirsquos two-square theorem aninformal accountrdquo Integers vol 10 no 1 pp 83ndash100 2010
[12] P C Toh ldquoOn representations by figurate numbers a uniformapproach to the conjectures of Melhamrdquo International Journalof Number Theory vol 9 no 4 pp 1055ndash1071 2013
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 International Journal of Mathematics and Mathematical Sciences
Theorem 4 Considerinfin
sum
119899=0
119903 Δ + 5Δ (119899) 119902119899=
infin
sum
119899=0
(1199023119899+ 1199027119899+1
1 minus 11990220119899+5minus11990213119899+9+ 11990217119899+12
1 minus 11990220119899+15
)
(5)infin
sum
119899=0
119903 Δ + 6Δ (119899) 119902119899=
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9
minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
(6)infin
sum
119899=0
119903 2Δ + 3Δ (119899) 119902119899=
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3+1199027119899+2
1 minus 11990224119899+9
minus11990217119899+10
1 minus 11990224119899+15minus11990219119899+16
1 minus 11990224119899+21)
(7)
The next section of this paper is devoted to notationsdefinitions and preliminary results
2 Notations and Preliminary Results
Ramanujanrsquos general theta-function 119891(119886 119887) is defined by [5page 34 (181)]
119891 (119886 119887) =
infin
sum
119896=minusinfin
119886119896(119896+1)2119887119896(119896minus1)2 |119886119887| lt 1 (8)
Jacobirsquos famous triple product identity takes the simple form[5 page 35 Entry 19]
119891 (119886 119887) = (minus119886 119886119887)infin(minus119887 119886119887)
infin(119886119887 119886119887)
infin (9)
where here and throughout the paper for |119902| lt 1
(119886 119902)119899=
119899
prod
119896=0
(1 minus 119886119902119896minus1) (119886 119902)
infin=
infin
prod
119896=0
(1 minus 119886119902119896)
(10)
We also use the following notation for the sake of brevityof expressions
[1198861 1198862 1198863 119886
119899 119902]infin
= (1198861 119902)infin(1198862 119902)infin(1198863 119902)infinsdot sdot sdot (119886119899 119902)infin
(11)
One important special case of 119891(119886 119887) is
120595 (119902) = 119891 (119902 1199023) =
infin
sum
119896=0
119902119896(119896+1)2=
(1199022 1199022)infin
(119902 1199022)infin
(12)
where the product representations in (12) arise from (9)From [5 page 46 Entry 30] we find that
119891 (119886 119887) = 119891 (1198863119887 1198861198873) + 119886119891 (119886
minus1119887 11988651198873) (13)
Also if 119886119887 = 119888119889 then [5 page 45 Entry 29]
119891 (119886 119887) (119888 119889) = 119891 (119886119888 119887119889) 119891 (119886119889 119887119888)
+ 119886119891 (119887119888minus1 1198861198882119889)119891 (119887119889
minus1 1198861198881198892)
(14)
We recall Ramanujanrsquos11205951formula in the form [5 page
34 (176)]infin
sum
119899=minusinfin
(119886 119902)119899
(119887 119902)119899
119911119899=
(119886119911 119902)infin(119902119886minus1119911minus1 119902)infin(119902 119902)infin(119887119886minus1 119902)infin
(119911 119902)infin(119902119886minus1119911minus1 119902)infin(119887 119902)infin(119902119886minus1 119902)infin
(15)
where10038161003816100381610038161003816119887119886minus110038161003816100381610038161003816lt |119911| lt 1 (16)
Finally we note thatinfin
sum
119899=0
119903 119886Δ + 119887Δ (119899) 119902119899= 120595 (119902
119886) 120595 (119902
119887) (17)
3 Proof of the Conjectures (5)ndash(7)Proof of Conjecture (5) In view of (17) we rewrite (5) as
120595 (119902) 120595 (1199025) =
infin
sum
119899=0
(1199023119899+ 1199027119899+1
1 minus 11990220119899+5minus11990213119899+9+ 11990217119899+12
1 minus 11990220119899+15
) (18)
Replacing 119902 by 11990220 and then setting 119911 = 1199023 119886 = 1199025 and119887 = 11990225 in (15) we have
infin
sum
119899=minusinfin
(1199025 11990220)119899
(11990225 11990220)119899
1199023119899
=
(1199028 11990220)infin(11990212 11990220)infin(11990220 11990220)2
infin
(1199023 11990220)infin(11990217 11990220)infin(11990225 11990220)infin(11990215 11990220)infin
(19)
Expanding the 119902-products on the left side of (19) and notingthat (1 minus 1199025)(11990225 11990220)
infin= (1199025 11990220)infin we obtain
infin
sum
119899=minusinfin
1199023119899
1 minus 11990220119899+5
=
(1199028 11990220)infin(11990212 11990220)infin(11990220 11990220)2
infin
(1199023 11990220)infin(11990217 11990220)infin(1199025 11990220)infin(11990215 11990220)infin
(20)
Alsoinfin
sum
119899=minusinfin
1199023119899
1 minus 11990220119899+5=
infin
sum
119899=0
1199023119899
1 minus 11990220119899+5+
minus1
sum
119899=minusinfin
1199023119899
1 minus 11990220119899+5
=
infin
sum
119899=0
1199023119899
1 minus 11990220119899+5+
infin
sum
119899=1
119902minus3119899
1 minus 119902minus20119899+5
=
infin
sum
119899=0
1199023119899
1 minus 11990220119899+5minus
infin
sum
119899=1
11990217119899minus5
1 minus 11990220119899minus5
=
infin
sum
119899=0
1199023119899
1 minus 11990220119899+5minus
infin
sum
119899=0
11990217119899+12
1 minus 11990220119899+15
(21)
International Journal of Mathematics and Mathematical Sciences 3
Thus from (20) and (21) we haveinfin
sum
119899=0
1199023119899
1 minus 11990220119899+5minus
infin
sum
119899=0
11990217119899+12
1 minus 11990220119899+15
=
(1199028 11990220)infin(11990212 11990220)infin(11990220 11990220)2
infin
(1199023 11990220)infin(11990217 11990220)infin(1199025 11990220)infin(11990215 11990220)infin
(22)
Replacing 119902 by 11990220 and then setting 119911 = 1199023 119886 = 1199025 and 119887 = 11990225in (15) and proceeding as in case of (22) we find thatinfin
sum
119899=0
1199027119899
1 minus 11990220119899+5minus
infin
sum
119899=0
11990213119899+8
1 minus 11990220119899+15
=
(11990212 11990220)infin(1199028 11990220)infin(11990220 11990220)2
infin
(1199027 11990220)infin(11990213 11990220)infin(1199025 11990220)infin(11990215 11990220)infin
(23)
Multiplying (23) by 119902 and adding to (22) we obtaininfin
sum
119899=0
(1199023119899+ 1199027119899+1
1 minus 11990220119899+5minus11990213119899+9+ 11990217119899+12
1 minus 11990220119899+15
)
=
(1199028 11990220)infin(11990212 11990220)infin(11990220 11990220)2
infin
(1199023 11990220)infin(11990217 11990220)infin(1199025 11990220)infin(11990215 11990220)infin
+ 119902
(11990212 11990220)infin(1199028 11990220)infin(11990220 11990220)2
infin
(1199027 11990220)infin(11990213 11990220)infin(1199025 11990220)infin(11990215 11990220)infin
=
[1199028 11990212 11990220 11990220]infin
[1199023 1199025 1199027 11990213 11990215 11990217 11990220]infin
times ((1199027 11990220)infin(11990213 11990220)infin(11990220 11990220)infin
+ 119902(1199023 11990220)infin(11990217 11990220)infin(11990220 11990220)infin)
=
[1199028 11990212 11990220 11990220]infin
[1199023 1199025 1199027 11990213 11990215 11990217 11990220]infin
times (119891 (minus1199027 minus11990213) + 119902119891 (minus119902
3 minus11990217))
(24)
Setting 119886 = 119902 and 119887 = minus1199024 in (13) we find that
119891 (minus1199027 minus11990213) + 119902119891 (minus119902
3 minus11990217) = 119891 (119902 minus119902
4) (25)
Thus (24) reduces toinfin
sum
119899=0
(1199023119899+ 1199027119899+1
1 minus 11990220119899+5minus11990213119899+9+ 11990217119899+12
1 minus 11990220119899+15
)
=
[1199028 11990212 11990220 11990220]infin
[1199023 1199025 1199027 11990213 11990215 11990217 11990220]infin
119891 (119902 minus1199024)
(26)
Now
119891 (119902 minus1199024) = (minus119902 minus119902
5)infin(1199024 minus1199025)infin(minus1199025 minus1199025)infin (27)
We have
(minus119902 minus1199025)infin= (minus119902 119902
10)infin(1199026 11990210)infin
=
(1199022 11990220)infin(1199026 11990210)infin
(119902 11990210)infin
=
[1199022 1199026 11990216 11990220]infin
[119902 11990211 11990220]infin
(28)
Similarly
(1199024 minus1199025)infin=
[1199024 11990214 11990218 11990220]infin
[1199029 11990219 11990220]infin
(minus1199025 minus1199025)infin=
[11990210 11990210 11990220 11990220]infin
[1199025 11990215 11990220]infin
(29)
Hence from (27) (28) and (29) we have
119891 (119902 minus1199024) =
[1199022 1199024 1199026 11990210 11990210 11990214 11990216 11990218 11990220 11990220]infin
[119902 1199025 1199029 11990211 11990215 11990219 11990220]infin
(30)
Using (30) in (26) we find that
infin
sum
119899=0
(1199023119899+ 1199027119899+1
1 minus 11990220119899+5minus11990213119899+9+ 11990217119899+12
1 minus 11990220119899+15
)
=
[1199022 1199024 1199026 1199028 11990210 11990210 11990212 11990214 11990216 11990218 11990220 11990220 11990220]infin
[119902 1199023 1199025 1199025 1199027 1199029 11990211 11990213 11990215 11990215 11990217 11990219 11990220]infin
= ([1199022 1199024 1199026 1199028 11990210 11990212 11990214 11990216 11990218 11990220 11990220]infin
times[11990210 11990220 11990220]infin)
times ([119902 1199023 1199025 1199027 1199029 11990211 11990213 11990215 11990217 11990219 11990220]infin
times[1199025 11990215 11990220]infin)minus1
=
(1199022 1199022)infin(11990210 11990210)infin
(119902 1199022)infin(1199025 11990210)infin
= 120595 (119902) 120595 (1199025)
(31)
This completes the proof of (18)
Proof of Conjecture (6) In view of (17) we rewrite (6) as
120595 (119902) 120595 (1199026) =
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9
minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
(32)
4 International Journal of Mathematics and Mathematical Sciences
Replacing 119902 by 11990224 and then setting 119911 = 1199027 119886 = 1199023 and 119887 = 11990227in (15) and proceeding as in case of (22) we find that
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3minus11990217119899+14
1 minus 11990224119899+21)
=
(11990210 11990224)infin(11990214 11990224)infin(11990224 11990224)2
infin
(1199027 11990224)infin(11990217 11990224)infin(1199023 11990224)infin(11990221 11990224)infin
(33)
Replacing 119902 by 11990224 and then setting 119911 = 1199025 119886 = 1199029 and119887 = 11990233 in (15) and proceeding in a way similar to obtaining
(22) we have
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+9minus11990219119899+10
1 minus 11990224119899+15)
=
(11990214 11990224)infin(11990210 11990224)infin(11990224 11990224)2
infin
(1199025 11990224)infin(11990219 11990224)infin(1199029 11990224)infin(11990215 11990224)infin
(34)
Multiplying (34) by 119902 and then adding to (33) we find that
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
(11990210 11990224)infin(11990214 11990224)infin(11990224 11990224)2
infin
(1199027 11990224)infin(11990217 11990224)infin(1199023 11990224)infin(11990221 11990224)infin
+ 119902
(11990214 11990224)infin(11990210 11990224)infin(11990224 11990224)2
infin
(1199025 11990224)infin(11990219 11990224)infin(1199029 11990224)infin(11990215 11990224)infin
=
[11990210 11990214 11990224]infin
[1199023 1199025 1199027 1199029 11990215 11990217 11990219 11990221 11990224]infin
times (119891 (minus1199025 minus11990219) 119891 (minus119902
9 minus11990215)
+ 119902119891 (minus1199027 minus11990217) 119891 (minus119902
3 minus11990221))
(35)
Setting 119886 = minus119902 119887 = minus11990211 119888 = minus1199024 and 119889 = minus1199028 in (14) wefind that
119891 (minus1199025 minus11990219) 119891 (minus119902
9 minus11990215) + 119902119891 (minus119902
7 minus11990217) 119891 (minus119902
3 minus11990221)
= 119891 (119902 11990211) 119891 (minus119902
4 minus1199028)
(36)
Using (36) in (35) we have
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
[11990210 11990214 11990224]infin
[1199023 1199025 1199027 1199029 11990215 11990217 11990219 11990221 11990224]infin
times 119891 (119902 11990211) 119891 (minus119902
4 minus1199028)
(37)
Now
119891 (119902 11990211) = (minus119902 119902
12)infin(minus11990211 11990212)infin(11990212 11990212)infin
=
(1199022 11990224)infin(11990222 11990224)infin(11990212 11990212)infin
(119902 11990212)infin(11990211 11990212)infin
=
[1199022 11990212 11990222 11990224 11990224]infin
[119902 11990211 11990213 11990223 11990224]infin
119891 (minus1199024 minus1199028) = (119902
4 11990212)infin(1199028 11990212)infin(11990212 11990212)infin
= [1199024 1199028 11990212 11990216 11990220 11990224 11990224]infin
(38)
Using (38) in (37) we have
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
[1199022 1199024 1199028 11990210 11990212 11990212 11990214 11990216 11990220 11990222 11990224 11990224 11990224]infin
[119902 1199023 1199025 1199027 1199029 11990211 11990213 11990215 11990217 11990219 11990221 11990223 11990224]infin
= ([1199022 1199024 1199026 1199028 11990210 11990212 11990214 11990216 11990218 11990220 11990222 11990224 11990224]infin
times [11990212 11990224 11990224]infin)
times ([119902 1199023 1199025 1199027 1199029 11990211 11990213 11990215 11990217 11990219 11990221 11990223 11990224]infin
times [1199026 11990212 11990224]infin)minus1
=
(1199022 1199022)infin(11990212 11990212)infin
(119902 1199022)infin(1199026 11990212)infin
= 120595 (119902) 120595 (1199026)
(39)
This completes the proof of (32)
Proof of Conjecture (7) In view of (17) we rewrite (7) as
120595 (1199022) 120595 (119902
3) =
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3+1199027119899+2
1 minus 11990224119899+9
minus11990217119899+10
1 minus 11990224119899+15minus11990219119899+16
1 minus 11990224119899+21)
(40)
International Journal of Mathematics and Mathematical Sciences 5
Replacing 119902 by 11990224 and then setting 119911 = 1199025 119886 = 1199023 and119887 = 11990227 in (15) and proceeding as in case of (22) we find that
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3minus11990219119899+16
1 minus 11990224119899+21)
=
(1199028 11990224)infin(11990216 11990224)infin(11990224 11990224)2
infin
(1199025 11990224)infin(11990219 11990224)infin(1199023 11990224)infin(11990221 11990224)infin
(41)
Also replacing 119902 by 11990224 and then setting 119911 = 1199027 119886 = 1199029and 119887 = 11990233 in (15) and proceeding similar to (22) we obtain
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+9minus11990217119899+8
1 minus 11990224119899+15)
=
(11990216 11990224)infin(1199028 11990224)infin(11990224 11990224)2
infin
(1199027 11990224)infin(11990217 11990224)infin(1199029 11990224)infin(11990215 11990224)infin
(42)
Multiplying (42) by 1199022 and then adding to (41) we find that
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3+1199027119899+2
1 minus 11990224119899+9minus11990217119899+10
1 minus 11990224119899+15minus11990219119899+16
1 minus 11990224119899+21)
=
(1199028 11990224)infin(11990216 11990224)infin(11990224 11990224)2
infin
(1199025 11990224)infin(11990219 11990224)infin(1199023 11990224)infin(11990221 11990224)infin
+ 1199022(11990216 11990224)infin(1199028 11990224)infin(11990224 11990224)2
infin
(1199027 11990224)infin(11990217 11990224)infin(1199029 11990224)infin(11990215 11990224)infin
=
[1199028 11990216 11990224]infin
[1199023 1199025 1199027 1199029 11990215 11990217 11990219 11990221 11990224]infin
times (119891 (minus1199027 minus11990217) 119891 (minus119902
9 minus11990215)
+ 1199022119891 (minus119902
5 minus11990219) 119891 (minus119902
3 minus11990221))
(43)
Setting 119886 = 1199022 119887 = minus11990210 119888 = minus1199025 and 119889 = minus1199027 in (14) we findthat
119891 (minus1199027 minus11990217) 119891 (minus119902
9 minus11990215) + 1199022119891 (minus119902
5 minus11990219) 119891 (minus119902
3 minus11990221)
= 119891 (1199022 11990210) 119891 (minus119902
5 minus1199027)
(44)
Hence (43) reduces toinfin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
[1199028 11990216 11990224]infin
[1199023 1199025 1199027 1199029 11990215 11990217 11990219 11990221 11990224]infin
times 119891 (1199022 11990210) 119891 (minus119902
5 minus1199027)
(45)
Now
119891 (1199022 11990210) = (minus119902
2 11990212)infin(minus11990210 11990212)infin(11990212 11990212)infin
=
(1199024 11990224)infin(11990220 11990224)infin(11990212 11990212)infin
(1199022 11990212)infin(11990210 11990212)infin
=
[1199024 11990212 11990220 11990224 11990224]infin
[1199022 11990210 11990214 11990222 11990224]infin
119891 (minus1199025 minus1199027) = (119902
5 11990212)infin(1199027 11990212)infin(11990212 11990212)infin
= [1199025 1199027 11990212 11990217 11990219 11990224 11990224]infin
(46)
Using (46) in (45) we have
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
[1199024 1199025 1199027 1199028 11990212 11990212 11990216 11990217 11990219 11990220 11990224 11990224 11990224]infin
[1199022 1199023 1199025 1199027 1199029 11990210 11990214 11990215 11990217 11990219 11990221 11990222 11990224]infin
=
[1199024 1199028 11990212 11990216 11990220 11990224 11990224]infin[1199026 11990212 11990218 11990224 11990224]infin
[1199022 1199026 11990210 11990214 11990218 11990222 11990224]infin[1199023 1199029 11990215 11990221 11990224]infin
=
(1199024 1199024)infin(1199026 1199026)infin
(1199022 1199024)infin(1199023 1199026)infin
= 120595 (1199022) 120595 (119902
3)
(47)
This completes the proof of (40)
4 Concluding Remarks
All conjectures in [11] can easily be reformulated as thetafunction identity using Ramanujanrsquos
11205951formula However
these identities might be too complicated to actually have aproof
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] C G J Jacobi Fundamenta Nova Theoriae Functionum Ellipti-carum vol 107 of Werke I 162-163 Letter to Legendre 991828Werke I 424 1829
[2] P G L Dirichlet J Math 21 (1840) 3 6 Werke 463 466[3] L Lorenz ldquoBidrag til tallenes theorirdquo Tidsskrift for Mathematik
vol 3 no 1 pp 97ndash114 1871[4] A M Legendre Traite des Fonctions Elliptiques et des Integrales
Euleriennes III Huzard-Courcier Paris France 1828[5] B C Berndt Ramanujanrsquos Notebooks Part III Springer New
York NY USA 1991
6 International Journal of Mathematics and Mathematical Sciences
[6] M D Hirschhorn ldquoThe number of representations of a numberby various formsrdquo Discrete Mathematics vol 298 no 1ndash3 pp205ndash211 2005
[7] M D Hirschhorn ldquoThe number of representation of a numberby various forms involving triangles squares pentagons andoctagonsrdquo in Ramanujan Rediscovered N D Baruah B CBerndt S Cooper T Huber and M Schlosser Eds RMS Lec-ture Note Series No 14 pp 113ndash124 Ramanujan MathematicalSociety 2010
[8] ND Baruah and B K Sarmah ldquoThenumber of representationsof a number as sums of various polygonal numbersrdquo Integersvol 12 2012
[9] H Y Lam ldquoThe number of representations by sums of squaresand triangular numbersrdquo Integers vol 7 article A28 2007
[10] R S Melham ldquoAnalogues of two classical theorems on therepresentations of a numberrdquo Integers vol 8 article A51 2008
[11] R S Melham ldquoAnalogues of Jacobirsquos two-square theorem aninformal accountrdquo Integers vol 10 no 1 pp 83ndash100 2010
[12] P C Toh ldquoOn representations by figurate numbers a uniformapproach to the conjectures of Melhamrdquo International Journalof Number Theory vol 9 no 4 pp 1055ndash1071 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Mathematics and Mathematical Sciences 3
Thus from (20) and (21) we haveinfin
sum
119899=0
1199023119899
1 minus 11990220119899+5minus
infin
sum
119899=0
11990217119899+12
1 minus 11990220119899+15
=
(1199028 11990220)infin(11990212 11990220)infin(11990220 11990220)2
infin
(1199023 11990220)infin(11990217 11990220)infin(1199025 11990220)infin(11990215 11990220)infin
(22)
Replacing 119902 by 11990220 and then setting 119911 = 1199023 119886 = 1199025 and 119887 = 11990225in (15) and proceeding as in case of (22) we find thatinfin
sum
119899=0
1199027119899
1 minus 11990220119899+5minus
infin
sum
119899=0
11990213119899+8
1 minus 11990220119899+15
=
(11990212 11990220)infin(1199028 11990220)infin(11990220 11990220)2
infin
(1199027 11990220)infin(11990213 11990220)infin(1199025 11990220)infin(11990215 11990220)infin
(23)
Multiplying (23) by 119902 and adding to (22) we obtaininfin
sum
119899=0
(1199023119899+ 1199027119899+1
1 minus 11990220119899+5minus11990213119899+9+ 11990217119899+12
1 minus 11990220119899+15
)
=
(1199028 11990220)infin(11990212 11990220)infin(11990220 11990220)2
infin
(1199023 11990220)infin(11990217 11990220)infin(1199025 11990220)infin(11990215 11990220)infin
+ 119902
(11990212 11990220)infin(1199028 11990220)infin(11990220 11990220)2
infin
(1199027 11990220)infin(11990213 11990220)infin(1199025 11990220)infin(11990215 11990220)infin
=
[1199028 11990212 11990220 11990220]infin
[1199023 1199025 1199027 11990213 11990215 11990217 11990220]infin
times ((1199027 11990220)infin(11990213 11990220)infin(11990220 11990220)infin
+ 119902(1199023 11990220)infin(11990217 11990220)infin(11990220 11990220)infin)
=
[1199028 11990212 11990220 11990220]infin
[1199023 1199025 1199027 11990213 11990215 11990217 11990220]infin
times (119891 (minus1199027 minus11990213) + 119902119891 (minus119902
3 minus11990217))
(24)
Setting 119886 = 119902 and 119887 = minus1199024 in (13) we find that
119891 (minus1199027 minus11990213) + 119902119891 (minus119902
3 minus11990217) = 119891 (119902 minus119902
4) (25)
Thus (24) reduces toinfin
sum
119899=0
(1199023119899+ 1199027119899+1
1 minus 11990220119899+5minus11990213119899+9+ 11990217119899+12
1 minus 11990220119899+15
)
=
[1199028 11990212 11990220 11990220]infin
[1199023 1199025 1199027 11990213 11990215 11990217 11990220]infin
119891 (119902 minus1199024)
(26)
Now
119891 (119902 minus1199024) = (minus119902 minus119902
5)infin(1199024 minus1199025)infin(minus1199025 minus1199025)infin (27)
We have
(minus119902 minus1199025)infin= (minus119902 119902
10)infin(1199026 11990210)infin
=
(1199022 11990220)infin(1199026 11990210)infin
(119902 11990210)infin
=
[1199022 1199026 11990216 11990220]infin
[119902 11990211 11990220]infin
(28)
Similarly
(1199024 minus1199025)infin=
[1199024 11990214 11990218 11990220]infin
[1199029 11990219 11990220]infin
(minus1199025 minus1199025)infin=
[11990210 11990210 11990220 11990220]infin
[1199025 11990215 11990220]infin
(29)
Hence from (27) (28) and (29) we have
119891 (119902 minus1199024) =
[1199022 1199024 1199026 11990210 11990210 11990214 11990216 11990218 11990220 11990220]infin
[119902 1199025 1199029 11990211 11990215 11990219 11990220]infin
(30)
Using (30) in (26) we find that
infin
sum
119899=0
(1199023119899+ 1199027119899+1
1 minus 11990220119899+5minus11990213119899+9+ 11990217119899+12
1 minus 11990220119899+15
)
=
[1199022 1199024 1199026 1199028 11990210 11990210 11990212 11990214 11990216 11990218 11990220 11990220 11990220]infin
[119902 1199023 1199025 1199025 1199027 1199029 11990211 11990213 11990215 11990215 11990217 11990219 11990220]infin
= ([1199022 1199024 1199026 1199028 11990210 11990212 11990214 11990216 11990218 11990220 11990220]infin
times[11990210 11990220 11990220]infin)
times ([119902 1199023 1199025 1199027 1199029 11990211 11990213 11990215 11990217 11990219 11990220]infin
times[1199025 11990215 11990220]infin)minus1
=
(1199022 1199022)infin(11990210 11990210)infin
(119902 1199022)infin(1199025 11990210)infin
= 120595 (119902) 120595 (1199025)
(31)
This completes the proof of (18)
Proof of Conjecture (6) In view of (17) we rewrite (6) as
120595 (119902) 120595 (1199026) =
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9
minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
(32)
4 International Journal of Mathematics and Mathematical Sciences
Replacing 119902 by 11990224 and then setting 119911 = 1199027 119886 = 1199023 and 119887 = 11990227in (15) and proceeding as in case of (22) we find that
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3minus11990217119899+14
1 minus 11990224119899+21)
=
(11990210 11990224)infin(11990214 11990224)infin(11990224 11990224)2
infin
(1199027 11990224)infin(11990217 11990224)infin(1199023 11990224)infin(11990221 11990224)infin
(33)
Replacing 119902 by 11990224 and then setting 119911 = 1199025 119886 = 1199029 and119887 = 11990233 in (15) and proceeding in a way similar to obtaining
(22) we have
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+9minus11990219119899+10
1 minus 11990224119899+15)
=
(11990214 11990224)infin(11990210 11990224)infin(11990224 11990224)2
infin
(1199025 11990224)infin(11990219 11990224)infin(1199029 11990224)infin(11990215 11990224)infin
(34)
Multiplying (34) by 119902 and then adding to (33) we find that
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
(11990210 11990224)infin(11990214 11990224)infin(11990224 11990224)2
infin
(1199027 11990224)infin(11990217 11990224)infin(1199023 11990224)infin(11990221 11990224)infin
+ 119902
(11990214 11990224)infin(11990210 11990224)infin(11990224 11990224)2
infin
(1199025 11990224)infin(11990219 11990224)infin(1199029 11990224)infin(11990215 11990224)infin
=
[11990210 11990214 11990224]infin
[1199023 1199025 1199027 1199029 11990215 11990217 11990219 11990221 11990224]infin
times (119891 (minus1199025 minus11990219) 119891 (minus119902
9 minus11990215)
+ 119902119891 (minus1199027 minus11990217) 119891 (minus119902
3 minus11990221))
(35)
Setting 119886 = minus119902 119887 = minus11990211 119888 = minus1199024 and 119889 = minus1199028 in (14) wefind that
119891 (minus1199025 minus11990219) 119891 (minus119902
9 minus11990215) + 119902119891 (minus119902
7 minus11990217) 119891 (minus119902
3 minus11990221)
= 119891 (119902 11990211) 119891 (minus119902
4 minus1199028)
(36)
Using (36) in (35) we have
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
[11990210 11990214 11990224]infin
[1199023 1199025 1199027 1199029 11990215 11990217 11990219 11990221 11990224]infin
times 119891 (119902 11990211) 119891 (minus119902
4 minus1199028)
(37)
Now
119891 (119902 11990211) = (minus119902 119902
12)infin(minus11990211 11990212)infin(11990212 11990212)infin
=
(1199022 11990224)infin(11990222 11990224)infin(11990212 11990212)infin
(119902 11990212)infin(11990211 11990212)infin
=
[1199022 11990212 11990222 11990224 11990224]infin
[119902 11990211 11990213 11990223 11990224]infin
119891 (minus1199024 minus1199028) = (119902
4 11990212)infin(1199028 11990212)infin(11990212 11990212)infin
= [1199024 1199028 11990212 11990216 11990220 11990224 11990224]infin
(38)
Using (38) in (37) we have
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
[1199022 1199024 1199028 11990210 11990212 11990212 11990214 11990216 11990220 11990222 11990224 11990224 11990224]infin
[119902 1199023 1199025 1199027 1199029 11990211 11990213 11990215 11990217 11990219 11990221 11990223 11990224]infin
= ([1199022 1199024 1199026 1199028 11990210 11990212 11990214 11990216 11990218 11990220 11990222 11990224 11990224]infin
times [11990212 11990224 11990224]infin)
times ([119902 1199023 1199025 1199027 1199029 11990211 11990213 11990215 11990217 11990219 11990221 11990223 11990224]infin
times [1199026 11990212 11990224]infin)minus1
=
(1199022 1199022)infin(11990212 11990212)infin
(119902 1199022)infin(1199026 11990212)infin
= 120595 (119902) 120595 (1199026)
(39)
This completes the proof of (32)
Proof of Conjecture (7) In view of (17) we rewrite (7) as
120595 (1199022) 120595 (119902
3) =
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3+1199027119899+2
1 minus 11990224119899+9
minus11990217119899+10
1 minus 11990224119899+15minus11990219119899+16
1 minus 11990224119899+21)
(40)
International Journal of Mathematics and Mathematical Sciences 5
Replacing 119902 by 11990224 and then setting 119911 = 1199025 119886 = 1199023 and119887 = 11990227 in (15) and proceeding as in case of (22) we find that
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3minus11990219119899+16
1 minus 11990224119899+21)
=
(1199028 11990224)infin(11990216 11990224)infin(11990224 11990224)2
infin
(1199025 11990224)infin(11990219 11990224)infin(1199023 11990224)infin(11990221 11990224)infin
(41)
Also replacing 119902 by 11990224 and then setting 119911 = 1199027 119886 = 1199029and 119887 = 11990233 in (15) and proceeding similar to (22) we obtain
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+9minus11990217119899+8
1 minus 11990224119899+15)
=
(11990216 11990224)infin(1199028 11990224)infin(11990224 11990224)2
infin
(1199027 11990224)infin(11990217 11990224)infin(1199029 11990224)infin(11990215 11990224)infin
(42)
Multiplying (42) by 1199022 and then adding to (41) we find that
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3+1199027119899+2
1 minus 11990224119899+9minus11990217119899+10
1 minus 11990224119899+15minus11990219119899+16
1 minus 11990224119899+21)
=
(1199028 11990224)infin(11990216 11990224)infin(11990224 11990224)2
infin
(1199025 11990224)infin(11990219 11990224)infin(1199023 11990224)infin(11990221 11990224)infin
+ 1199022(11990216 11990224)infin(1199028 11990224)infin(11990224 11990224)2
infin
(1199027 11990224)infin(11990217 11990224)infin(1199029 11990224)infin(11990215 11990224)infin
=
[1199028 11990216 11990224]infin
[1199023 1199025 1199027 1199029 11990215 11990217 11990219 11990221 11990224]infin
times (119891 (minus1199027 minus11990217) 119891 (minus119902
9 minus11990215)
+ 1199022119891 (minus119902
5 minus11990219) 119891 (minus119902
3 minus11990221))
(43)
Setting 119886 = 1199022 119887 = minus11990210 119888 = minus1199025 and 119889 = minus1199027 in (14) we findthat
119891 (minus1199027 minus11990217) 119891 (minus119902
9 minus11990215) + 1199022119891 (minus119902
5 minus11990219) 119891 (minus119902
3 minus11990221)
= 119891 (1199022 11990210) 119891 (minus119902
5 minus1199027)
(44)
Hence (43) reduces toinfin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
[1199028 11990216 11990224]infin
[1199023 1199025 1199027 1199029 11990215 11990217 11990219 11990221 11990224]infin
times 119891 (1199022 11990210) 119891 (minus119902
5 minus1199027)
(45)
Now
119891 (1199022 11990210) = (minus119902
2 11990212)infin(minus11990210 11990212)infin(11990212 11990212)infin
=
(1199024 11990224)infin(11990220 11990224)infin(11990212 11990212)infin
(1199022 11990212)infin(11990210 11990212)infin
=
[1199024 11990212 11990220 11990224 11990224]infin
[1199022 11990210 11990214 11990222 11990224]infin
119891 (minus1199025 minus1199027) = (119902
5 11990212)infin(1199027 11990212)infin(11990212 11990212)infin
= [1199025 1199027 11990212 11990217 11990219 11990224 11990224]infin
(46)
Using (46) in (45) we have
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
[1199024 1199025 1199027 1199028 11990212 11990212 11990216 11990217 11990219 11990220 11990224 11990224 11990224]infin
[1199022 1199023 1199025 1199027 1199029 11990210 11990214 11990215 11990217 11990219 11990221 11990222 11990224]infin
=
[1199024 1199028 11990212 11990216 11990220 11990224 11990224]infin[1199026 11990212 11990218 11990224 11990224]infin
[1199022 1199026 11990210 11990214 11990218 11990222 11990224]infin[1199023 1199029 11990215 11990221 11990224]infin
=
(1199024 1199024)infin(1199026 1199026)infin
(1199022 1199024)infin(1199023 1199026)infin
= 120595 (1199022) 120595 (119902
3)
(47)
This completes the proof of (40)
4 Concluding Remarks
All conjectures in [11] can easily be reformulated as thetafunction identity using Ramanujanrsquos
11205951formula However
these identities might be too complicated to actually have aproof
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] C G J Jacobi Fundamenta Nova Theoriae Functionum Ellipti-carum vol 107 of Werke I 162-163 Letter to Legendre 991828Werke I 424 1829
[2] P G L Dirichlet J Math 21 (1840) 3 6 Werke 463 466[3] L Lorenz ldquoBidrag til tallenes theorirdquo Tidsskrift for Mathematik
vol 3 no 1 pp 97ndash114 1871[4] A M Legendre Traite des Fonctions Elliptiques et des Integrales
Euleriennes III Huzard-Courcier Paris France 1828[5] B C Berndt Ramanujanrsquos Notebooks Part III Springer New
York NY USA 1991
6 International Journal of Mathematics and Mathematical Sciences
[6] M D Hirschhorn ldquoThe number of representations of a numberby various formsrdquo Discrete Mathematics vol 298 no 1ndash3 pp205ndash211 2005
[7] M D Hirschhorn ldquoThe number of representation of a numberby various forms involving triangles squares pentagons andoctagonsrdquo in Ramanujan Rediscovered N D Baruah B CBerndt S Cooper T Huber and M Schlosser Eds RMS Lec-ture Note Series No 14 pp 113ndash124 Ramanujan MathematicalSociety 2010
[8] ND Baruah and B K Sarmah ldquoThenumber of representationsof a number as sums of various polygonal numbersrdquo Integersvol 12 2012
[9] H Y Lam ldquoThe number of representations by sums of squaresand triangular numbersrdquo Integers vol 7 article A28 2007
[10] R S Melham ldquoAnalogues of two classical theorems on therepresentations of a numberrdquo Integers vol 8 article A51 2008
[11] R S Melham ldquoAnalogues of Jacobirsquos two-square theorem aninformal accountrdquo Integers vol 10 no 1 pp 83ndash100 2010
[12] P C Toh ldquoOn representations by figurate numbers a uniformapproach to the conjectures of Melhamrdquo International Journalof Number Theory vol 9 no 4 pp 1055ndash1071 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 International Journal of Mathematics and Mathematical Sciences
Replacing 119902 by 11990224 and then setting 119911 = 1199027 119886 = 1199023 and 119887 = 11990227in (15) and proceeding as in case of (22) we find that
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3minus11990217119899+14
1 minus 11990224119899+21)
=
(11990210 11990224)infin(11990214 11990224)infin(11990224 11990224)2
infin
(1199027 11990224)infin(11990217 11990224)infin(1199023 11990224)infin(11990221 11990224)infin
(33)
Replacing 119902 by 11990224 and then setting 119911 = 1199025 119886 = 1199029 and119887 = 11990233 in (15) and proceeding in a way similar to obtaining
(22) we have
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+9minus11990219119899+10
1 minus 11990224119899+15)
=
(11990214 11990224)infin(11990210 11990224)infin(11990224 11990224)2
infin
(1199025 11990224)infin(11990219 11990224)infin(1199029 11990224)infin(11990215 11990224)infin
(34)
Multiplying (34) by 119902 and then adding to (33) we find that
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
(11990210 11990224)infin(11990214 11990224)infin(11990224 11990224)2
infin
(1199027 11990224)infin(11990217 11990224)infin(1199023 11990224)infin(11990221 11990224)infin
+ 119902
(11990214 11990224)infin(11990210 11990224)infin(11990224 11990224)2
infin
(1199025 11990224)infin(11990219 11990224)infin(1199029 11990224)infin(11990215 11990224)infin
=
[11990210 11990214 11990224]infin
[1199023 1199025 1199027 1199029 11990215 11990217 11990219 11990221 11990224]infin
times (119891 (minus1199025 minus11990219) 119891 (minus119902
9 minus11990215)
+ 119902119891 (minus1199027 minus11990217) 119891 (minus119902
3 minus11990221))
(35)
Setting 119886 = minus119902 119887 = minus11990211 119888 = minus1199024 and 119889 = minus1199028 in (14) wefind that
119891 (minus1199025 minus11990219) 119891 (minus119902
9 minus11990215) + 119902119891 (minus119902
7 minus11990217) 119891 (minus119902
3 minus11990221)
= 119891 (119902 11990211) 119891 (minus119902
4 minus1199028)
(36)
Using (36) in (35) we have
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
[11990210 11990214 11990224]infin
[1199023 1199025 1199027 1199029 11990215 11990217 11990219 11990221 11990224]infin
times 119891 (119902 11990211) 119891 (minus119902
4 minus1199028)
(37)
Now
119891 (119902 11990211) = (minus119902 119902
12)infin(minus11990211 11990212)infin(11990212 11990212)infin
=
(1199022 11990224)infin(11990222 11990224)infin(11990212 11990212)infin
(119902 11990212)infin(11990211 11990212)infin
=
[1199022 11990212 11990222 11990224 11990224]infin
[119902 11990211 11990213 11990223 11990224]infin
119891 (minus1199024 minus1199028) = (119902
4 11990212)infin(1199028 11990212)infin(11990212 11990212)infin
= [1199024 1199028 11990212 11990216 11990220 11990224 11990224]infin
(38)
Using (38) in (37) we have
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
[1199022 1199024 1199028 11990210 11990212 11990212 11990214 11990216 11990220 11990222 11990224 11990224 11990224]infin
[119902 1199023 1199025 1199027 1199029 11990211 11990213 11990215 11990217 11990219 11990221 11990223 11990224]infin
= ([1199022 1199024 1199026 1199028 11990210 11990212 11990214 11990216 11990218 11990220 11990222 11990224 11990224]infin
times [11990212 11990224 11990224]infin)
times ([119902 1199023 1199025 1199027 1199029 11990211 11990213 11990215 11990217 11990219 11990221 11990223 11990224]infin
times [1199026 11990212 11990224]infin)minus1
=
(1199022 1199022)infin(11990212 11990212)infin
(119902 1199022)infin(1199026 11990212)infin
= 120595 (119902) 120595 (1199026)
(39)
This completes the proof of (32)
Proof of Conjecture (7) In view of (17) we rewrite (7) as
120595 (1199022) 120595 (119902
3) =
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3+1199027119899+2
1 minus 11990224119899+9
minus11990217119899+10
1 minus 11990224119899+15minus11990219119899+16
1 minus 11990224119899+21)
(40)
International Journal of Mathematics and Mathematical Sciences 5
Replacing 119902 by 11990224 and then setting 119911 = 1199025 119886 = 1199023 and119887 = 11990227 in (15) and proceeding as in case of (22) we find that
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3minus11990219119899+16
1 minus 11990224119899+21)
=
(1199028 11990224)infin(11990216 11990224)infin(11990224 11990224)2
infin
(1199025 11990224)infin(11990219 11990224)infin(1199023 11990224)infin(11990221 11990224)infin
(41)
Also replacing 119902 by 11990224 and then setting 119911 = 1199027 119886 = 1199029and 119887 = 11990233 in (15) and proceeding similar to (22) we obtain
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+9minus11990217119899+8
1 minus 11990224119899+15)
=
(11990216 11990224)infin(1199028 11990224)infin(11990224 11990224)2
infin
(1199027 11990224)infin(11990217 11990224)infin(1199029 11990224)infin(11990215 11990224)infin
(42)
Multiplying (42) by 1199022 and then adding to (41) we find that
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3+1199027119899+2
1 minus 11990224119899+9minus11990217119899+10
1 minus 11990224119899+15minus11990219119899+16
1 minus 11990224119899+21)
=
(1199028 11990224)infin(11990216 11990224)infin(11990224 11990224)2
infin
(1199025 11990224)infin(11990219 11990224)infin(1199023 11990224)infin(11990221 11990224)infin
+ 1199022(11990216 11990224)infin(1199028 11990224)infin(11990224 11990224)2
infin
(1199027 11990224)infin(11990217 11990224)infin(1199029 11990224)infin(11990215 11990224)infin
=
[1199028 11990216 11990224]infin
[1199023 1199025 1199027 1199029 11990215 11990217 11990219 11990221 11990224]infin
times (119891 (minus1199027 minus11990217) 119891 (minus119902
9 minus11990215)
+ 1199022119891 (minus119902
5 minus11990219) 119891 (minus119902
3 minus11990221))
(43)
Setting 119886 = 1199022 119887 = minus11990210 119888 = minus1199025 and 119889 = minus1199027 in (14) we findthat
119891 (minus1199027 minus11990217) 119891 (minus119902
9 minus11990215) + 1199022119891 (minus119902
5 minus11990219) 119891 (minus119902
3 minus11990221)
= 119891 (1199022 11990210) 119891 (minus119902
5 minus1199027)
(44)
Hence (43) reduces toinfin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
[1199028 11990216 11990224]infin
[1199023 1199025 1199027 1199029 11990215 11990217 11990219 11990221 11990224]infin
times 119891 (1199022 11990210) 119891 (minus119902
5 minus1199027)
(45)
Now
119891 (1199022 11990210) = (minus119902
2 11990212)infin(minus11990210 11990212)infin(11990212 11990212)infin
=
(1199024 11990224)infin(11990220 11990224)infin(11990212 11990212)infin
(1199022 11990212)infin(11990210 11990212)infin
=
[1199024 11990212 11990220 11990224 11990224]infin
[1199022 11990210 11990214 11990222 11990224]infin
119891 (minus1199025 minus1199027) = (119902
5 11990212)infin(1199027 11990212)infin(11990212 11990212)infin
= [1199025 1199027 11990212 11990217 11990219 11990224 11990224]infin
(46)
Using (46) in (45) we have
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
[1199024 1199025 1199027 1199028 11990212 11990212 11990216 11990217 11990219 11990220 11990224 11990224 11990224]infin
[1199022 1199023 1199025 1199027 1199029 11990210 11990214 11990215 11990217 11990219 11990221 11990222 11990224]infin
=
[1199024 1199028 11990212 11990216 11990220 11990224 11990224]infin[1199026 11990212 11990218 11990224 11990224]infin
[1199022 1199026 11990210 11990214 11990218 11990222 11990224]infin[1199023 1199029 11990215 11990221 11990224]infin
=
(1199024 1199024)infin(1199026 1199026)infin
(1199022 1199024)infin(1199023 1199026)infin
= 120595 (1199022) 120595 (119902
3)
(47)
This completes the proof of (40)
4 Concluding Remarks
All conjectures in [11] can easily be reformulated as thetafunction identity using Ramanujanrsquos
11205951formula However
these identities might be too complicated to actually have aproof
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] C G J Jacobi Fundamenta Nova Theoriae Functionum Ellipti-carum vol 107 of Werke I 162-163 Letter to Legendre 991828Werke I 424 1829
[2] P G L Dirichlet J Math 21 (1840) 3 6 Werke 463 466[3] L Lorenz ldquoBidrag til tallenes theorirdquo Tidsskrift for Mathematik
vol 3 no 1 pp 97ndash114 1871[4] A M Legendre Traite des Fonctions Elliptiques et des Integrales
Euleriennes III Huzard-Courcier Paris France 1828[5] B C Berndt Ramanujanrsquos Notebooks Part III Springer New
York NY USA 1991
6 International Journal of Mathematics and Mathematical Sciences
[6] M D Hirschhorn ldquoThe number of representations of a numberby various formsrdquo Discrete Mathematics vol 298 no 1ndash3 pp205ndash211 2005
[7] M D Hirschhorn ldquoThe number of representation of a numberby various forms involving triangles squares pentagons andoctagonsrdquo in Ramanujan Rediscovered N D Baruah B CBerndt S Cooper T Huber and M Schlosser Eds RMS Lec-ture Note Series No 14 pp 113ndash124 Ramanujan MathematicalSociety 2010
[8] ND Baruah and B K Sarmah ldquoThenumber of representationsof a number as sums of various polygonal numbersrdquo Integersvol 12 2012
[9] H Y Lam ldquoThe number of representations by sums of squaresand triangular numbersrdquo Integers vol 7 article A28 2007
[10] R S Melham ldquoAnalogues of two classical theorems on therepresentations of a numberrdquo Integers vol 8 article A51 2008
[11] R S Melham ldquoAnalogues of Jacobirsquos two-square theorem aninformal accountrdquo Integers vol 10 no 1 pp 83ndash100 2010
[12] P C Toh ldquoOn representations by figurate numbers a uniformapproach to the conjectures of Melhamrdquo International Journalof Number Theory vol 9 no 4 pp 1055ndash1071 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Mathematics and Mathematical Sciences 5
Replacing 119902 by 11990224 and then setting 119911 = 1199025 119886 = 1199023 and119887 = 11990227 in (15) and proceeding as in case of (22) we find that
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3minus11990219119899+16
1 minus 11990224119899+21)
=
(1199028 11990224)infin(11990216 11990224)infin(11990224 11990224)2
infin
(1199025 11990224)infin(11990219 11990224)infin(1199023 11990224)infin(11990221 11990224)infin
(41)
Also replacing 119902 by 11990224 and then setting 119911 = 1199027 119886 = 1199029and 119887 = 11990233 in (15) and proceeding similar to (22) we obtain
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+9minus11990217119899+8
1 minus 11990224119899+15)
=
(11990216 11990224)infin(1199028 11990224)infin(11990224 11990224)2
infin
(1199027 11990224)infin(11990217 11990224)infin(1199029 11990224)infin(11990215 11990224)infin
(42)
Multiplying (42) by 1199022 and then adding to (41) we find that
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3+1199027119899+2
1 minus 11990224119899+9minus11990217119899+10
1 minus 11990224119899+15minus11990219119899+16
1 minus 11990224119899+21)
=
(1199028 11990224)infin(11990216 11990224)infin(11990224 11990224)2
infin
(1199025 11990224)infin(11990219 11990224)infin(1199023 11990224)infin(11990221 11990224)infin
+ 1199022(11990216 11990224)infin(1199028 11990224)infin(11990224 11990224)2
infin
(1199027 11990224)infin(11990217 11990224)infin(1199029 11990224)infin(11990215 11990224)infin
=
[1199028 11990216 11990224]infin
[1199023 1199025 1199027 1199029 11990215 11990217 11990219 11990221 11990224]infin
times (119891 (minus1199027 minus11990217) 119891 (minus119902
9 minus11990215)
+ 1199022119891 (minus119902
5 minus11990219) 119891 (minus119902
3 minus11990221))
(43)
Setting 119886 = 1199022 119887 = minus11990210 119888 = minus1199025 and 119889 = minus1199027 in (14) we findthat
119891 (minus1199027 minus11990217) 119891 (minus119902
9 minus11990215) + 1199022119891 (minus119902
5 minus11990219) 119891 (minus119902
3 minus11990221)
= 119891 (1199022 11990210) 119891 (minus119902
5 minus1199027)
(44)
Hence (43) reduces toinfin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
[1199028 11990216 11990224]infin
[1199023 1199025 1199027 1199029 11990215 11990217 11990219 11990221 11990224]infin
times 119891 (1199022 11990210) 119891 (minus119902
5 minus1199027)
(45)
Now
119891 (1199022 11990210) = (minus119902
2 11990212)infin(minus11990210 11990212)infin(11990212 11990212)infin
=
(1199024 11990224)infin(11990220 11990224)infin(11990212 11990212)infin
(1199022 11990212)infin(11990210 11990212)infin
=
[1199024 11990212 11990220 11990224 11990224]infin
[1199022 11990210 11990214 11990222 11990224]infin
119891 (minus1199025 minus1199027) = (119902
5 11990212)infin(1199027 11990212)infin(11990212 11990212)infin
= [1199025 1199027 11990212 11990217 11990219 11990224 11990224]infin
(46)
Using (46) in (45) we have
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
=
[1199024 1199025 1199027 1199028 11990212 11990212 11990216 11990217 11990219 11990220 11990224 11990224 11990224]infin
[1199022 1199023 1199025 1199027 1199029 11990210 11990214 11990215 11990217 11990219 11990221 11990222 11990224]infin
=
[1199024 1199028 11990212 11990216 11990220 11990224 11990224]infin[1199026 11990212 11990218 11990224 11990224]infin
[1199022 1199026 11990210 11990214 11990218 11990222 11990224]infin[1199023 1199029 11990215 11990221 11990224]infin
=
(1199024 1199024)infin(1199026 1199026)infin
(1199022 1199024)infin(1199023 1199026)infin
= 120595 (1199022) 120595 (119902
3)
(47)
This completes the proof of (40)
4 Concluding Remarks
All conjectures in [11] can easily be reformulated as thetafunction identity using Ramanujanrsquos
11205951formula However
these identities might be too complicated to actually have aproof
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] C G J Jacobi Fundamenta Nova Theoriae Functionum Ellipti-carum vol 107 of Werke I 162-163 Letter to Legendre 991828Werke I 424 1829
[2] P G L Dirichlet J Math 21 (1840) 3 6 Werke 463 466[3] L Lorenz ldquoBidrag til tallenes theorirdquo Tidsskrift for Mathematik
vol 3 no 1 pp 97ndash114 1871[4] A M Legendre Traite des Fonctions Elliptiques et des Integrales
Euleriennes III Huzard-Courcier Paris France 1828[5] B C Berndt Ramanujanrsquos Notebooks Part III Springer New
York NY USA 1991
6 International Journal of Mathematics and Mathematical Sciences
[6] M D Hirschhorn ldquoThe number of representations of a numberby various formsrdquo Discrete Mathematics vol 298 no 1ndash3 pp205ndash211 2005
[7] M D Hirschhorn ldquoThe number of representation of a numberby various forms involving triangles squares pentagons andoctagonsrdquo in Ramanujan Rediscovered N D Baruah B CBerndt S Cooper T Huber and M Schlosser Eds RMS Lec-ture Note Series No 14 pp 113ndash124 Ramanujan MathematicalSociety 2010
[8] ND Baruah and B K Sarmah ldquoThenumber of representationsof a number as sums of various polygonal numbersrdquo Integersvol 12 2012
[9] H Y Lam ldquoThe number of representations by sums of squaresand triangular numbersrdquo Integers vol 7 article A28 2007
[10] R S Melham ldquoAnalogues of two classical theorems on therepresentations of a numberrdquo Integers vol 8 article A51 2008
[11] R S Melham ldquoAnalogues of Jacobirsquos two-square theorem aninformal accountrdquo Integers vol 10 no 1 pp 83ndash100 2010
[12] P C Toh ldquoOn representations by figurate numbers a uniformapproach to the conjectures of Melhamrdquo International Journalof Number Theory vol 9 no 4 pp 1055ndash1071 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 International Journal of Mathematics and Mathematical Sciences
[6] M D Hirschhorn ldquoThe number of representations of a numberby various formsrdquo Discrete Mathematics vol 298 no 1ndash3 pp205ndash211 2005
[7] M D Hirschhorn ldquoThe number of representation of a numberby various forms involving triangles squares pentagons andoctagonsrdquo in Ramanujan Rediscovered N D Baruah B CBerndt S Cooper T Huber and M Schlosser Eds RMS Lec-ture Note Series No 14 pp 113ndash124 Ramanujan MathematicalSociety 2010
[8] ND Baruah and B K Sarmah ldquoThenumber of representationsof a number as sums of various polygonal numbersrdquo Integersvol 12 2012
[9] H Y Lam ldquoThe number of representations by sums of squaresand triangular numbersrdquo Integers vol 7 article A28 2007
[10] R S Melham ldquoAnalogues of two classical theorems on therepresentations of a numberrdquo Integers vol 8 article A51 2008
[11] R S Melham ldquoAnalogues of Jacobirsquos two-square theorem aninformal accountrdquo Integers vol 10 no 1 pp 83ndash100 2010
[12] P C Toh ldquoOn representations by figurate numbers a uniformapproach to the conjectures of Melhamrdquo International Journalof Number Theory vol 9 no 4 pp 1055ndash1071 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
