This article was downloaded by: [University of Haifa Library]On: 23 October 2013, At: 11:46Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Integral Transforms and SpecialFunctionsPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/gitr20

A general class of Voronoi's andKoshliakov–Ramanujan's summationformulas involving d k (n)Semyon Yakubovich aa Department of Mathematics, Faculty of Sciences , University ofPorto , Campo Alegre st., 687 4169-007 , Porto , PortugalPublished online: 23 Feb 2011.

To cite this article: Semyon Yakubovich (2011) A general class of Voronoi's andKoshliakov–Ramanujan's summation formulas involving d k (n), Integral Transforms and SpecialFunctions, 22:11, 801-821, DOI: 10.1080/10652469.2010.540447

To link to this article: http://dx.doi.org/10.1080/10652469.2010.540447

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Integral Transforms and Special FunctionsVol. 22, No. 11, November 2011, 801–821

A general class of Voronoi’s and Koshliakov–Ramanujan’ssummation formulas involving dk(n)

Semyon Yakubovich*

Department of Mathematics, Faculty of Sciences, University of Porto, Campo Alegre st., 687 4169-007Porto, Portugal

(Received 11 October 2010 )

By using the theory of the Mellin and Mellin convolution type transforms, we prove a general summationformula of Voronoi involving sums of the form

∑dk(n)f (n), where dk(n), k = 2, 3, . . . , d2(n) ≡ d(n) is

the number of ways of expressing n as a product of k factors. These sums are related to the famous Dirichletdivisor problem of determining the asymptotic behaviour as x → ∞ of the sum Dk(x) = ∑

n≤x dk(n). Inparticular, we generalize Koshliakov’s formula and certain identities from Ramanujan’s lost notebook to thecase of hyper-Bessel functions and Jacobi elliptic theta functions. New examples of Voronoi’s summationformulas involving Bessel, exponential functions and their products, which are based on a comprehensiveMarichev’s table of Mellin’s transforms are given. The equivalence of these relations to the functionalequation for the Riemann Zeta -function is discussed. An extension of the Koshliakov formula involvingthe Kontorovich–Lebedev transform is obtained.

Keywords: Voronoi summation formula; Dirichlet divisor problem; Mellin transform; Mellin convolutiontype transforms; Fourier–Watson transforms; Riemann Zeta-function; Kontorovich–Lebedev transform;modified Bessel and hyper-Bessel functions; hypergeometric functions; Ramanujan’s lost notebook

AMS Subject Classifications: 44A15; 33C05; 33C10; 11M06; 11M36; 11N37

1. Introduction and auxiliary results

The main goal of this paper is to extend the Voronoi summation formula, which was proved in [7]

limN→∞

[N∑

n=1

d(n)f (n) −∫ N

1/N

(log x + 2γ )f (x) dx

]

= limN→∞

[N∑

n=1

d(n)g(n) −∫ N

1/N

(log x + 2γ )g(x) dx

], (1.1)

on the sums involving the divisor function dk(n), k = 2, 3, . . . and d2(n) ≡ d(n). In the latter caseto prove certain extensions of (1.1) (cf. [15]) the theory of the Fourier–Watson transformations in

*Email: syakubov@fc.up.pt

ISSN 1065-2469 print/ISSN 1476-8291 online© 2011 Taylor & Francishttp://dx.doi.org/10.1080/10652469.2010.540447http://www.tandfonline.com

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

802 S. Yakubovich

L2(R+) [12] was applied. We will show below that this theory does not work for the sums withdk(n), k = 3, 4, . . . and we will call the general theory of the Mellin convolution type transforms(see in [6,12,16]). As usual [13], dk(n) means the number of ways of expressing n as a product of k

factors.We note that γ in (1.1) is Euler’s constant and functions f, g are defined on R+ representinga Fourier–Watson transformation in L2(R+) of each other. By using similar methods, we will givein the sequel various extensions of the Koshliakov formula [1,2,4,7,8,11] involving the modifiedBessel and hyper-Bessel functions (cf. [3, Vol. II; 6]). Further generalizations will be obtainedintegrating with respect to a pure imaginary index of the modified Bessel function and employingthe Kontorovich–Lebedev transformation [5,10,14,16].

As it is known [12], the Mellin transform in L2(R+) is defined by the integral

F(s) ≡ f ∗(s) = l.i.m.N→∞

∫ N

1/N

f (x)xs−1 dx, (1.2)

where s ∈ σ , σ = {s ∈ C : Re s = 1/2} and the convergence of the integral is in the mean squarewith respect to the norm of the space L2(σ ). Moreover, the inversion formula takes place

f (x) = l.i.m.N→∞

1

2πi

∫ 1/2+iN

1/2−iN

F (s)x−s ds, (1.3)

where integral (1.3) is convergent in mean with respect to the norm in L2(R+) and the generalizedParseval equality holds

∫ ∞

0f (xt)g(t) dt = 1

2πi

∫σ

F (s)G(1 − s)x−s ds. (1.4)

However, when f (x)xμ−1 ∈ L1(R+), then integral (1.2) exists as a Lebesgue integral and can bewritten in the form

F(s) =∫ ∞

0f (x)xs−1 dx, Re s = μ. (1.5)

Moreover, if f has a bounded variation in the neighbourhood of a point x, then the inversionformula like (1.3) holds and can be represented by the pointwise limit (see in [12])

1

2[f (x + 0) + f (x − 0)] = 1

2πilim

N→∞

∫ μ+iN

μ−iN

F (s)x−s ds,

where the latter integral converges in the principal value sense. It is easy to verify that undercondition f (x)xμ−1 ∈ L1(R+), μ0 < μ < μ1, one can differentiate with respect to s in (1.5) andwe get straightforward the representation of the derivative F ′(s)

F ′(s) =∫ ∞

0f (x)xs−1 log x dx, (1.6)

where integral (1.6) is absolutely convergent in the open strip {s ∈ C, μ0 < Re s < μ1}. Further,the left-hand side of (1.4) is called the Mellin convolution and for instance, considering f as akernel, we get a class of the Mellin convolution type transforms. In the L2-case, it corresponds to

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

Integral Transforms and Special Functions 803

the Fourier–Watson transformations

gk(x) = d

dx

∫ ∞

0

k(xu)

uf (u) du, (1.7)

gh(x) = d

dx

∫ ∞

0

h(xu)

uf (u) du, (1.8)

which are automorphisms in L2(R+; dx) and have reciprocal inversion formulas for almost allx ∈ R+ in the form

f (x) = d

dx

∫ ∞

0

h(xu)

ugk(u) du, (1.9)

f (x) = d

dx

∫ ∞

0

k(xu)

ugh(u) du, (1.10)

if and only if a continual analogue of the biorthogonality for sequences

∫ ∞

0k(ξx)h(ηx)

dx

x2= min(ξ, η), (1.11)

holds. Moreover, in this case, the Parseval type equality takes place

∫ ∞

0gk(x)gh(x) dx =

∫ ∞

0[f (x)]2 dx. (1.12)

Functions k, h are called the conjugate Watson kernels. These kernels possess the followingproperties: functions

k(x)

x∈ L2(R+; dx) and

K(s)

1 − s∈ L2(σ ),

h(x)

x∈ L2(R+; dx) and

H(s)

1 − s∈ L2(σ )

are reciprocal Mellin pairs in L2 (1.2), (1.3), where

sup−∞<t<∞

∣∣∣∣K(

1

2+ it

)∣∣∣∣2

≤ Ck < ∞,

sup−∞<t<∞

∣∣∣∣H(

1

2+ it

)∣∣∣∣2

≤ Ch < ∞.

Furthermore, conditions (1.11) and

K(s)H(1 − s) = 1, Re s = 1

2(1.13)

are equivalent. In the case h(x) = k(x) condition (1.13) becomes |K(1/2 + it)| = 1 and k(x) iscalled the Watson kernel. We note that a class of transformations (1.7), (1.8) involves classical sineand cosine Fourier transforms, the Hankel transform, the Hilbert transform, etc. (cf. [10,12,16]).

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

804 S. Yakubovich

Let us introduce the Kontorovich–Lebedev transformation in the form (see in [5,10,14,16])

gλ(x) = l.i.m.N→∞

∫ N

−N

Kiτ (λx)f (τ) dτ, x > 0, (1.14)

where λ > 0 is a parameter. It involves an integration with respect to the pure imaginary index iτ

of the real-valued modified Bessel function Kiτ (x) given by the following Fourier integral [10]

Kiτ (x) =∫ ∞

0e−x cosh u cos τu du. (1.15)

The transform (1.14) forms an isometric isomorphism

gλ : L2

(R; dτ

τ sinh πτ

)↔ L2

(R+; dx

x

),

where integral (1.14) converges with respect to the norm in L2(R+; (dx/x)). Furthermore, thereciprocal inversion is given by the formula

f (τ) = l.i.m.N→∞

1

π2τ sinh πτ

∫ ∞

1/N

Kiτ (λx)gλ(x)dx

x, (1.16)

where integral (1.16) converges with respect to the norm inL2(R; (dτ/τ sinh πτ)) and the Parsevalequality holds ∫ ∞

0|gλ(x)|2 dx

x= π2

∫ ∞

−∞|f (τ)|2

τ sinh πτdτ. (1.17)

However, integral (1.14) exists in the Lebesgue sense for a wide class of functions. Indeed, thisfact can be easily verified using the Schwarz inequality and the uniform inequality for the modifiedBessel functions [14,16]

|Kiτ (x)| ≤ e−π |τ |/3K0

(x

2

), x > 0, τ ∈ R. (1.18)

Generally, the modified Bessel function Kμ(z) [3, Vol. II] satisfies the differential equation

z2 d2u

dz2+ z

du

dz− (z2 + μ2)u = 0, (1.19)

and has the asymptotic behaviour

Kμ(z) =(

π

2z

)1/2

e−z

[1 + O

(1

z

)], z → ∞, (1.20)

and near the origin

z|Reμ|Kμ(z) = 2μ−1(μ) + o(1), z → 0, μ = 0, (1.21)

K0(z) = − log z + O(1), z → 0. (1.22)

It can be represented by the inverse Mellin transform (1.3) of the product of two Euler’s gamma-functions (see in [6])

Kν(x) = 1

πi

∫ μ+i∞

μ−i∞2s−3

(s + ν

2

)

(s − ν

2

)x−s ds, x > 0, (1.23)

where μ > |Reν|. This integral can be calculated by the Slater theorem (cf. [6]) as a sum ofresidues in the left-hand simple poles (ν = 0) of the gamma- function ((s + ν)/2). So the result

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

Integral Transforms and Special Functions 805

will be written in terms of the hypergeometric functions. However, when ν = 0, we arrive at theso-called logarithmic case [6]

K0(x) = 1

πi

∫ μ+i∞

μ−i∞2s−32

( s

2

)x−s ds, x > 0, (1.24)

where the value of the integral will be a sum of residues in the double poles s = −2k, k ∈ N0 ofthe function 2(s/2). Therefore by straightforward calculations employing a series expansion ofthe gamma-function in the neighbourhood of the pole (see in [6]), we get the result

K0(x) =∞∑

k=0

(x

2

)2k ψ(1 + k) − log(x/2)

[k!]2, (1.25)

where ψ(z) = (d/dz)[log (z)] is Euler’s psi-function [3, Vol. I]. In the sequel, we will appealto the hyper-Bessel functions, which generalize, for instance, integral (1.24), namely

Uk(x) = 1

2πi

∫ μ+i∞

μ−i∞k

( s

2

)x−s ds, x > 0, k ∈ N. (1.26)

An analogue of the series expansion (1.25) in this case is quite complicated because we shouldcalculate the residue of the function k(s/2) in the multiple pole s = −2m, m ∈ N0 of the orderk. However, we can represent Uk(x), k = 2, 3, . . . as a k − 1-fold Mellin convolution of theexponential function [12], since each gamma-function (s/2) is the Mellin transform of 2e−x2

.Thus denoting by u1 = u1 u2 . . . uk−1, du = du1 du2 . . . duk−1 and making elementary changesof variables, we obtain

Uk(x) = 2∫

Rk−1+

exp

(−

k−1∑m=1

um − x2

u1

)du

u1, k ∈ N\{1}, U1(x) = 2e−x2

. (1.27)

Differentiating with respect to x under the integral sign in (1.26) owing to the absolute and uniformconvergence, it is not difficult to deduce a kth order differential equation whose particular solutionis Uk(x). So we derive (

−x

2

d

dx

)k

ω − x2ω = 0, ω ≡ ω(x).

As it is known [13], the Dirichlet function Dk(x) = ∑n≤x dk(n), k = 2, 3, . . . can be represented

as the inverse Mellin transform

Dk(x) = 1

2πi

∫ μ+i∞

μ−i∞ζ k(s)

sxs ds, μ > 1, (1.28)

where ζ(s) is the Riemann zeta-function satisfying the familiar functional equation

ζ(s) = 2sπs−1 sin(πs

2

)(1 − s)ζ(1 − s). (1.29)

It has the following asymptotic behaviour on the critical line [13]

ζ

(1

2+ it

)= O(t1/6 log t), |t | → ∞ (1.30)

and the integral representation in the critical strip

ζ(s) = s

∫ ∞

0

[x] − x

xs+1dx, 0 < Re s < 1, (1.31)

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

806 S. Yakubovich

where [ ] denotes the integer part of a number. Its kth power is represented by the series

ζ k(s) =∞∑

n=1

dk(n)

ns, Re s > 1. (1.32)

Further, the integrand in (1.28) has a pole of order k at s = 1, and the residue is of the formxPk(log x), where Pk is a polynomial of degree k − 1. Denoting by

�k(x) = Dk(x) − xPk(log x), (1.33)

it behaves as (see in [13])

�k(x) = O(x1−1/k logk−2 x), x → ∞, k = 2, 3, . . . , (1.34)

�k(x) = O(x logk−1 x), x → 0, k = 2, 3, . . . . (1.35)

2. General expansions

Let f (x) be defined on R+ and its Mellin transform (1.2) be such that sF (s) ∈ L2(σ ), σ = {s ∈C, Re s = 1/2}. Hence, it is easily seen that f is continuous on R+, f (x)x1/2 tends to zero whenx → 0 and x → ∞, f ∈ L2(R+), F (s) ∈ L2(σ ) ∩ L1(σ ) and the integral (1.3) can be written inthe form of the absolutely convergent integral (1.6), namely

f (x) = 1

2πi

∫σ

F (s)x−s ds, x > 0. (2.1)

In fact, we obtain ∫σ

|F(s)|2|ds| < 2∫

σ

|sF (s)|2| ds| < ∞,

∫σ

|F(s) ds| ≤(∫

σ

|sF (s)|2 ds

)1/2 (∫σ

|ds||s|2

)1/2

< ∞.

Therefore formula (2.1) yields the estimate

x1/2|f (x)| ≤ 1

2π

∫σ

|F(s) ds| = O(1), x > 0.

Moreover, it tends to zero when x approaches to zero and infinity via the Riemann–Lebesguelemma. Finally, condition sF (s) ∈ L2(σ ) implies that f is equivalent to some absolutelycontinuous function ρ such that for almost all x > 0, we have(

−xd

dx

)ρ(x) =

(−x

d

dx

)f (x) ∈ L2(R+).

In [15], we have proved main theorems characterizing the classical Voronoi summation formula(1.1) (k = 2), which can be reformulated as in the following

Theorem 1 Let sF (s) ∈ L2(σ ). Then the following statements are equivalent:

(i) Riemann’s zeta-function ζ(s) satisfies the functional equation (1.29);

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

Integral Transforms and Special Functions 807

(ii) There exist two functions f, g ∈ L2(R+) such that f and g are Fourier–Watson transforms(1.7) of each other

g(x) = d

dx

∫ ∞

0

χ(xu)

uf (u) du, (2.2)

f (x) = d

dx

∫ ∞

0

χ(xu)

ug(u) du, (2.3)

with the Watson kernel

χ(x) = l.i.m.N→∞

1

2πi

∫ (1/2)+iN

12 −iN

π1−2s

[

(s2

)

(1−s

2

)]2

x1−s

1 − sds

= 2∫ x

0[2K0(4π

√t) − πY0(4π

√t)] dt,

and for all x > 0, the following identity holds∫σ

ζ 2(s)F (s)x−s ds =∫

σ

ζ 2(s)G(s)xs−1 ds, (2.4)

where

G(s) = π1−2s

[(s/2)

((1 − s)/2)

]2

F(1 − s)

is the Mellin transform (1.2) of g in L2, sG(s) ∈ L2(σ ) and both integrals are absolutelyconvergent;

(iii) An analogue of the Voronoi formula holds

limN→∞

[N∑

n=1

d(n)f (xn) −∫ N

1/N

(log t + 2γ )f (xt) dt

]

= 1

xlim

N→∞

[N∑

n=1

d(n)g(n

x

)−

∫ N

1/N

(log t + 2γ )g

(t

x

)dt

], x > 0, (2.5)

where f, g are reciprocal Fourier–Watson transforms (2.2), (2.3) and the convergence in(2.5) is pointwise.

Corollary 1 Let in addition to conditions of Theorem 1 series in (2.5) be convergent andf, g ∈ L1(R+; | log x| dx). Then it can be rewritten in a form involving Mellin’s transforms offunctions f, g and their derivatives, which are evaluated at the point s = 1

∞∑n=1

d(n)

[f (xn) − 1

xg

(n

x

)]

= 1

x[(2γ − log x)F (1) + F ′(1)] − (2γ + log x)G(1) − G′(1), x > 0. (2.6)

Corollary 2 Formula (2.6) holds if F(μ + it), G(μ + it) ∈ L1(R) are analytic in an openvertical strip containing the line μ = 1 and tend to zero when |t | → ∞ uniformly in its closure.

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

808 S. Yakubovich

Proof In fact, in this case, we get that inverse Mellin transforms (2.1) f, g are continuous onR+ and the estimates

|f (x)| ≤ x−μ

2π

∫ μ+i∞

μ−i∞|F(s) ds| = O(x−μ), x → ∞, μ > 1,

|g(x)| ≤ x−μ

2π

∫ μ+i∞

μ−i∞|G(s) ds| = O(x−μ), x → ∞, μ > 1

are valid. Therefore conditions of Corollary 1 are easily satisfied and series in (2.6) convergeabsolutely (see (1.32)).

In order to extend formulas (2.5) and (2.6) for the sums with dk(n), k = 2, 3, 4, . . . , we willappeal to the theory of Mellin’s transform (1.5) and the Mellin convolution type transforms in theweighted Lebesgue spaces. We begin with �

Lemma 1 Let 0 < μ < 1/k, k = 2, 3, . . . ,. Then

ζ k(1 − s)

1 − s=

∫ ∞

0�k(x)xs−2 dx, s = μ + it, (2.7)

where integral (2.7) converges absolutely.

Proof In fact, the absolute convergence easily follows from estimates (1.34), (1.35) and arestriction on μ. Further, using (1.33) we write∫ ∞

0�k(x)xs−2 dx =

∫ 1

0�k(x)xs−2 dx +

∫ ∞

1�k(x)xs−2 dx

= −∫ 1

0Pk(log x)xs−1 dx +

∫ ∞

1�k(x)xs−2 dx

=∫ ∞

1�k(x)xs−2 dx −

k−1∑j=0

aj

dj

dsj[s−1], (2.8)

where we use notation Pk(z) = ∑k−1j=0 ajz

j . But the right-hand side of the latter equality is analyticfor μ < 0. Thus,it gives the analytic continuation of the integral (2.7) into the left half-plane μ < 0.

In the meantime, appealing to (1.32), splitting the range of integration (1, ∞) into(1, 2), (2, 3), . . . and making straightforward calculations, we get (μ < 0)∫ ∞

1�k(x)xs−2 dx =

∫ ∞

1Dk(x)xs−2 dx −

∫ ∞

1Pk(log x)xs−1 dx

= 1

1 − s

∞∑n=1

dk(n)ns−1 +k−1∑j=0

aj

dj

dsj[s−1] = ζ k(1 − s)

1 − s+

k−1∑j=0

aj

dj

dsj[s−1].

Substituting the latter expression into (2.8) and returning to the strip 0 < μ < (1/k) by analyticcontinuation, we come out with (2.6). Lemma 1 is proved.

From Lemma 1, we obtain that ζ k(1 − s)/(1 − s) is uniformly bounded in the strip 0 < μ <

(1/k). Moreover, integration by parts in (2.7) yields the equality

ζ k(1 − s) = limN→∞

∫ N

0xs−1d[�k(x)], s = μ + iτ, 0 < μ <

1

k, (2.9)

where the convergence is uniform by τ ∈ R. �

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

Integral Transforms and Special Functions 809

Lemma 2 Let sF (s) ∈ L1(1 − μ − i∞, 1 − μ + i∞), 0 < μ < (1/k), k = 2, 3, . . . ,. Then forall x > 0

1

2πi

∫ 1−μ+i∞

1−μ−i∞ζ k(s)F (s)x−s ds = lim

N→∞

[N∑

n=1

dk(n)f (nx) −∫ N

0Qk(log t)f (xt) dt

], (2.10)

where f is defined by the inverse Mellin transform (2.1), Qk(z) = Pk(z) + P ′k(z), ‘′’ denotes a

first derivative of Pk and the integral in the left-hand side of (2.10) is absolutely convergent.

Proof Indeed, the absolute convergence of the integral in the left-hand side of (2.10) easilyfollows from Lemma 1 and condition of the present lemma. Moreover, it yields that F(s) isintegrable and therefore by (2.1) f (x) is continuous for all x > 0. Hence, using the uniformconvergence in (2.9), equalities (1.33)and (2.1), the condition of the lemma and the Lebesgue–Stieljes integration, we find for all x > 0

1

2πi

∫ 1−μ+i∞

1−μ−i∞ζ k(s)F (s)x−s ds = lim

N→∞1

2πi

∫ N

0d[�k(t)]

∫ μ+i∞

μ−i∞F(1 − s)(xt)s−1 ds dt

= limN→∞

[N∑

n=1

dk(n)f (nx)

× −∫ N

0[Pk(log t) + P ′

k(log t)]f (xt) dt

].

Thus, it leads to (2.10) and completes the proof of Lemma 2.Now calling the functional equation (1.29) for the Riemann zeta-function and the supplement

and duplication formulas for gamma-functions [3, Vol. I], we write the left-hand side of (2.9) inthe form

1

2πi

∫ 1−μ+i∞

1−μ−i∞ζ k(s)F (s)x−s ds = 1

2πi

∫ μ+i∞

μ−i∞

[π1/2−sζ(s)

(s/2)

(1 − s/2)

]k

F (1 − s)xs−1 ds.

(2.11)

Meanwhile the right-hand side of (2.11) can be treated employing representation (1.31),differential properties of the Mellin transform and the value of an elementary integral∫ ∞

0t s−1 cos(2πt) dt = 1

2π−s+1/2 (s/2)

((1 − s)/2), 0 < Re s < 1.

Hence taking into account the uniform convergence by x ≥ x0 > 0, we derive

1

2πi

∫ μ+i∞

μ−i∞

[π1/2−sζ(s)

(s/2)

((1 − s)/2))

]k

F (1 − s)xs−1 ds

= 2k−1

πix

(x

d

dx

)k ∫ μ+i∞

μ−i∞[V ∗(s)]kF (1 − s)xs ds, (2.12)

where

V ∗(s) = π1/2−s 1

2

(s/2)

((1 − s)/2)

∫ ∞

0

([1

x

]− 1

x

)xs−1 dx =

∫ ∞

0V (x)xs−1dx,

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

810 S. Yakubovich

and a function V (x) is defined as a Mellin’s convolution [12] by the convergent integral

V (x) =∫ ∞

0

[t] − t

tcos(2πxt) dt, x > 0.

On the other hand, calling the Stirling asymptotic formula for gamma-functions [3, Vol. I], we eas-ily find [V ∗(s)]k = O(|s|k(μ−1/2)), |s| → ∞. Therefore, it belongs to L2(μ − i∞, μ + i∞) ∩L1(μ − i∞, μ + i∞) when 0 < μ < (1/k), k = 4, 5, . . . , and for k = 3 when 0 < μ < (1/6).The case k = 2 corresponds to the space L2(μ − i∞, μ + i∞) when 0 < μ < (1/4). Further, thecondition skF (1 − s) ∈ L2(μ − i∞, μ + i∞), k = 2, 3, . . . will evidently guarantee the integra-bility of sF (1 − s) on (μ − i∞, μ + i∞). Hence using the generalized Parseval equality (1.4),we can write the right-hand side of (2.12) in the form

2k−1

πix

(x

d

dx

)k ∫ μ+i∞

μ−i∞[V ∗(s)]kF (1 − s)xs ds = 2k

x

(x

d

dx

)k ∫ ∞

0Vk(t)f

(t

x

)dt, x > 0,

(2.13)

where

Vk(x) = 1

2πi

∫ μ+i∞

μ−i∞[V ∗(s)]kx−s ds, x > 0 (2.14)

and the convergence in (2.14) is absolute for 0 < μ < (1/k), k = 4, 5, . . . , and for 0 < μ <

(1/6), (k = 3). In the case k = 2 integral (2.14) the converges in the mean square sense for0 < μ < 1/4.

Thus, combining with Lemma 2, we have proved the following result. �

Theorem 2 Let skF (1 − s) ∈ L2(μ − i∞, μ + i∞), k = 2, 3, . . . , 0 < μ < (1/k). Then forall x > 0, the following Voronoi type formula holds

limN→∞

[N∑

n=1

dk(n)f (nx) −∫ N

0Qk(log t)f (xt) dt

]= x−1

(2x

d

dx

)k ∫ ∞

0Vk(t)f

(t

x

)dt,

(2.15)

where f is defined by the inverse Mellin transform (2.1) and Vk(x) is given by (2.14) with thecorresponding convergence of the integral:

(i) it converges absolutely for k = 4, 5, . . . , and for k = 3, if μ ∈ (0, (1/6));(ii) when k = 2, it converges in mean square for all μ ∈ (0, (1/4)).

3. Various examples and extensions of the Voronoi and Koshliakov formulas and certainidentities from Ramanujan’s lost notebook

This section will be devoted to examples of general Voronoi’s summation formula and its partic-ular case such as Koshliakov’s [4] formula associated with the modified Bessel function K0(x),and as it says in [1] was proved by Ramanujan about 10 years earlier. We will also generalizecertain identities from Ramanujan’s lost notebook recently exhibited in [1]. We note here that thisinvestigation would be impossible without the use of invaluable comprehensive table of Mellin’stransforms of hypergeometric functions created by Marichev [6] in 1978 and extended latter in[9, Vol. 3].

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

Integral Transforms and Special Functions 811

Koshliakov proved the following summation formula

∞∑n=1

d(n)K0(2πzn) − 1

z

∞∑n=1

d(n)K0

(2πn

z

)

= 1

4z(γ − log 4πz) − 1

4

(γ − log

(4π

z

)), Re z > 0. (3.1)

Here, we will give another simple proof of (3.1) basing on the equality (2.4) and the inverse Mellintransform (2.1). In fact, appealing to the relation (2.16.2.2) in [9, Vol. 2]∫ ∞

0Kν(cx)xs−1 dx = 2s−2c−s

(s + ν

2

)

(s − ν

2

), Re c > 0, Re s > |Reν|, (3.2)

we have reciprocally

Kν(cx) = 1

2πi

∫ α+i∞

α−i∞2s−2

(s + ν

2

)

(s − ν

2

)(cx)−s ds, Re s = α, (3.3)

where both integrals (3.2) and (3.3) are absolutely convergent via the asymptotic behaviour of themodified Bessel functions (1.20), (1.21) and Euler’s gamma-functions (see [3, Vol. I]). Hence,letting ν = 0 and taking f (x) = K0(2πzx), we find its Mellin transform (see (3.2), (1.26)) asF(s) = (1/4)(πz)−s2(s/2) = (1/4)(πz)−sU ∗

2 (s). Therefore, the corresponding function G(s)

in (2.4) is equal to G(s) = (1/4)π−szs−1U ∗2 (s). So reciprocally g(x) = z−1K0(2πx/z) and by

straightforward calculations of the related absolutely convergent integrals in (2.5) (see (2.6)), wecome out with (3.1), which can be written in terms of the function U2(x) = 4K0(2x) (see (1.24),(1.26)) as follows:

∞∑n=1

d(n)U2(πzn) − 1

z

∞∑n=1

d(n)U2

(πn

z

)= 1

z(γ − log 4πz) − γ + log

(4π

z

), Re z > 0.

(3.4)On the other hand, taking (1.27) for k = 2, we write by straightforward calculations for any a > 0

∞∑n=1

d(n)U2(2π√

an) = 4∞∑

n=1

d(n)K0(4π√

an) = 2∞∑

n=1

d(n)

∫ ∞

0e−2π(u+(an/u)) du

u

= 2∞∑

m=1

∞∑k=1

∫ ∞

0e−2π(kv+(am/v)) dv

v= 2

∫ ∞

0

dv

v(e2πv − 1)(e2πa/v − 1).

So we deduce an identity, which was found in Ramanujan’s lost book and discussed in [1],Theorem 4.1, namely

2∞∑

n=1

d(n)K0(4π√

an) =∫ ∞

0

dv

v(e2πv − 1)(e2πa/v − 1). (3.5)

Moreover, the corresponding identity (2.6) for this case can be done dealing with the relation(2.6.5.15) in [9, Vol. 1]. So we have finally the following Koshliakov type formula

2∞∑

n=1

d(n)K0(4π√

an) = a

π2

∞∑n=1

d(n) log(a/n)

a2 − n2− γ

2− log a

4

(1 + 1

π2a

)− log(2π)

2π2a, (3.6)

which was established for the first time in [11] (see also in [1,15]).

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

812 S. Yakubovich

Considering F(s) = (s) = U ∗1 (2s) we have correspondingly in (2.4), G(s) = (π3/2(2π2)−s/

sin(πs/2))((s/2)/((1 − s)/2)). Therefore, f (x) = e−x = (1/2)U1(√

x) and g(x) is calcu-lated with the use of the (1.4) and relation (2.5.9.12) in [9, Vol. 1]. Therefore

g(x) = −2[e−4π2xEi(4π2x) + e4π2xEi(−4π2x)],

where Ei(z) is the exponential integral. So appealing to the values of the related Mellin transformsand its derivatives at the point s = 1 in (2.6), we arrive at the followingVoronoi summation formula

∞∑n=1

d(n)

[e−xn + 2

x

[e−4π2n/xEi

(4π2n

x

)+ e4π2n/xEi

(−4π2n

x

)]]= 1

4+ γ − log x

x, (3.7)

which was proved in [8] by a different approach.Another Voronoi’s type formula involving the exponential function can be obtained employing

the function F(s) = (s/2), which forms a Mellin pair with f (x) = U1(x) = 2e−x2. Calculating

the corresponding elementary integrals, the left-hand side of (2.5) in this case becomes

2∞∑

n=1

d(n)e−(xn)2 −√

π

x

(2γ − log x + 1

2ψ

(1

2

)), x > 0. (3.8)

Meanwhile, G(s) = π1−2s2(s/2)/((1 − s)/2) and its inverse Mellin transform can be calcu-lated as a sum of residues in double poles of the square of a gamma-function. Therefore after asimple change of variable, we get

g(x) = 1

2πi

∫ μ+i∞

μ−i∞2π

2(s)

((1/2) − s)(π4x2)−s ds

= 2√

π

∞∑k=0

(π2x)2k

[k!]2 (1/2)k

[2ψ(1 + k) + ψ

(1

2+ k

)− 2 log(π2x)

], x > 0, (3.9)

where (a)k is Pochhammer’s symbol [3, Vol. I]. Moreover, since G(1) = 0, and G′(1) =−(1/2), ψ(1/2) = −γ − log 4, we keep in mind (2.6), (3.8), and (3.9) to arrive at the followingsummation formula:

2∞∑

n=1

d(n)

[e−(xn)2 +

√π

x

∞∑k=0

(π2n/x)2k

[k!]2 (1/2)k

[2ψ(1 + k) + ψ

(1

2+ k

)− 2 log

(π2 n

x

)]]

= 1

2+

√π

x

(3

2γ − log(2x)

), x > 0.

As it was mentioned in [1], the Koshliakov formula (3.4) can be considered as an analogue ofthe transformation formula for the Jacobi elliptic theta function [13], namely

ϑ(0, ix) = 1√x

ϑ

(0,

i

x

),

where

ϑ(0; ix) = 1 + 2∞∑

n=1

e−πn2x, x > 0. (3.10)

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

Integral Transforms and Special Functions 813

It has the following representation [13]

ϑ(0; ix) = 1 + 1

2πi

∫ μ+i∞

μ−i∞π−s/2

( s

2

)ζ(s)x−s/2 ds, μ > 1. (3.11)

In the sequel, we will follow in a similar direction to generalize the Koshliakov formula (3.4),Ramanujan’s identity (3.5) and Soni’s formula (3.6) to functions Uk(x), k = 2, 3, . . . involvingthe divisor function dk(n). We have

Theorem 3 Let a > 0, k = 2, 3, . . .. Then

1

2πi

∫ μ+i∞

μ−i∞

[(2π)−s/2ζ

( s

2

)

( s

2

)]k

a−s/2 ds

=∞∑

n=1

dk(n)Uk((2π)k/2√an)

= 2∫

Rk−1+

[exp

(2πa

u1u2 . . . uk−1

)− 1

]−1 k−1∏j=1

(e2πuj − 1)−1 duj

uj

, μ > 2,

× 1

2πi

∫ μ+i∞

μ−i∞

[π−s/2ζ(s)

( s

2

)]k

a−s/2 ds =∞∑

n=1

dk(n)Uk((aπk)1/2n) (3.12)

= 21−k

∫R

k−1+

[ϑ

(0; ia

u1u2 . . . uk−1

)− 1

] k−1∏j=1

(ϑ(0; iuj ) − 1)duj

uj

, μ > 1,

× 1

2πi

∫ μ+i∞

μ−i∞

[πζ(s)

sin(πs/2)

]k

a−s ds = 1

a2

∞∑n=1

dk(n)

∫ ∞

0Uk(nt)Uk

(t

a

)t dt (3.13)

= πk

∫R

k−1+

[a−1 u1u2 . . . uk−1cotanh

(π

au1u2 . . . uk−1

)− 1

]

×k−1∏j=1

(1

uj

cotanh

(π

uj

)− 1

)duj

uj

, 1 < μ < 2. (3.14)

Proof Indeed, calling representations (1.27) and (1.32), the definition of dk(n) with elementaryseries expansions and substitutions, we change the order of integration and summation owing tothe absolute convergence of series and integrals to derive

1

2πi

∫ μ+i∞

μ−i∞

[(2π)−s/2ζ

( s

2

)

( s

2

)]k

a−s/2 ds =∞∑

n=1

dk(n)Uk((2π)k/2√an)

= 2∞∑

m1=1

· · ·∞∑

mk=1

∫R

k−1+exp

⎛⎝−

k−1∑j=1

uj − (2π)ka m1m2 . . . mk

u1u2 . . . uk−1

⎞⎠ du

u1u2 . . . uk−1

= 2∞∑

m1=1

· · ·∞∑

mk=1

∫R

k−1+exp

⎛⎝−2π

k−1∑j=1

mjuj − 2πamk

u1u2 . . . uk−1

⎞⎠ du

u1u2 . . . uk−1

= 2∫

Rk−1+

[exp

(2πa

u1u2 . . . uk−1

)− 1

] k−1∏j=1

(e2πuj − 1)−1 duj

uj

.

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

814 S. Yakubovich

This leads to (3.12). In order to prove (3.13), we write similarly, taking into account (3.11)

1

2πi

∫ μ+i∞

μ−i∞

[π−s/2ζ(s)

( s

2

)]k

a−s/2 ds =∞∑

n=1

dk(n)Uk((aπk)1/2n)

= 2∞∑

m1=1

· · ·∞∑

mk=1

∫R

k−1+exp

⎛⎝−

k−1∑j=1

uj − aπk(m1m2 . . . mk)2

u1u2 . . . uk−1

⎞⎠ du

u1u2 . . . uk−1

× 21−k

∫R

k−1+

[ϑ

(0; ia

u1u2 . . . uk−1

)− 1

] k−1∏j=1

(ϑ(0; iuj ) − 1)duj

uj

.

Finally, we establish (3.14). Calling series expansion (5.4.5.1) in [9,Vol]. 1 and relation (8.4.2.5)in [9, Vol. 3], we obtain the equalities

1

2πi

∫ μ+i∞

μ−i∞πζ(s)

sin(πs/2)a−s ds = 2

∞∑n=1

1

1 + (an)2= π

(1

acotanh

(π

a

)− 1

). (3.15)

In the meantime, representations (1.26) and (1.32) and the factorization property of Mellin’sconvolution immediately imply the first equality in (3.14). On the other hand, invoking with(3.15) and the absolute convergence of the series and integrals, we deduce

1

a2

∞∑n=1

dk(n)

∫ ∞

0Uk(nt)Uk

(t

a

)t dt

= 1

a2

∞∑m1=1

· · ·∞∑

mk=1

∫ ∞

0Uk(m1m2 . . . mkt)Uk

(t

a

)t dt

=∞∑

m1=1

. . .

∞∑mk=1

1

2πi

∫ μ+i∞

μ−i∞

[

( s

2

)

(1 − s

2

)]k

(m1m2 . . . mka)−s ds

= 2k

∫R

k−1+

∞∑m1=1

· · ·∞∑

mk=1

[1 +

(m1m2 . . . mka

u1u2 . . . uk−1

)2]−1 k−1∏

j=1

1

1 + u2j

duj

uj

= 2πk−1∫

Rk−1+

∞∑mk=1

[1 +

(mka

u1u2 . . . uk−1

)2]−1 k−1∏

j=1

(1

uj

cotanh

(π

uj

)− 1

)duj

uj

= πk

∫R

k−1+

[a−1 u1u2 . . . uk−1 cotanh

(π

au1u2 . . . uk−1

)− 1

]

×k−1∏j=1

(1

uj

cotanh

(π

uj

)− 1

)duj

uj

.

This gives (3.14) and completes the proof of Theorem 3. �

Remark 1 We note that formulas (3.12), (3.13), and (3.14) are k − 1-fold Mellin convolutions ofthe functions 21/k(e2πx − 1)−1, 2(1−k)/k(ϑ(0; ix) − 1) and π(x−1 cotanh(π/x) − 1), respectively.

Using (3.13), we get straightforward

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

Integral Transforms and Special Functions 815

Corollary 3 The following analogue of the Koshliakov–Ramanujan formula (3.4) holds

∞∑n=1

dk(n)Uk(aπk/2n) − 1

a

∞∑n=1

dk(n)Uk

(πk/2

an

)

= 21−k

∫R

k−1+

[ϑ

(0; ia

u1u2 . . . uk−1

)− 1 − 1

a

(ϑ

(0; i

au1u2 . . . uk−1

)− 1

)]

×k−1∏j=1

(ϑ(0; iuj ) − 1)duj

uj

, a > 0, μ > 1, k ∈ N0\{1}.

Corollary 4 Letting k = 2, moving the contour to the left in the integral with respect to s

and taking into account the residue of the double pole at s = 1, identity (3.12) becomes Soni’ssummation formula (3.6).

Finally in this section, we give more new examples of Voronoi’s summation formulas involvingBessel and exponential functions, which are based on the table of Mellin’s transform in [9, Vol. 3].

Example 1 Let F(s) = s−2. Then G(s) = ((π1−2s)/4)[((s/2)/(3 − s/2))]2 and corre-spondingly, by (1.4), (2.1), and relations (2.5.24.2) in [9, Vol. 1], (8.4.5.1) in [9, Vol. 3], wefind

f (x) =⎧⎨⎩log

(1

x

), if x ∈ (0, 1],

0, if x ∈ (1, ∞),

g(x) = (1/xπ)[(1/2)Y0(4π√

x) + (1/π)K0(4π√

x)]. This gives

∑n≤[x−1]

d(n) log(nx) + 1

π

∞∑n=1

d(n)

n

[1

2Y0

(4π

√n

x

)+ 1

πK0

(4π

√n

x

)]

= 1

x(2(1 − γ ) + log x) + 1

4log

( x

16π8

)− 3

2γ, x > 0.

Example 2 Let F(s) = ((s/2))(ν − (s/2)), ν > (1/2), ν = (1/2) + m, m ∈ N. Then byrelation (8.4.2.5) in [9, Vol. 3], we obtain f (x) = 2(ν)(1 + x2)−ν . Meanwhile, G(s) =π1−2s2(s/2)((ν − 1/2 + s/2)/(1 − s/2)). Hence by Slater’s theorem in the logarithmiccase, we derive

g(x) = 1

2πi

∫σ

π1−2s2( s

2

) (ν − (1/2) + (s/2))

(1 − s/2)x−s ds

= 2π4ν−1x2ν−1 2((1/2) − nu)

(ν)0F3

(1

2+ ν; 1

2+ ν, ν; −x2π4

)

− 2π√

π

cos πν((3/2) − ν)

∞∑k=0

(−4π4x2)k

k!(2k)!((3/2) − ν)k

×[

2ψ(1 + k) + ψ

(1

2+ k

)+ ψ

(ν − 1

2− k

)− 2 log(π2x)

],

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

816 S. Yakubovich

where 0F3(a, b, c; z) is the generalized hypergeometric function [3, Vol. I]. Calculating the valuesin the right-hand side of (2.6), we arrive at the summation formula for all x > 0

4x

∞∑n=1

d(n)

[(ν)

(1 + (xn)2)ν− 1

πn

(π2n

x

)2ν2 ((1/2) − ν)

(ν)0F3

(1

2+ ν; 1

2+ ν, ν; −

(π2n

x

)2)

+ π√

π

cos πν (

32 − ν

) ∞∑k=0

[k!(2k)!

(3

2− ν

)k

]−1 (−2

π2n

x

)2k

×[

2ψ(1 + k) + ψ

(1

2+ k

)+ ψ

(ν − 1

2− k

)− 2 log

(π2n

x

)]]

= √π(ν − 1/2)

[3γ − 2 log(2x) − ψ

(ν − 1

2

)]+ x(ν).

In particular, the case ν = 1, x = a−1 with the use of relation (2.5.37.2) in [9, Vol. 1] leads to theknown identity

∞∑n=1

d(n)

[a

a2 + n2− 2π(K0(4π

√ina) + K0(−4π

√ina))

]= πγ + π

2log a + 1

4a,

which is proved in [8] by another method.

Example 3 Let F(s) = (2/π5/2)3(s/2)((1 − s)/2). Then by relation (8.4.20.35) in [9,Vol. 3], we obtain the so-called Nicholson kernel [3, Vol. II] f (x) = 2[J 2

0 (x) + Y 20 (x)]. Cor-

respondingly, G(s) = 2π−(3/2)−2s3(s/2)((1 − s)/2) and g(x) = 2π [J 20 (π2x) + Y 2

0 (π2x)]. Itis easily seen by asymptotic of the Bessel functions that f, g ∈ L2(R+) and not L1(R+). Thereforethe Voronoi formula for this case will be written in the form (2.5). Precisely, we have the identity

limN→∞

[N∑

n=1

d(n)(J 20 (nx) + Y 2

0 (nx)) −∫ N

1/N

(log t + 2γ )(J 20 (xt) + Y 2

0 (xt)) dt

]= π

x

× limN→∞

[N∑

n=1

d(n)

(J 2

0

(π2n

x

)+ Y 2

0

(π2n

x

))

−∫ N

1/N

(log t + 2γ )

(J 2

0

(π2t

x

)+ Y 2

0

(π2t

x

))dt

], x > 0,

which is trivial when x = π .

Example 4 Let F(s) = 3(s/2). Then by (1.26) f (x) = U3(x) and G(s) = π1−2s2(s/2)

((1 − s)/2). Hence relation (8.4.23.5) in [9, Vol. 3] says that g(x) = 2π√

πe(πx)2/2K0(π2x2/2) ∈

L2(R+) and not L1(R+) (see (1.20), (1.21), and (1.22)). Thus, it gives

limN→∞

2π√

π

x

[N∑

n=1

d(n) exp

(1

2

(πn

x

)2)

K0

(1

2

(πn

x

)2)

−∫ N

1/N

(log t + 2γ ) exp

(1

2

(πt

x

)2)

K0

(1

2

(πt

x

)2)

dt

]

=∞∑

n=1

d(n)U3(nx) − π√

π

x

[γ

2− log(8x)

], x > 0.

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

Integral Transforms and Special Functions 817

Example 5 Let F(s) = −2(s/2)(1 − (s/2)). Then by relation (8.4.11.3) in [9, Vol. 3],f (x) = 2ex2

Ei(−x2), and correspondingly, G(s) = −π1−2s2(s/2)((1 + s)/2), F (1) =−π3/2, F ′(1) = (1/2)π3/2(γ + log 4), G(1) = −1, G′(1) = (1/

√π)((3γ /2) + log(2

√2π)).

Calculating g(x) by Slater’s theorem, we find

g(x) = i

∫ (1/4)+i∞

(1/4)−i∞2(s)

(1

2+ s

)(xπ2)−2s ds = 2π

√π

∞∑k=0

(−1)k(xπ2)2k

[k!]2(1/2)k

×[

2 log(xπ2) − 2ψ(1 + k) − ψ

(1

2− k

)]− 8π4x 0F2

(3

2,

3

2; −x2π4

).

So,

2∞∑

n=1

d(n)

[e(xn)2

Ei(−(xn)2) + 4π4n

x2 0F2

(3

2,

3

2; −

(nπ2

x

)2)]

+ π3/2

x

(2γ − log x + 1

2ψ

(1

2

))

= 2γ + log x − 1√π

(3γ

2+ log(2

√2π)

)+ 2π3/2

x

∞∑n=1

d(n)

∞∑k=0

(−1)k

[k!]2(1/2)k

×(

n π2

x

)2k [2 log

(nπ2

x

)− ψ(1 + k) − ψ

(1

2− k

)], x > 0.

Example 6 Let F(s) = √π2(s/2)[((1 + s)/2)]−1. Then by relation (8.4.23.3) in [9,

Vol. 3], f (x) = 2e−x2/2K0(x2/2). Hence G(s) = −π(3/2)−2s2(s/2)[(1 − (s/2))]−1, F (1) =

π3/2, F ′(1) = −π3/2((γ /2) + log 4), G(1) = 1, G′(1) = −(3/2)γ − log(8π2). Further,

g(x) = −i√

π

∫ (1/4)+i∞

(1/4)−i∞2(s)

(1 − s)(xπ2)−2s ds

= 2π√

π

∞∑k=0

(xπ2)2k

[k!]3[3ψ(1 + k) − 2 log(xπ2)].

Therefore,

2∞∑

n=1

d(n)

[e−(nx)2/2K0

((nx)2

2

)− π3/2

x

∞∑k=0

1

[k!]3

(nπ2

x

)2k [3ψ(1 + k) − 2 log

(nπ2

x

)]]

= π3/2

x

(3

2γ − log(4x)

)− γ

2+ log

(8π2

x

), x > 0.

Example 7 Let F(s) = (√

π/2)3(s/2)[((1 + s)/2)]−1. Then by relation (8.4.23.27) in [9,Vol. 3], f (x) = K2

0 (x). Hence G(s) = (1/2)π(3/2)−2s2(s/2)((1 − s)/2)[(1 − (s/2))]−1, andrelation (8.4.23.23) in [9, Vol. 3] gives immediately g(x) = 2π2I0(xπ2)K0(xπ2). Moreover,F(1) = (π2/2), F ′(1) = −(π2/2)(γ + 3 log 2). Consequently,

limN→∞

[N∑

n=1

d(n)I0

(nπ2

x

)K0

(nπ2

x

)−

∫ N

1/N

(log t + 2γ )I0

(tπ2

x

)K0

(tπ2

x

)dt

]

= x

π2

∞∑n=1

d(n)K20 (nx) − 1

4(γ − log(8x)), x > 0.

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

818 S. Yakubovich

4. An application of the Kontorovich–Lebedev transform

In this section, we will use the Kontorovich–Lebedev transformation (1.14) (see also in [15]),which involves an integration with respect to a pure imaginary index of the modified Besselfunction in order to obtain a new class of Voronoi’s summation formulas. In fact, taking Fτ (s) =2s−2((s + iτ )/2)((s − iτ )/2), τ ∈ R, we appeal to (1.23) and we get fτ (x) = Kiτ (x). Thenas usual

Gτ(s) = π

2(2π2)−s

((s/2)

(1 − s/2)

)2

(1 − s + iτ

2

)

(1 − s − iτ

2

)

and after straightforward calculation of additional terms with the use of (2.4), it becomes

x

∞∑n=1

d(n)Kiτ (nx) − π

2 cosh(πτ/2)

(2γ − log

(x

2

)+ Re

[ψ

(1 + iτ

2

)])

= 1

2πi

∫σ

ζ 2(s)Gτ (s)xs ds, x > 0, τ ∈ R. (4.1)

Hence, assuming τ = 0 and substituting Gτ(s) into the right-hand side of (4.1), we make anelementary change of the variable and we move the line of integration to the right, taking thecontour σ̂ = {s ∈ C, Res = 1}. Doing this, we encounter simple poles s = (1/2) ± (iτ/2) ofthe respective gamma-functions. Therefore, invoking with (1.32), we write (2.5) for this case asfollows:

x

∞∑n=1

d(n)Kiτ (nx) − π

2 cosh(πτ/2)

(2γ − log

(x

2

)+ Re

[ψ

(1 + iτ

2

)])

= 1

2πi

∫ 1+i∞

1−i∞πζ 2(2s)

((s)

( 12 − s)

)2

(1

2− s + iτ

2

)

(1

2− s − iτ

2

) (2π2

x

)−2s

ds

+ x

2√

π cosh(πτ/2)Re

[ζ 2(1 + iτ )

((1 + iτ )/2)

(−iτ/2)

(4π2

x

)−iτ]

=∞∑

n=1

d(n) Kτ

(n

x

)+ x

2√

π cosh(πτ/2)Re

[ζ 2(1 + iτ )

((1 + iτ )/2)

(−iτ/2)

(4π2

x

)−iτ]

,

where

Kτ (x) = 1

2πi

∫ 1+i∞

1−i∞π

((s)

((1/2) − s)

)2

(1

2− s + iτ

2

)

(1

2− s − iτ

2

) (2π2x

)−2sds

= 2∞∑

k=0

(2π2x)2k

[k!(1/2)k]2

∣∣∣∣(

1

2+ k + iτ

2

)∣∣∣∣2 [

ψ(1 + k) + ψ

(1

2+ k

)

−Re

[ψ

(1

2+ k + iτ

2

)]− log(2π2x)

]+ 1

2√

π cosh(πτ/2)

× Re

[ ((1 + iτ )/2)

(−iτ/2)

(4π2

x

)−iτ]

, x > 0, τ ∈ R\{0}. (4.2)

It can be extended continuously for all τ ∈ R via the absolute and uniform convergence by τ ∈ R

of the integral in (4.2) and its value K0(x) = 2πK0(4π2x) due to (1.25). So letting x = 2πz, we

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

Integral Transforms and Special Functions 819

derive the following extension of the Koshliakov formula (3.1)

∞∑n=1

d(n)Kiτ (2πzn) − π

4πz cosh(πτ/2)

(2γ − log(πz) + Re

[ψ

(1 + iτ

2

)])

= 1

2πz

∞∑n=1

d(n) Kτ

(n

2πz

)+ 1

2√

π cosh(πτ/2)

× Re

[ζ 2(1 + iτ )

((1 + iτ )/2)

(−iτ/2)

(4π2

x

)−iτ]

, z > 0. (4.3)

Letting τ → 0 in (4.3) and taking into account the value

limτ→0

Re

[ζ 2(1 + iτ )

((1 + iτ )/2)

(−iτ/2)

(4π2

x

)−iτ]

= −√

π

2

[γ − log

(4π

z

)],

we easily come out again with the Koshliakov formula (3.1).On the other hand, taking an arbitrary function f (τ) ∈ L2(R), we multiply by f both sides of

equality (4.1) and integrate with respect to τ . Changing the order of integration and summationvia the absolute convergence (see (1.18)), we call (1.14), where the integral exists in the Lebesguesense to obtain

x

∞∑n=1

d(n)gx(n) − π

2

∫R

f (τ)

cosh(πτ/2)

(2γ − log

(x

2

)+ Re

[ψ

(1 + iτ

2

)])dτ

= 1

2πi

∫σ

ζ 2(s)(Hf )∗(s)xs ds, (4.4)

where

(Hf )∗(s) = π

2(2π2)−s

( (s/2)

(1 − s/2)

)2 ∫R

(1 − s + iτ

2

)

(1 − s − iτ

2

)f (τ) dτ. (4.5)

Meanwhile, integrals with respect to τ in (4.4) and (4.5) can be treated employing the Parsevalequality for the Fourier transform [12]. In fact, appealing to relations (2.5.46.5) in [9, Vol. 1] and(1.104) in [14] with the duplication formula for gamma-functions, we derive

∫R

f (τ)

cosh(πτ/2)dτ =

√2

π

∫R

Ff )(y)

cosh ydy,

(Hf )∗(s) = π√2(2π2)−s2

( s

2

) (1 − (s/2))

((1 − s)/2)

∫R

(Ff )(y)

cosh1−s ydy, s ∈ σ, (4.6)

where

(Ff )(y) = 1√2π

∫R

f (τ)eiτy dτ (4.7)

is the Fourier transform with the convergence in the mean square sense. Further, by definition ofthe psi-function and making a change of the integration order, it is not difficult to calculate the

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

820 S. Yakubovich

following Fourier transform

∫R

eiτy

∣∣∣∣(

1 + iτ

2

)∣∣∣∣2

Re

[ψ

(1 + iτ

2

)]dτ = − 2π

cosh y

[γ + y + log(2 cosh y)

].

Therefore, via the Parseval equality for Fourier transforms

∫R

f (τ)

cosh(πτ/2)Re

[ψ

(1 + iτ

2

)]dτ = −

√2

π

∫R

(Ff )(y)

cosh y[γ + y + log(2 cosh y)] dy

and taking into account our calculations above∫R

f (τ)

cosh(πτ/2)

(2γ − log

(x

2

)+ Re

[ψ

(1 + iτ

2

)])dτ

=√

2

π

∫R

(Ff )(y)

cosh y

[γ − log

(x

2

)− y − log(2 cosh y)

]dy.

Hence, the left-hand side of (4.4) becomes

x

∞∑n=1

d(n)gx(n) −√

π

2

∫R

(Ff )(y)

cosh y

[γ − log

(x

2

)− y − log(2 cosh y)

]dy.

In the meantime, calculating the inverse Mellin transform of (Hf )∗(s), we use values above toget the summation formula (3.7). So calling the result as a transformation (Hf )(x), we find

(Hf )(x) = − 1

π

√2

π

∫R

(Ff )(y)

cosh y

×[e−4π2x cosh yEi(4π2x cosh y) + e4π2x cosh yEi(−4π2x cosh y)

]dy. (4.8)

Moreover, if f is absolutely continuous then similar to [15] involving the inversion theorem forthe Fourier transform [12], we prove from (4.6) that (Hf )∗(1) = 0, ((Hf )∗)′(1) = −(π/4)f (0).

Therefore, we find from (2.6) and (4.4)

1

2πi

∫σ

ζ 2(s)(Hf )∗(s)xs ds =∞∑

n=1

d(n)(Hf )(n

x

)+ πx

4f (0)

and combining with our calculations above, we derive the following Voronoi type summationformula

∞∑n=1

d(n)(xgx(n) − (Hf )

(n

x

))

= πx

4f (0) +

√π

2

∫R

(Ff )(y)

cosh y

[γ − log

(x

2

)− y − log(2 cosh y)

]dy, x > 0. (4.9)

Thus, we prove

Theorem 4 Let f be absolutely continuous on R and belong to L2(R). Then the summationformula (4.9) holds for all x > 0, where gx is the Kontorovich–Lebedev transform (1.14), Hf isa transformation (4.8) and Ff is the Fourier transform (4.7).

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013

Integral Transforms and Special Functions 821

Acknowledgements

The present investigation was supported, in part, by the ‘Centro de Matemática’ of the University of Porto.

References

[1] B.C. Berndt,Y. Lee, and J. Sohn, The formulas of Koshliakov and Guinand in Ramanujan’s lost notebook, in Surveysin Number Theory, Developments in Mathematics, Vol. 17, K. Alladi, ed., Springer, New York, 2008, pp. 21–42.

[2] A.L. Dixon and W.L. Ferrar, On the summation formulae of Voronoi and Poisson, Q. J. Math. 8 (1937), pp. 66–74.[3] A. Erdélyi,W. Magnus, F. Oberhettinger, and F.G. Tricomi, Higher Transcendental Functions,Vols. I and II, McGraw-

Hill, New York, London and Toronto, 1953.[4] N.S. Koshliakov, On Voronoi’s sum-formula, Mes. Math. 58 (1929), pp. 30–32.[5] N.N. Lebedev, Sur une formule d’inversion, C.R. Acad. Sci. URSS 52 (1946), pp. 655–658.[6] O.I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions: Theory and Algorithmic

Tables, Ellis Horwood, Chichester, 1983.[7] C. Nasim, On the summation formula of Voronoi, Trans. Amer. Math. Soc. 163 (1972), pp. 35–45.[8] F. Oberhettinger and K.L. Soni, On some relations which are equivalent to functional equations involving the

Riemann zeta-function, Math. Z. 127 (1972), pp. 17–34.[9] A.P. Prudnikov, Yu.A. Brychkov, and O.I. Marichev, Integrals and Series: Vol. 1: Elementary Functions, Gordon

and Breach, New York, 1986; Vol. 2: Special Functions, Gordon and Breach, New York, 1986; Vol. 3: More SpecialFunctions, Gordon and Breach, New York, 1989.

[10] I.N. Sneddon, The Use of Integral Transforms, McGray Hill, New York, 1972.[11] K. Soni, Some relations associated with an extension of Koshliakov’s formula, Proc. Amer. Math. Soc. 17 (1966),

pp. 543–551.[12] E.C. Titchmarsh, An Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford, 1937.[13] E.C. Titchmarsh, The Theory of The Riemann Zeta-Function, 2nd ed., The Clarendon Press, Oxford, 1986.[14] S.B. Yakubovich, Index Transforms, World Scientific Publishing Company, Singapore, New Jersey, London and

Hong Kong, 1996.[15] S.B.Yakubovich, Voronoi-Nasim Summation Formulas and Index Transforms, preprint (2010), Center of Math. Univ.

Porto, 19 pp.[16] S.B. Yakubovich and Yu.F. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions,

Mathematics and its Applications, 287, Kluwer Academic Publishers Group, Dordrecht, 1994.

Dow

nloa

ded

by [

Uni

vers

ity o

f H

aifa

Lib

rary

] at

11:

46 2

3 O

ctob

er 2

013