All YouCan Sum(part 2) · Tactic 1: Expand and simplify 10 Sigma.m is based on diﬀerence...

1

Computer Algebra and Elementary Particle Theory at the Large Scale:

10 Years of JKU-DESY Collaboration, Febrary 7, 2017

All You Can Sum (part 2)

Carsten Schneider

Research Institute for Symbolic Computation (RISC)Johannes Kepler University Linz

1 / 90

2

Evaluation of Feynman Integrals

Behavior of particles

2 / 90

2



//

∫

Φ(N, ǫ, x)dx

Feynman integrals

3 / 90

2



//

∫

Φ(N, ǫ, x)dx

Feynman integrals

DESY

��∑

f(N, ǫ, k)

complicatedmulti-sums

4 / 90

2



//

∫

Φ(N, ǫ, x)dx

Feynman integrals

DESY

��

expression inspecial functions

∑

f(N, ǫ, k)


RISC

(Sigma-package)oo

5 / 90

2



//

∫

Φ(N, ǫ, x)dx

Feynman integrals

DESY

��

LHC at CERN

expression inspecial functions

applicable

;;

∑

f(N, ǫ, k)


RISC

(Sigma-package)oo

6 / 90

7

F (ε,N) =

∫d4+εk1

(2π)4+ε

d4+εk2

(2π)4+ε

d4+εk3

(2π)4+ε

(∆.k3)N

k42((k1−k3)2−m2)(k1−k2)2((k3−p)2−m2)

||?

F−3(N)ε−3 + F−2(N)ε−2 + F−1(N)ε−1 + F0(N)ε0 + . . .

7 / 90

7

F (ε,N) =

∫d4+εk1

(2π)4+ε

d4+εk2

(2π)4+ε

d4+εk3

(2π)4+ε

(∆.k3)N

k42((k1−k3)2−m2)(k1−k2)2((k3−p)2−m2)

||

N∑

k=1

(−1)ke−3εγ2 Γ(− 1−

3ε

2

)×

×B(2 + k,

ε

2

)B(−ε+ k,−ε)B

(1−

ε

2+ k, 1 +

ε

2

)(N

k

)

where

B(a, b) =Γ(a)Γ(b)

Γ(a+ b).

8 / 90

7

F (ε,N) =

∫d4+εk1

(2π)4+ε

d4+εk2

(2π)4+ε

d4+εk3

(2π)4+ε

(∆.k3)N

k42((k1−k3)2−m2)(k1−k2)2((k3−p)2−m2)

||

N∑

k=1

(−1)ke−3εγ2 Γ(− 1−

3ε

2

)×

×B(2 + k,

ε

2

)B(−ε+ k,−ε)B

(1−

ε

2+ k, 1 +

ε

2

)(N

k

)

︸︷︷︸

= f−3(N, k)ε−3 + f−2(N, k)ε−2 + f−1(N, k)ε−1 + . . .

G.E. Andrews, R. Askey, R. Roy: Special Functions

9 / 90

7

F (ε,N) =

∫d4+εk1

(2π)4+ε

d4+εk2

(2π)4+ε

d4+εk3

(2π)4+ε

(∆.k3)N

k42((k1−k3)2−m2)(k1−k2)2((k3−p)2−m2)

||

N∑

k=1

(−1)ke−3εγ2 Γ(− 1−

3ε

2

)×

×B(2 + k,

ε

2

)B(−ε+ k,−ε)B

(1−

ε

2+ k, 1 +

ε

2

)(N

k

)

︸︷︷︸

= f−3(N, k)ε−3 + f−2(N, k)ε−2 + f−1(N, k)ε−1 + . . .

||

( N∑

k=1

f−3(N, k)

)

ε−3 +( N∑

k=1

f−2(N, k)

)

ε−2 +( N∑

k=1

f−1(N, k)

)

ε−1 + . . .

10 / 90

7

F (ε,N) =

∫d4+εk1

(2π)4+ε

d4+εk2

(2π)4+ε

d4+εk3

(2π)4+ε

(∆.k3)N

k42((k1−k3)2−m2)(k1−k2)2((k3−p)2−m2)

||

N∑

k=1

(−1)ke−3εγ2 Γ(− 1−

3ε

2

)×

×B(2 + k,

ε

2

)B(−ε+ k,−ε)B

(1−

ε

2+ k, 1 +

ε

2

)(N

k

)

︸︷︷︸

= f−3(N, k)ε−3 + f−2(N, k)ε−2 + f−1(N, k)ε−1 + . . .

||

( N∑

k=1

f−3(N, k)

)

ε−3 +( N∑

k=1

f−2(N, k)

)

ε−2 +( N∑

k=1

f−1(N, k)

)

ε−1 + . . .

11 / 90

Tactic 1: Expand and simplify 8

Simplify

F−1(N) =N∑

k=1

(−1)k+1

(

N

k

)

( (2 + 3k)(

− 2 + 3k + 7k2 + 3k3)

3k2(1 + k)3+

2S2(k)

1 + k+

ζ2

2(1 + k)

)

where

Sa(N) =

N∑

i=1

sign(a)i

iaand ζa =

∞∑

i=1

1

ia

12 / 90


Simplify

F−1(N) =N∑

k=1

(−1)k+1

(

N

k

)

( (2 + 3k)(

− 2 + 3k + 7k2 + 3k3)

3k2(1 + k)3+

2S2(k)

1 + k+

ζ2

2(1 + k)

)

↓ (summation package Sigma.m)

(

16N3 + 144N2 + 413N + 384)

(N + 1)2F−1(N)

− (N + 2)(2N + 5)(

16N3 + 112N2 + 221N + 113)

F−1(N + 1)

+ (N + 3)2(

16N3 + 96N2 + 173N + 99)

F−1(N + 2)

=1

2

(

4N2 + 21N + 29)

ζ2 +−64N5−500N4−1133N3+203N2+3516N+3090

3(N+2)(N+3)

13 / 90


Simplify

F−1(N) =N∑

k=1

(−1)k+1

(

N

k

)

( (2 + 3k)(

− 2 + 3k + 7k2 + 3k3)

3k2(1 + k)3+

2S2(k)

1 + k+

ζ2

2(1 + k)

)


(

16N3 + 144N2 + 413N + 384)

(N + 1)2F−1(N)

− (N + 2)(2N + 5)(

16N3 + 112N2 + 221N + 113)

F−1(N + 1)

+ (N + 3)2(

16N3 + 96N2 + 173N + 99)

F−1(N + 2)

=1

2

(

4N2 + 21N + 29)

ζ2 +−64N5−500N4−1133N3+203N2+3516N+3090

3(N+2)(N+3)


{

c11− 4N

N + 1+ c2

−14N − 13

(N + 1)2

+(4N − 1)S1(N)

N + 1+

(1− 4N)S1(N)2

6(N + 1)+

(14N + 13)S1(N)

3(N + 1)2

+175N2 + 334N + 155

12(N + 1)3+

(1− 4N)S2(N)

6(N + 1)+

ζ2

8(N + 1)|c1, c2 ∈ Q}

14 / 90


Simplify

F−1(N) =N∑

k=1

(−1)k+1

(

N

k

)

( (2 + 3k)(

− 2 + 3k + 7k2 + 3k3)

3k2(1 + k)3+

2S2(k)

1 + k+

ζ2

2(1 + k)

)

∈{

c11− 4N

N + 1+ c2

−14N − 13

(N + 1)2

+(4N − 1)S1(N)

N + 1+

(1− 4N)S1(N)2

6(N + 1)+

(14N + 13)S1(N)

3(N + 1)2

+175N2 + 334N + 155

12(N + 1)3+

(1− 4N)S2(N)

6(N + 1)+

ζ2

8(N + 1)|c1, c2 ∈ Q}

15 / 90


Simplify

F−1(N) =N∑

k=1

(−1)k+1

(

N

k

)

( (2 + 3k)(

− 2 + 3k + 7k2 + 3k3)

3k2(1 + k)3+

2S2(k)

1 + k+

ζ2

2(1 + k)

)

=

(recurrence finding and solving)

( 1

12−

1

8ζ2) 1− 4N

N + 1+ 1

−14N − 13

(N + 1)2

+(4N − 1)S1(N)

N + 1+

(1− 4N)S1(N)2

6(N + 1)+

(14N + 13)S1(N)

3(N + 1)2

+175N2 + 334N + 155

12(N + 1)3+

(1− 4N)S2(N)

6(N + 1)+

ζ2

8(N + 1)

16 / 90


1. Creative telescoping (for the special case of hypergeometric terms see Zeilberger’s algorithm (1991))

GIVEN a definite sum

F (N) =

N∑

k=0

f(N, k); f(N, k): indefinite nested product-sum in k;N : extra parameter

FIND a recurrence for F (N)

17 / 90




F (N) =

N∑

k=0



2. Recurrence solving

GIVEN a recurrence a0(N), . . . , ad(N), h(N):indefinite nested product-sum expressions.

a0(N)F (N) + · · ·+ ad(N)F (N + d) = h(N);

FIND all solutions expressible by indefinite nested products/sums(Abramov/Bronstein/Petkovsek/CS, in preparation)

18 / 90




F (N) =

N∑

k=0



2. Recurrence solving

GIVEN a recurrence a0(N), . . . , ad(N), h(N):indefinite nested product-sum expressions.

a0(N)F (N) + · · ·+ ad(N)F (N + d) = h(N);

FIND all solutions expressible by indefinite nested products/sums(Abramov/Bronstein/Petkovsek/CS, in preparation)

3. Find a “closed form”

F(N)=combined solutions in terms of indefinite nested sums.

19 / 90


Sigma.m is based on difference ring/field theory1. M. Karr. Summation in finite terms. J. ACM, 28:305–350, 1981.

2. P. Paule. Greatest factorial factorization and symbolic summation. J. Symbolic Comput. 20(3), 235–268 (1995)

3. M. Petkovsek, H. S. Wilf, and D. Zeilberger. A = B. A. K. Peters, Wellesley, MA, 1996.

4. P. A. Hendriks and M. F. Singer. Solving difference equations in finite terms. J. Symbolic Comput., 27(3):239–259, 1999.

5. M. Bronstein. On solutions of linear ordinary difference equations in their coefficient field. J. Symbolic Comput., 29(6):841–877, 2000.

6. CS. Symbolic summation in difference fields. J. Kepler University, May 2001. PhD Thesis.

7. CS. A collection of denominator bounds to solve parameterized linear difference equations in ΠΣ-extensions. An. Univ. Timisoara Ser.Mat.-Inform., 42(2):163–179, 2004.

8. CS. Symbolic summation with single-nested sum extensions. In J. Gutierrez, editor, Proc. ISSAC’04, pages 282–289. ACM Press, 2004.

9. CS. Degree bounds to find polynomial solutions of parameterized linear difference equations in ΠΣ-fields. Appl. Algebra Engrg. Comm. Comput.,16(1):1–32, 2005.

10. CS. Product representations in ΠΣ-fields. Ann. Comb., 9(1):75–99, 2005.

11. CS. Solving parameterized linear difference equations in terms of indefinite nested sums and products. J. Differ. Equations Appl., 11(9):799–821,2005.

12. CS. Finding telescopers with minimal depth for indefinite nested sums and product expressions. In Proc. ISSAC’05, pages 285–292. ACM Press,2005.

13. CS. Simplifying Sums in ΠΣ-Extensions. J. Algebra Appl., 6(3):415–441, 2007.

14. CS. A refined difference field theory for symbolic summation. J. Symbolic Comput., 43(9):611–644, 2008. [arXiv:0808.2543v1].

15. CS. A Symbolic Summation Approach to Find Optimal Nested Sum Representations. In A. Carey, D. Ellwood, S. Paycha, and S. Rosenberg,editors, Motives, Quantum Field Theory, and Pseudodifferential Operators, pages 285–308. 2010.

16. CS. Parameterized Telescoping Proves Algebraic Independence of Sums. Ann. Comb., 14(4):533–552, 2010. [arXiv:0808.2596].

17. CS. Structural Theorems for Symbolic Summation. Appl. Algebra Engrg. Comm. Comput., 21(1):1–32, 2010.

18. C. Schneider. Simplifying Multiple Sums in Difference Fields. In: Computer Algebra in Quantum Field Theory: Integration, Summation andSpecial Functions, J. Blumlein, C. Schneider (ed.), Texts and Monographs in Symbolic Computation, pp. 325-360. Springer, 2013.

19. CS. Fast Algorithms for Refined Parameterized Telescoping in Difference Fields. To appear in Computer Algebra and Polynomials, Lecture Notesin Computer Science (LNCS), Springer, 2014. arXiv:1307.7887 [cs.SC].

20. CS. A Difference Ring Theory for Symbolic Summation. J. Symb. Comput. 72, pp. 82-127. 2016.

21. CS. Summation Theory II: Characterizations of RΠΣ-extensions and algorithmic aspects. J. Symb. Comput. 80(3), pp. 616-664. 2017.

22. S.A. Abramov, M. Bronstein, M. Petkovsek, CS, in preparation

20 / 90


In[1]:= << Sigma.m

Sigma - A summation package by Carsten Schneider c© RISC-Linz

In[2]:= << HarmonicSums.m

HarmonicSums by Jakob Ablinger c© RISC-Linz

In[3]:= << EvaluateMultiSums.m

EvaluateMultiSums by Carsten Schneider c© RISC-Linz

21 / 90


In[1]:= << Sigma.m






In[4]:= mySum =N∑

k=1

(−1)ke−3εγ

2

(

− 2−3ε

2

)

!B(

2+k,ε

2

)

B(−ε+k,−ε)B(

1−ε

2+k, 1+

ε

2

)

(N

k

)

;

In[5]:= EvaluateMultiSum[mySum, {}, {N}, {1},ExpandIn → {ε,−3, − 3}]

22 / 90


In[1]:= << Sigma.m






In[4]:= mySum =N∑

k=1

(−1)ke−3εγ

2

(

− 2−3ε

2

)

!B(

2+k,ε

2

)

B(−ε+k,−ε)B(

1−ε

2+k, 1+

ε

2

)

(N

k

)

;


Out[5]= {59N

2 + 120N + 49

9(N + 1)2−

2(N+ 3)S1[N]

3(N+ 1)}

23 / 90


In[1]:= << Sigma.m






In[4]:= mySum =N∑

k=1

(−1)ke−3εγ

2

(

− 2−3ε

2

)

!B(

2+k,ε

2

)

B(−ε+k,−ε)B(

1−ε

2+k, 1+

ε

2

)

(N

k

)

;


Out[5]= {59N

2 + 120N + 49

9(N + 1)2−

2(N+ 3)S1[N]

3(N+ 1),

−2(20N3 + 58N

2 + 57N+ 22)

3(N + 1)3+

2(N + 2)(2N − 1)S1[N]

3(N+ 1)2−

S1[N]2

N+ 1−

S2[N]

N+ 1}

24 / 90


In[1]:= << Sigma.m






In[4]:= mySum =N∑

k=1

(−1)ke−3εγ

2

(

− 2−3ε

2

)

!B(

2+k,ε

2

)

B(−ε+k,−ε)B(

1−ε

2+k, 1+

ε

2

)

(N

k

)

;


Out[5]= {59N

2 + 120N + 49

9(N + 1)2−

2(N+ 3)S1[N]

3(N+ 1),

−2(20N3 + 58N

2 + 57N+ 22)

3(N + 1)3+

2(N + 2)(2N − 1)S1[N]

3(N+ 1)2−

S1[N]2

N+ 1−

S2[N]

N+ 1,

( 1

12−

1

8ζ(2)

) 1− 4N

N+ 1+

−14N− 13

(N + 1)2+

(4N− 1)S1(N)

N+ 1+

(1− 4N)S1(N)2

6(N + 1)+

(14N + 13)S1(N)

3(N + 1)2+

175N2 + 334N + 155

12(N + 1)3+

(1 − 4N)S2(N)

6(N + 1)+

ζ(2)

8(N+ 1)}

25 / 90


n−2∑

j=0

j+1∑

r=0

n−j+r−2∑

s=0

(−1)r+s(j+1r

)(−j+n+r−2s

)(−j + n− 2)!r!

(n− s)(s+ 1)(−j + n+ r)!

Simple sum

26 / 90


n−2∑

j=0

j+1∑

r=0

n−j+r−2∑

s=0

(−1)r+s(j+1r

)(−j+n+r−2s

)(−j + n− 2)!r!

(n− s)(s+ 1)(−j + n+ r)!

||

n−2∑

j=0

j+1∑

r=0

n−j+r−2∑

s=0

(−1)r+s(j+1r

)(−j+n+r−2s

)(−j + n− 2)!r!

(n− s)(s+ 1)(−j + n+ r)!

27 / 90


n−2∑

j=0

j+1∑

r=0

n−j+r−2∑

s=0

(−1)r+s(j+1r

)(−j+n+r−2s

)(−j + n− 2)!r!

(n− s)(s+ 1)(−j + n+ r)!

||

n−2∑

j=0

j+1∑

r=0

n−j+r−2∑

s=0

(−1)r+s(j+1r

)(−j+n+r−2s

)(−j + n− 2)!r!

(n− s)(s+ 1)(−j + n+ r)!

||(j + 1

r

)( (−1)r(−j + n− 2)!r!

(n + 1)(−j + n+ r − 1)(−j + n+ r)!+

(−1)n+r(j + 1)!(−j + n− 2)!(−j + n− 1)rr!

(n− 1)n(n+ 1)(−j + n+ r)!(−j − 1)r(2− n)j

)

28 / 90


n−2∑

j=0

j+1∑

r=0

n−j+r−2∑

s=0

(−1)r+s(j+1r

)(−j+n+r−2s

)(−j + n− 2)!r!

(n− s)(s+ 1)(−j + n+ r)!

||

n−2∑

j=0

j+1∑

r=0

(j + 1

r

)( (−1)r(−j + n− 2)!r!

(n+ 1)(−j + n+ r − 1)(−j + n+ r)!+

(−1)n+r(j + 1)!(−j + n− 2)!(−j + n− 1)rr!

(n− 1)n(n + 1)(−j + n+ r)!(−j − 1)r(2− n)j

)

29 / 90


n−2∑

j=0

j+1∑

r=0

n−j+r−2∑

s=0

(−1)r+s(j+1r

)(−j+n+r−2s

)(−j + n− 2)!r!

(n− s)(s+ 1)(−j + n+ r)!

||

n−2∑

j=0

j+1∑

r=0

(j + 1

r

)( (−1)r(−j + n− 2)!r!

(n + 1)(−j + n+ r − 1)(−j + n+ r)!+

(−1)n+r(j + 1)!(−j + n− 2)!(−j + n− 1)rr!

(n− 1)n(n+ 1)(−j + n+ r)!(−j − 1)r(2− n)j

)

||

( n2 − n+ 1

(n− 1)2n2(n + 1)(2 − n)j+

j∑

i=1

(2− n)i(−i+ n− 1)2(i+ 1)!

(n+ 1)(2 − n)j+

(−1)j+n(−j − 2)(−j + n− 2)!

(j − n+ 1)(n + 1)2n!

)

(j + 1)! −1

(n + 1)2(−j + n− 1)

30 / 90


n−2∑

j=0

j+1∑

r=0

n−j+r−2∑

s=0

(−1)r+s(j+1r

)(−j+n+r−2s

)(−j + n− 2)!r!

(n− s)(s+ 1)(−j + n+ r)!

||

n−2∑

j=0

(( n2 − n+ 1

(n− 1)2n2(n+ 1)(2 − n)j+

j∑

i=1

(2− n)i(−i+ n− 1)2(i+ 1)!

(n + 1)(2 − n)j+

(−1)j+n(−j − 2)(−j + n− 2)!

(j − n+ 1)(n + 1)2n!

)

(j + 1)!−1

(n+ 1)2(−j + n− 1)

)

31 / 90


n−2∑

j=0

j+1∑

r=0

n−j+r−2∑

s=0

(−1)r+s(j+1r

)(−j+n+r−2s

)(−j + n− 2)!r!

(n− s)(s+ 1)(−j + n+ r)!

||

n−2∑

j=0

(( n2 − n+ 1

(n− 1)2n2(n+ 1)(2 − n)j+

j∑

i=1

(2− n)i(−i+ n− 1)2(i+ 1)!

(n + 1)(2 − n)j+

(−1)j+n(−j − 2)(−j + n− 2)!

(j − n+ 1)(n + 1)2n!

)

(j + 1)!−1

(n+ 1)2(−j + n− 1)

)

||

−n2 − n− 1

n2(n+ 1)3+

(−1)n(n2 + n+ 1

)

n2(n+ 1)3−

2S−2(n)

n+ 1+

S1(n)

(n + 1)2+

S2(n)

−n− 1

Note: Sa(n) =∑N

i=1sign(a)i

i|a|, a ∈ Z \ {0}.

32 / 90


= F−3(N)ε−3 + F−2(N)ε−2 + F−1(N)ε−1 + F0(N)

33 / 90


= F−3(N)ε−3 + F−2(N)ε−2 + F−1(N)ε−1 + F0(N)

Simplify ||

N−3∑

j=0

j∑

k=0

k∑

l=0

−j+N−3∑

q=0

−l+N−q−3∑

s=1

−l+N−q−s−3∑

r=0

(−1)−j+k−l+N−q−3×

×(j+1k+1)(

kl)(

N−1j+2 )(

−j+N−3q )(−l+N−q−3

s ) (−l+N−q−s−3r )r!(−l+N−q−r−s−3)!(s−1)!

(−l+N−q−2)!(−j+N−1)(N−q−r−s−2)(q+s+1)[

4S1(−j +N − 1)− 4S1(−j +N − 2)− 2S1(k)

− (S1(−l +N − q − 2) + S1(−l +N − q − r − s− 3)− 2S1(r + s))

+ 2S1(s − 1)− 2S1(r + s)

]

+ 3 further 6–fold sums

34 / 90


F0(N) =

7

12S1(N)4 +

(17N + 5)S1(N)3

3N(N + 1)+(35N2 − 2N − 5

2N2(N + 1)2+

13S2(N)

2+

5(−1)N

2N2

)

S1(N)2

+(

−4(13N + 5)

N2(N + 1)2+(4(−1)N (2N + 1)

N(N + 1)−

13

N

)

S2(N) +(29

3− (−1)N

)

S3(N)

+(

2 + 2(−1)N)

S2,1(N) − 28S−2,1(N) +20(−1)N

N2(N + 1)

)

S1(N) +( 3

4+ (−1)N

)

S2(N)2

− 2(−1)NS−2(N)2 + S−3(N)( 2(3N − 5)

N(N + 1)+(

26 + 4(−1)N)

S1(N) +4(−1)N

N + 1

)

+( (−1)N (5 − 3N)

2N2(N + 1)−

5

2N2

)

S2(N) + S−2(N)(

10S1(N)2 +(8(−1)N (2N + 1)

N(N + 1)

+4(3N − 1)

N(N + 1)

)

S1(N) +8(−1)N (3N + 1)

N(N + 1)2+(

− 22 + 6(−1)N)

S2(N)−16

N(N + 1)

)

+( (−1)N (9N + 5)

N(N + 1)−

29

3N

)

S3(N) +(19

2− 2(−1)N

)

S4(N) +(

− 6 + 5(−1)N)

S−4(N)

+(

−2(−1)N (9N + 5)

N(N + 1)−

2

N

)

S2,1(N) +(

20 + 2(−1)N)

S2,−2(N) +(

− 17 + 13(−1)N)

S3,1(N)

−8(−1)N (2N + 1) + 4(9N + 1)

N(N + 1)S−2,1(N) −

(

24 + 4(−1)N)

S−3,1(N) +(

3− 5(−1)N)

S2,1,1(N)

+ 32S−2,1,1(N) +

(

3

2S1(N)2 −

3S1(N)

N+

3

2(−1)NS−2(N)

)

ζ(2)

35 / 90


F0(N) =

7

12S1(N)4 +

(17N + 5)S1(N)3

3N(N + 1)+(35N2 − 2N − 5

2N2(N + 1)2+

13S2(N)

2+

5(−1)N

2N2

)

S1(N)2

+(

−4(13N + 5)

N2(N + 1)2+(4(−1)N (2N + 1)

N(N + 1)−

13

N

)

S2(N) +(29

3− (−1)N

)

S3(N)

+(

2 + 2(−1)N)

S2,1(N) − 28S−2,1(N) +20(−1)N

N2(N + 1)

)

S1(N) +( 3

4+ (−1)N

)

S2(N)2

− 2(−1)NS−2(N)2 + S−3(N)( 2(3N − 5)

N(N + 1)+(

26 + 4(−1)N)

S1(N) +4(−1)N

N + 1

)

+( (−1)N (5 − 3N)

2N2(N + 1)−

5

2N2

)

S2(N) + S−2(N)(

10S1(N)2 +(8(−1)N (2N + 1)

N(N + 1)

+4(3N − 1)

N(N + 1)

)

S1(N) +8(−1)N (3N + 1)

N(N + 1)2+(

− 22 + 6(−1)N)

S2(N)−16

N(N + 1)

)

+( (−1)N (9N + 5)

N(N + 1)−

29

3N

)

S3(N) +(19

2− 2(−1)N

)

S4(N) +(

− 6 + 5(−1)N)

S−4(N)

+(

−2(−1)N (9N + 5)

N(N + 1)−

2

N

)

S2,1(N) +(

20 + 2(−1)N)

S2,−2(N) +(

− 17 + 13(−1)N)

S3,1(N)

−8(−1)N (2N + 1) + 4(9N + 1)

N(N + 1)S−2,1(N) −

(

24 + 4(−1)N)

S−3,1(N) +(

3− 5(−1)N)

S2,1,1(N)

+ 32S−2,1,1(N)+

(

3

2S1(N)2 −

3S1(N)

N+

3

2(−1)NS−2(N)

)

ζ(2)

S1(N) =N∑

i=1

1

i

36 / 90


F0(N) =

7

12S1(N)4 +

(17N + 5)S1(N)3

3N(N + 1)+(35N2 − 2N − 5

2N2(N + 1)2+

13S2(N)

2+

5(−1)N

2N2

)

S1(N)2

+(

−4(13N + 5)

N2(N + 1)2+(4(−1)N (2N + 1)

N(N + 1)−

13

N

)

S2(N) +(29

3− (−1)N

)

S3(N)

+(

2 + 2(−1)N)

S2,1(N) − 28S−2,1(N) +20(−1)N

N2(N + 1)

)

S1(N) +( 3

4+ (−1)N

)

S2(N)2

− 2(−1)NS−2(N)2 + S−3(N)( 2(3N − 5)

N(N + 1)+(

26 + 4(−1)N)

S1(N) +4(−1)N

N + 1

)

+( (−1)N (5 − 3N)

2N2(N + 1)−

5

2N2

)

S2(N) + S−2(N)(

10S1(N)2 +(8(−1)N (2N + 1)

N(N + 1)

+4(3N − 1)

N(N + 1)

)

S1(N) +8(−1)N (3N + 1)

N(N + 1)2+(

− 22 + 6(−1)N)

S2(N)−16

N(N + 1)

)

+( (−1)N (9N + 5)

N(N + 1)−

29

3N

)

S3(N) +(19

2− 2(−1)N

)

S4(N) +(

− 6 + 5(−1)N)

S−4(N)

+(

−2(−1)N (9N + 5)

N(N + 1)−

2

N

)

S2,1(N) +(

20 + 2(−1)N)

S2,−2(N) +(

− 17 + 13(−1)N)

S3,1(N)

−8(−1)N (2N + 1) + 4(9N + 1)

N(N + 1)S−2,1(N) −

(

24 + 4(−1)N)

S−3,1(N) +(

3− 5(−1)N)

S2,1,1(N)

+ 32S−2,1,1(N) +

(

3

2S1(N)2 −

3S1(N)

N+

3

2(−1)NS−2(N)

)

ζ(2)

S1(N) =N∑

i=1

1

i

S2(N) =N∑

i=1

1

i2

37 / 90


F0(N) =

7

12S1(N)4 +

(17N + 5)S1(N)3

3N(N + 1)+(35N2 − 2N − 5

2N2(N + 1)2+

13S2(N)

2+

5(−1)N

2N2

)

S1(N)2

+(

−4(13N + 5)

N2(N + 1)2+(4(−1)N (2N + 1)

N(N + 1)−

13

N

)

S2(N) +(29

3− (−1)N

)

S3(N)

+(

2 + 2(−1)N)

S2,1(N) − 28S−2,1(N) +20(−1)N

N2(N + 1)

)

S1(N) +( 3

4+ (−1)N

)

S2(N)2

− 2(−1)NS−2(N)2 + S−3(N)( 2(3N − 5)

N(N + 1)+(

26 + 4(−1)N)

S1(N) +4(−1)N

N + 1

)

+( (−1)N (5 − 3N)

2N2(N + 1)−

5

2N2

)

S2(N) + S−2(N)(

10S1(N)2 +(8(−1)N (2N + 1)

N(N + 1)

+4(3N − 1)

N(N + 1)

)

S1(N) +8(−1)N (3N + 1)

N(N + 1)2+(

− 22 + 6(−1)N)

S2(N)−16

N(N + 1)

)

+( (−1)N (9N + 5)

N(N + 1)−

29

3N

)

S3(N) +(19

2− 2(−1)N

)

S4(N) +(

− 6 + 5(−1)N)

S−4(N)

+(

−2(−1)N (9N + 5)

N(N + 1)−

2

N

)

S2,1(N) +(

20 + 2(−1)N)

S2,−2(N) +(

− 17 + 13(−1)N)

S3,1(N)

−8(−1)N (2N + 1) + 4(9N + 1)

N(N + 1)S−2,1(N) −

(

24 + 4(−1)N)

S−3,1(N) +(

3− 5(−1)N)

S2,1,1(N)

+ 32S−2,1,1(N) +

(

3

2S1(N)2 −

3S1(N)

N+

3

2(−1)NS−2(N)

)

ζ(2)

S1(N) =N∑

i=1

1

i

S2(N) =N∑

i=1

1

i2

S−2,1,1(N) =N∑

i=1

(−1)ii∑

j=1

j∑

k=1

1

k

j

i2

38 / 90


The general tactic

Feynman integrals

39 / 90


The general tactic

Feynman integrals

↓ non-trivial transformations (DESY)

multiple sums

40 / 90


The general tactic

Feynman integrals


multiple sums

↓ symbolic summation

compact expression in terms

of special functions

41 / 90


The general tactic

Feynman integrals


multiple sums




Tactic 1: Expand the summand and simplifyAblinger, Blumlein, Klein, CS, LL2010, arXiv:1006.4797 [math-ph]Blumlein, Hasselhuhn, CS, RADCOR’10, arXiv:1202.4303 [math-ph]CS, ACAT 2013, arXiv:1310.0160 [cs.SC]

42 / 90

Tactic 2: Expand the recurrence 16

The general tactic

Feynman integrals


multiple sums




Tactic 2: Expand a recurrence in εBlumlein, Klein, CS, Stan, J. Symbol. Comput. 2012; arXiv:1011.2656 [cs.SC]Ablinger, Blumlein, Round, CS, LL2012, arXiv:1210.1685 [cs.SC] 43 / 90


F (N) =N∑

k=1

(−1)ke−3εγ2 Γ(

−1−3ε

2

)

B(

2+k,ε

2

)

B(−ε+k,−ε)B(

1−ε

2+k, 1+

ε

2

)

(

N

k

)


2(N + 1)2F (N) +(

3ε2 + 3εN + 9ε− 4N2 − 12N − 8)

F (N + 1)

− (2ε−N−1)(ε+2N+6)F (N+2) = 0ε−3−16

3ε−2+

40

3ε−1−

(

2ζ2−68

3)ε0+ . . .

44 / 90


F (N) =N∑

k=1

(−1)ke−3εγ2 Γ(

−1−3ε

2

)

B(

2+k,ε

2

)

B(−ε+k,−ε)B(

1−ε

2+k, 1+

ε

2

)

(

N

k

)


2(N + 1)2F (N) +(

3ε2 + 3εN + 9ε− 4N2 − 12N − 8)

F (N + 1)

− (2ε−N−1)(ε+2N+6)F (N+2) = 0ε−3−16

3ε−2+

40

3ε−1−

(

2ζ2−68

3)ε0+ . . .

F (1) =2

3ε−3 −

11

6ε−2 +

(ζ2

4+

79

24

)

ε−1 + . . .

F (2) =8

9ε−3 −

73

27ε−2 +

(ζ2

3+

1415

324

)

ε−1 + . . .

↓

F (N) = F−3(N) ε−3 + F−2(N) ε−2 + F−1(N) ε−1 + . . .

45 / 90


Ansatz (for power series)

a0(ε,N)[

F (N)]

+a1(ε,N)[

F (N + 1)]

+

...

+ad(ε,N)[

F (N + d)]

= h0(N) + h1(N)ε+ h1(N)ε2 + . . .

given (in terms of indefinite nested sums and products)

46 / 90



a0(ε,N)[

F0(N) + F1(N)ε + F2(N)ε2 + . . .]

+a1(ε,N)[

F (N + 1)]

+

...

+ad(ε,N)[

F (N + d)]

= h0(N) + h1(N)ε+ h1(N)ε2 + . . .


47 / 90



a0(ε,N)[

F0(N) + F1(N)ε + F2(N)ε2 + . . .]

+a1(ε,N)[

F0(N + 1) + F1(N + 1)ε + F2(N + 1)ε2 + . . .]

+

...

+ad(ε,N)[

F (N + d)]

= h0(N) + h1(N)ε+ h1(N)ε2 + . . .


48 / 90



a0(ε,N)[

F0(N) + F1(N)ε + F2(N)ε2 + . . .]

+a1(ε,N)[

F0(N + 1) + F1(N + 1)ε + F2(N + 1)ε2 + . . .]

+

...

+ad(ε,N)[

F0(N + d) + F1(N + d)ε + F2(N + d)ε2 + . . .]

= h0(N) + h1(N)ε+ h1(N)ε2 + . . .


49 / 90



a0(ε,N)[

F0(N) + F1(N)ε + F2(N)ε2 + . . .]

+a1(ε,N)[

F0(N + 1) + F1(N + 1)ε + F2(N + 1)ε2 + . . .]

+

...

+ad(ε,N)[

F0(N + d) + F1(N + d)ε + F2(N + d)ε2 + . . .]

= h0(N) + h1(N)ε+ h1(N)ε2 + . . .

⇓ constant terms must agree

a0(0,N)F0(N) + a1(0, N)F0(N+1) + · · · + ad(0, N)F0(N+d) = h0(N)

50 / 90



a0(ε,N)[

F0(N) + F1(N)ε + F2(N)ε2 + . . .]

+a1(ε,N)[

F0(N + 1) + F1(N + 1)ε + F2(N + 1)ε2 + . . .]

+

...

+ad(ε,N)[

F0(N + d) + F1(N + d)ε + F2(N + d)ε2 + . . .]

= h0(N) + h1(N)ε+ h1(N)ε2 + . . .



REC solver: Given the initial values F0(1), F0(2), . . . , F0(d− 1),decide if F0(N) can be written in terms of indefinitenested sums and products. 51 / 90



a0(ε,N)[

F0(N) + F1(N)ε + F2(N)ε2 + . . .]

+a1(ε,N)[

F0(N + 1) + F1(N + 1)ε + F2(N + 1)ε2 + . . .]

+

...

+ad(ε,N)[

F0(N + d) + F1(N + d)ε + F2(N + d)ε2 + . . .]

= h0(N) + h1(N)ε+ h1(N)ε2 + . . .



52 / 90



a0(ε,N)[

F0(N) + F1(N)ε + F2(N)ε2 + . . .]

+a1(ε,N)[

F0(N + 1) + F1(N + 1)ε + F2(N + 1)ε2 + . . .]

+

...

+ad(ε,N)[

F0(N + d) + F1(N + d)ε + F2(N + d)ε2 + . . .]

= h0(N) + h1(N)ε+ h1(N)ε2 + . . .



53 / 90


a0(ε,N)[

F1(N)ε+ F2(N)ε2 + . . .]

+a1(ε,N)[

F1(N+1)ε+ F2(N+1)ε2 + . . .]

+

...

+ad(ε,N)[

F1(N+d)ε+ F2(N + d)ε2 + . . .]

= h′0(N) + h′1(N)ε+ h′2(N)ε2 + . . .

54 / 90


a0(ε,N)[

F1(N)ε+ F2(N)ε2 + . . .]

+a1(ε,N)[

F1(N+1)ε+ F2(N+1)ε2 + . . .]

+

...

+ad(ε,N)[

F1(N+d)ε+ F2(N + d)ε2 + . . .]

= h′0(N)︸︷︷︸

=0

+h′1(N)ε+ h′2(N)ε2 + . . .

Devide by ε

55 / 90


a0(ε,N)[

F1(N) + F2(N)ε+ . . .]

+a1(ε,N)[

F1(N + 1) + F2(N + 1)ε+ . . .]

+

...

+ad(ε,N)[

F1(N+d) + F2(N+d)ε+ . . .]

= h′1(N) + h′2(N)ε+ . . .

Now repeat for F1(N), F2(N), . . .

Remark: Works the same for Laurent series.

Blumlein, Klein, CS, Stan, J. Symbol. Comput. 2012; arXiv:1011.2656[cs.SC]

Ablinger, Blumlein, Round, CS, LL2012, arXiv:1210.1685 [cs.SC]

56 / 90


F (N) =N∑

k=1

(−1)ke−3εγ2 Γ(

−1−3ε

2

)

B(

2+k,ε

2

)

B(−ε+k,−ε)B(

1−ε

2+k, 1+

ε

2

)

(

N

k

)


2(N + 1)2F (N) +(

3ε2 + 3εN + 9ε− 4N2 − 12N − 8)

F (N + 1)

− (2ε−N−1)(ε+2N+6)F (N+2) = 0ε−3−16

3ε−2+

40

3ε−1−

(

2ζ2−68

3)ε0+ . . .

F (1) =2

3ε−3 −

11

6ε−2 +

(ζ2

4+

79

24

)

ε−1 + . . .

F (2) =8

9ε−3 −

73

27ε−2 +

(ζ2

3+

1415

324

)

ε−1 + . . .

↓

F (N) = F−3(N) ε−3 + F−2(N) ε−2 + F−1(N) ε−1 + . . .

57 / 90


F (N) =N∑

k=1

(−1)ke−3εγ2 Γ(

−1−3ε

2

)

B(

2+k,ε

2

)

B(−ε+k,−ε)B(

1−ε

2+k, 1+

ε

2

)

(

N

k

)


2(N + 1)2F (N) +(

3ε2 + 3εN + 9ε− 4N2 − 12N − 8)

F (N + 1)

− (2ε−N−1)(ε+2N+6)F (N+2) = 0ε−3−16

3ε−2+

40

3ε−1−

(

2ζ2−68

3)ε0+ . . .

F (1) =2

3ε−3 −

11

6ε−2 +

(ζ2

4+

79

24

)

ε−1 + . . .

F (2) =8

9ε−3 −

73

27ε−2 +

(ζ2

3+

1415

324

)

ε−1 + . . .


F (N) = 4N3(N+1)

ε−3 −

(

2(2N+1)3(N+1)

S1(N) + 2N(2N+3)

3(N+1)2

)

ε−2

(

(1−4N)6(N+1)

S1(N)2 −N

(

N2−2)

3(N+1)3+ (3N+2)(4N+5)

3(N+1)2S1(N) + (1−4N)

6(N+1)S2(N) + Nζ2

2(N+1)

)

ε−1 + . .

58 / 90


Find a recurrence for the integral/sum

F (N) =

∫ 1

0· · ·

∫ 1

0Φ(ε,N, x1, x2, . . . , x7)dx1dx2 . . . dx7

?= F−3(N)ε−3 + F−2(N)ε−2 + F−1(N)ε−1 + . . .

multivariateAlmquist/Zeilberger(J. Ablinger)

a0(ε,N)F (N) + . . .+ ad(ε,N)F (N + d) = h(ε,N)

59 / 90



F (N) =

∫ 1

0· · ·

∫ 1

0Φ(ε,N, x1, x2, . . . , x7)dx1dx2 . . . dx7

?= F−3(N)ε−3 + F−2(N)ε−2 + F−1(N)ε−1 + . . .

ε-recurrence solver



60 / 90



F (N) =

∫ 1

0· · ·

∫ 1

0Φ(ε,N, x1, x2, . . . , x7)dx1dx2 . . . dx7

?= F−3(N)ε−3 + F−2(N)ε−2 + F−1(N)ε−1 + . . .



∑

i1

· · ·∑

i7

f(ε,N, i1, i2, . . . , i7)

Wegschaider’s MultiSumPackage (F. Stan)


61 / 90



F (N) =

∫ 1

0· · ·

∫ 1

0Φ(ε,N, x1, x2, . . . , x7)dx1dx2 . . . dx7

?= F−3(N)ε−3 + F−2(N)ε−2 + F−1(N)ε−1 + . . .



∑

i1

· · ·∑

i7

f(ε,N, i1, i2, . . . , i7)

Wegschaider’s MultiSumPackage (F. Stan)

Holonomic/difference fieldapproach (M. Round)


62 / 90


The general tactic

Feynman integrals


multiple sums




Tactic 2: Expand a recurrence in εBlumlein, Klein, CS, Stan, J. Symbol. Comput. 2012; arXiv:1011.2656 [cs.SC]Ablinger, Blumlein, Round, CS, LL2012, arXiv:1210.1685 [cs.SC] 63 / 90

Tactic 3: Guess and solve 25

The general tactic

Feynman integrals


multiple sums




Tactic 3: Guess a recurrence and solve itJ. Blumlein, M. Kauers, S. Klein, CS, Comput. Phys. Comm. 180, arXiv:0902.4091 [hep-ph] 64 / 90


In the non-singlet (3-loop, massless) case ∼ 360 diagramscontribute. The integrals are of the form:

F (n, ε) =

∫ 1

0dx1· · ·

∫ 1

0dx7

K∑

i=1

pi(x1, x2, . . . , x7)n+···+riε+...

qi(x1, x2, . . . , x7)···+siε+...

where K ∈ N, ri, si ∈ Q, and pi, qi are polynomials in x1, . . . , x7.

Vermaseren, Moch: 3-5 CPU years (2004)

65 / 90



F (n, ε) =

∫ 1

0dx1· · ·

∫ 1

0dx7

K∑

i=1

pi(x1, x2, . . . , x7)n+···+riε+...

qi(x1, x2, . . . , x7)···+siε+...

= F−3(n)ε−3 + F−2(n)ε

−2 + F−1(n)ε−1 + F0(n) ε

0 + . . .

↓

Initial values F0(i), i = 1, . . . , 5114

66 / 90



F (n, ε) =

∫ 1

0dx1· · ·

∫ 1

0dx7

K∑

i=1

pi(x1, x2, . . . , x7)n+···+riε+...

qi(x1, x2, . . . , x7)···+siε+...

= F−3(n)ε−3 + F−2(n)ε

−2 + F−1(n)ε−1 + F0(n) ε

0 + . . .

↓


↓ Recurrence guesser (M. Kauers)

a0(n)F0(n) + a1(n)F0(n+ 1) + · · ·+ a35(n)F0(n+ 35) = 0

67 / 90


a0(n)F0(n) + a1(n)F0(n + 1) + · · · + a35(n) F0(n+ 35) = 0

68 / 90


a0(n)F0(n) + a1(n)F0(n + 1) + · · · + a35(n) F0(n+ 35) = 0

a35(n) = A0 +A1n+A2n2 + · · ·+A938n

983 ∈ Z[n]

69 / 90


a0(n)F0(n) + a1(n)F0(n + 1) + · · · + a35(n) F0(n+ 35) = 0

a35(n) = A0 +A1n+A2n2 + · · ·+A938n

983 ∈ Z[n]

A0 = 464094430921131367250398022371626412420040708599385400241246031519495765021269344971048446299722216293405285738333200767150194016391501666279502138073561097109520456039662733887577826975886022012779835605320173748759267144591132576514527194521425546215314730842059721076159532936551563452998613135384718911305253299053198893606401464021608911620974192090016680299516207801829472582629394508011545117745278325038743416618988916752210737846879797981026538551064393704386755756346752374040609465899100467933353731959645624977524424672990654427732309881685346483771128690208371474520244015281690794069336653444761812602433441720976916367066280305967553580902716969306447414771961021984962848689607964231297513620776876867741883488363846944854496482629372436829699055391369178850397003816380116123026795808974880766477213119306347353167877796207576599515202809978299053753901432067359626151

(885 decimal digits)

70 / 90



F (n, ε) =

∫ 1

0dx1· · ·

∫ 1

0dx7

K∑

i=1

pi(x1, x2, . . . , x7)n+···+riε+...

qi(x1, x2, . . . , x7)···+siε+...

= F−3(n)ε−3 + F−2(n)ε

−2 + F−1(n)ε−1 + F0(n) ε

0 + . . .

↓



a0(n)F0(n) + a1(n)F0(n+ 1) + · · ·+ a35(n)F0(n+ 35) = 0

↓ Sigma

CLOSED FORM

Blumlein, Kauers, Klein, CS, Comput. Phys. Comm. 180, arXiv:0902.4091 [hep-ph]

71 / 90

Tactic 4: Solve coupled systems 29

Tactic 4: Solve coupled systems

of differential equations

72 / 90


A coupled differential system for I1(x), I2(x), I3(x)

(produced by IBP [extension of REDUZE 2, A.v. Manteuffel])

Dx

I1(x)

I2(x)

I3(x)

=

−−1−ε+x

(x−1)x − 2(x−1)x 0

ε(3ε+2)4(x−1) −−2−ε+3x+3εx

2(x−1)x − ε+12(x−1)

− ε(3ε+2)(x−2)4(x−1)x

−2−5ε+x+3εx2(x−1)x

(−2ε−x+εx)2(x−1)x

I1(x)

I2(x)

I3(x)

+

R1(x)

R2(x)

−R2(x)

where

R1(x) =B4(x)

(x− 1)x,

R2(x) =−(ε+ 2)3

16(ε + 1)(x− 1)xB1(x) +

(ε+ 2)(3ε + 4)(19ε2 + 36ε + 16

)

16ε(5ε + 6)(x − 1)xB2(x)

+(ε+ 1)2(3ε + 4)2

2ε(5ε + 6)xB3(x) +

−24− 50ε− 25ε2 + 8x+ 14εx + 6ε2x

4(5ε+ 6)(x − 1)xB4(x)

B1(x), B2(x), B3(x) have been solved with symbolic summation.73 / 90


Tactic 4: the DE-REC approach

DE system

DI(x) = A I(x) + R(x)

74 / 90



DE system

DI(x) = A I(x) + R(x)

OreSys package (S. Gerhold)

uncoupling algorithm//

uncoupled DE system∑

i ai(x)DiI1(x) = r(x)

Ik(x) = exprk(I1(x)), k > 1

75 / 90



DE system

DI(x) = A I(x) + R(x)






I1(x) =

∞∑

N=0

I1(N)xN

76 / 90



DE system

DI(x) = A I(x) + R(x)






holonomic closure prop.

��

linear recurrence∑

i a′i(N)I1(N + i) = r′(N)

77 / 90



DE system

DI(x) = A I(x) + R(x)







��closed form solutions ofI1(N)in the class of nestedsums and products– if this is possible


i a′i(N)I1(N + i) = r′(N)

Sigma’s REC-solveroo

78 / 90


Tactic 4: the DE-REC approach (SolveCoupledSystem package)

DE system

DI(x) = A I(x) + R(x)





Ik(x) = exprk(I1(x)),k > 1


��

extract coefficientSumProduction package

rrrrrrrrrr

xxrrrrrrrrr

closed form solutions ofI1(N), I2(N), . . . , In(N)in the class of nestedsums and products– if this is possible


i a′i(N)I1(N + i) = r′(N)


J. Ablinger, A. Behring, J. Blmlein, A. De Freitas, A. von Manteuffel, CS. Comput. Phys. Comm. 202, pp. 33-112.79 / 90


Solving a coupled differential system

In[8]:= << OreSys.m

OreSys by Stefan Gerhold (optimized by C. Schneider) c© RISC-Linz

In[9]:= << SolveCoupledSystem.m

SolveCoupledSystem by Carsten Schneider c© RISC-Linz

In[10]:= coupledDESys = D[{I1(x), I2(x), I3(x)},x] − A.{I1(x), I2(x), I3(x)};

In[11]:= rhs = {R1(x), R2(x),−R2(x)} in power series representation;

In[12]:= SolveCoupledDESystem[coupledDESys, {I1[x], I2[x], I3[x]}, ε,−3,

{−2,−2,−2}, rhs, . . . ]

80 / 90


Solving a coupled differential system

In[8]:= << OreSys.m

OreSys by Stefan Gerhold (optimized by C. Schneider) c© RISC-Linz

In[9]:= << SolveCoupledSystem.m

SolveCoupledSystem by Carsten Schneider c© RISC-Linz

In[10]:= coupledDESys = D[{I1(x), I2(x), I3(x)},x] − A.{I1(x), I2(x), I3(x)};

In[11]:= rhs = {R1(x), R2(x),−R2(x)} in power series representation;

In[12]:= SolveCoupledDESystem[coupledDESys, {I1[x], I2[x], I3[x]}, ε,−3,

{−2,−2,−2}, rhs, . . . ]

Out[12]=

{

1

ε3

(

4

(

3N2 + 6N+ 4

)

3(N+ 1)2+

4S1[N]

3(N + 1)

)

+1

ε2

(

−2(20N3 + 58N

2 + 57N+ 22)

3(N+ 1)3+

2(N + 2)(2N − 1)S1[N]

3(N + 1)2−

S1[N]2

N+ 1−

S2[N]

N+ 1

)

,

4

3ε3−

2

ε2,

8

3ε3+

1

ε2

(

−4

(

4N2 + 7N+ 2

)

3(N + 1)2+

4(N + 2)S1[N]

3(N + 1)

)

}

81 / 90


Tactic 4: the DE-REC approach (SolveCoupledSystem package)

DE system

DI(x) = A I(x) + R(x)





Ik(x) = exprk(I1(x)),k > 1


��


rrrrrrrrrr

xxrrrrrrrrr

closed form solutions ofI1(N), I2(N), . . . , In(N)in the class of nestedsums and products– if this is possible


i a′i(N)I1(N + i) = r′(N)


J. Ablinger, A. Behring, J. Blmlein, A. De Freitas, A. von Manteuffel, CS. Comput. Phys. Comm. 202, pp. 33-112.82 / 90

Tactic 5: The methods of large moments 34

Tactic 5: compute large moments (SolveCoupledSystem package)

DE system

DI(x) = A I(x) + R(x)







��


rrrrrrrrrr

xxrrrrrrrrr

closed form solutions ofI1(N)in the class of nestedsums and products– if this is possible


i a′i(N)I1(N + i) = r′(N)


Given r′(0), r′(1), . . . , r′(5114),compute I1(0), I1(1), . . . , I1(5114)

J. Blumlein, CS, 2017, arXiv:1701.04614 [hep-ph].83 / 90



F (n, ε) =

∫ 1

0dx1· · ·

∫ 1

0dx7

K∑

i=1

pi(x1, x2, . . . , x7)n+···+riε+...

qi(x1, x2, . . . , x7)···+siε+...

= F−3(n)ε−3 + F−2(n)ε

−2 + F−1(n)ε−1 + F0(n) ε

0 + . . .

↓



a0(n)F0(n) + a1(n)F0(n+ 1) + · · ·+ a35(n)F0(n+ 35) = 0

↓ Sigma

CLOSED FORM


84 / 90



F (n, ε) =

∫ 1

0dx1· · ·

∫ 1

0dx7

K∑

i=1

pi(x1, x2, . . . , x7)n+···+riε+...

qi(x1, x2, . . . , x7)···+siε+...

= F−3(n)ε−3 + F−2(n)ε

−2 + F−1(n)ε−1 + F0(n) ε

0 + . . .

↓ a new method for large moments



a0(n)F0(n) + a1(n)F0(n+ 1) + · · ·+ a35(n)F0(n+ 35) = 0

↓ Sigma

CLOSED FORM


85 / 90

Summary 36

SUMMARY (of RISC packages)Backbone: a new difference ring/field approach implemented in Sigma.m

86 / 90

Summary 36


◮ symbolic summation

◮ CS’s EvaluateMultiSum.m

◮ M. Round’s RhoSum.m (difference ring/holonomic approach)

◮ K. Wegschaider’s MultiSum.m (enhanced Sister Celine/WZ approach,utilized by F. Stan)

87 / 90

Summary 36






◮ symbolic integration

◮ J. Ablinger’s MultiIntegrate.m (refined multivariateAlmkvist-Zeilberger approach)

88 / 90

Summary 36








◮ solving coupled systems

◮ S. Gerhold’s OreSys.m

◮ CS’s SolveCoupledSystem.m

89 / 90

Summary 36








◮ solving coupled systems

◮ S. Gerhold’s OreSys.m

◮ CS’s SolveCoupledSystem.m

◮ special function algorithms

◮ J. Ablinger’s HarmonicSums.m

90 / 90

All YouCan Sum(part 2) · Tactic 1: Expand and simplify 10 Sigma.m is based on diﬀerence...

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Transcript of All YouCan Sum(part 2) · Tactic 1: Expand and simplify 10 Sigma.m is based on diﬀerence...