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Novel mathematical . . .

Novel mathematical aspects of Feynman integrals

O. V. Tarasov

Sochi, Russia , September 25 , 2011

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 1


Novel mathematical . . .

Outline of the talk:
· Introduction · Mini review of useful features of Generalized recurrence relations Dimensional recurrences Gr¨ obner bases technique Functional equations · Finding new relationships for hypergeometric functions by evaluating Feynman integrals · Conclusions

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 2


Novel mathematical . . .

Radiative corrections to different physical quantities needed for the comparison of theoretical predictions with experimental data to be collected with the CERN Large Hadron Collider (LHC) and, in future, with an International Linear Collider (ILC) and other colliders are expressed in terms of complicated Feynman integrals. In many cases, radiative corrections must be evaluated analytically to achieve reliable accuracies in the calculations. Researches working for the LHC collider presented famous "wishlist" ­ a list of physical processes where next to leading order radiative corrections are needed. Practically all these corrections require evaluation of radiative corrections with 5-, 6- and more external legs. Characteristic features of these corrections: · masses of many particles must be taken into account · diagrams with many external lines, i.e.many kinematic variables must be calculated Therefore one should know how to calculate analytically and (or) numerically with very high precision functions of many variables.
O. V. Tarasov JINR, Dubna, Russia ­ Septemb er 25, 2011­ 3


Novel mathematical . . .

To perform such calculations new mathematical approaches are needed! Rather novel concepts for such calculations were proposed during last several years: · generalized recurrence relations · Gr¨bner bases technique o · the method of dimensional recurrences · functional equations These methods and techniques are either recursive or strongly connected with recurrence relations. They do not exploit traditional integral representations or differential equations. To extend applicability of these approaches their mathematical background should be further developed and certainly that will be useful in other fields of research like it was with computer algebra systems at the beginning of seventies.

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 4


Novel mathematical . . .

In general Feynman diagrams are sums of tensor integrals. These integrals can be expressed as combinations of scalar integrals multiplied by products of tensors made of Mankowski tensor and external momenta. There are essentially two different methods for reducing tensor integrals to scalar ones. One method (Passarino-Veltman) based on Ansatz for such integrals in terms of all possible combinations of Mankowski tensor and external momenta multiplied by unknown coefficients. For example, dd k kµ k k 2 (k - p1 )2 (k - p2 )2 = x1 gµ + x2 p1µ p1 + x3 p1µ p

2

+ x4 p2µ p

1

+ x5 p2µ p2

Contracting this Ansatz with gµ , p1µ p1 , ... one get system of equations for x1 , ..., x5 . Solution for x1 , ..., x5 will be given in terms of scalar integrals. In a similar way one can get representation for more complicated integrals. For higher rank tensor integrals such method leads to very big complicated systems of equations.

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 5


Novel mathematical . . .

Another method for reducing tensor integrals is attributed to introducing auxiliary vectors vj , representing products of integration vectors as derivatives w.r.t. these vectors, 1 ... exp[ivj kj ]|vs =0 , k1µ1 . . .kL,µr = r i vµ1 vµr transforming the resulting momentum integrals into integrals over Feynman parameters. From this parametric representation for an arbitrary tensor integral one can obtain the following formula: dd k1 . . . dd k k1µ . . .k
L

(k 1 - m2 )1 . . .( 1 dd k1 . . .

2

N 2 kN

- m2 ) N
2

N

= Tµ,..., (q , , d+ ) where d+ G
(d)

dd kL

1 (k 1 - m2 )1 . . .(k N - m2 ) 1 N m2 j
2 N

=G

(d+2)

and j =

A general formula for the polynomial tensor operator Tµ,..., (q , , d+ ) was given by O.T., Phys.Rev. D54 .
O. V. Tarasov JINR, Dubna, Russia ­ Septemb er 25, 2011­ 6


Novel mathematical . . .

This method is very efficient and it is easily implementable on computers. There is no need to solve huge systems of linear equations. However integrals with different shifts of the space-time dimension do appear. To reduce all such scalar integrals to basic set of integrals the method of generalized recurrence relations was developed O.V. T., Phys.Rev. D.54(1996) 6479. To obtain recurrence relations one can use 't Hooft and Veltman idea (Nucl.Phys. B44 (1972) 189) that dd k1 . . . dd kL lj µ kj µ (k 2 - m2 )1 . . .(k 2 - m2 ) 1 N 1 N = 0,
N

where k j are linear combinations of external and integration momenta, lµ - is either integration or external momentum. Scalar products emerging after differentiation w.r.t. k , can be represented as combinations of factors in denominators, masses and external momenta (IBP method): 1 2 2 k1 q1 = {[(k1 + q1 )2 - m2 ] - [k1 - m2 ] - q1 }, .... 1 1 2

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 7


Novel mathematical . . .

When it is not possible one should introduce into consideration artificial factors in denominators. As a result one gets recurrence relations connecting integrals with different powers of propagators j . One can represent emerging scalar products in terms of integrals with shifted space-time dimension by exploiting the formula: dd k1 . . . dd k k1µ . . .k
L

(k 1 - m2 )1 . . .( 1 d k1 . . .
d d

2

N 2 kN

- m2 ) N
2

N

= Tµ,..., (q , , d )

+

d kL

1 (k 1 - m2 )1 . . .(k N - m2 ) 1 N
2 N

As a result one gets recurrence relations connecting integrals with different powers of propagators j and also integrals with different dimensionality d. Important: These recurrence relations additionally to j have new recurrence parameter - d. For this reason we call them generalized recurrence relations With this new parameter one can construct very efficient algorithms for reducing scalar integrals to a set of bases integrals.

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 8


Novel mathematical . . .

The system of generalized recurrence relations is strongly overdetermined. To find minimal set of recurrence relations allowing to reduce scalar integrals to minimal set of integrals it was proposed to use Theory of Gr¨ ner bases and Buchberger algorithm. ob To use theory of Gr¨ obner bases for recurrence relations for Feynman integrals for the first time was proposed by 1. O.V. T Reduction of Feynman graph amplitudes to a minimal set of basic integrals , Acta Physica Polonica, v B29 (1998) 2655 2. O. V. T, Computation of Gr¨ obner bases for two-loop propagator type integrals, Talk at ACAT-2003 Nucl. Instrum. Meth. A 534 (2004) 293 [arXiv:hep-ph/0403253]

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 9


Novel mathematical . . .

Gr¨ obner bases for IBP relations for the two-loop self energy integrals
(d) J3

(1 , 2 , 3 ) =

1 (i
d/2 )2

dd k1 dd k2 2 - m2 )1 ((k - k )2 - m2 )2 (k 2 - m2 )3 . (k1 1 2 3 2 1 2

Gr¨ obner bases for IBP recurrence relations:
123

1 1+ J

(d) 3

(1 2 3 ) = u123 (d - 1 - 22 ) + 2m2 (1 - 2 ) 2
(d)

+u312 1 1+ (2- - 3- ) + 2m2 2 2+ (1- - 3- ) J3 (1 2 3 ). 2
123 2

2+ J

(d) 3

(1 2 3 ) = u213 (d - 2 - 21 ) + 2m2 (2 - 1 ) 1
(d)

+u321 2 2+ (1- - 3- ) + 2m2 1 1+ (2- - 3- ) J3 (1 2 3 ). 1
123 3

3+ J

(d) 3

(1 2 3 ) = u312 (d - 3 - 21 ) + 2m2 (3 - 1 ) 1
(d)

+u231 3 3+ (1- - 2- ) + 2m2 1 1+ (3- - 2- ) J3 (1 2 3 ). 1 where 1± J3 (1 , 2 3 ) = J
ij k (d) (d) 3

(1 ± 1, 2 , 3 ),..., uij k = mi - mj - mk and

= m4 + m4 + m4 - 2m2 m2 - 2m2 m2 - 2m2 m2 . i j ij ik jk k

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 10


Novel mathematical . . .

Gr¨ obner bases for generalized recurrence relations: (d - 2)1 1+ J3 (1 2 3 ) = -u123 - 1- + 2- + 3 (d - 2)2 2+ J3 (1 2 3 ) = -u213 - 2- + 1- + 3 (d - 2)3 3+ J3 (1 2 3 ) = -u321 - 3- + 2- + 1 (d - 2)(d - 1 - 2 - 3 )J -
123 (d) 3 (d) (d) (d) - - -

J J J

(d-2) 3 (d-2) 3 (d-2) 3

(1 , 2 , 3 ), (1 , 2 , 3 ), (1 , 2 , 3 ),

(1 , 2 , 3 ) =
(d-2)

+ u123 1- + u213 2- + u312 3- J3

(1 , 2 , 3 ).

By exploiting Gr¨ obner bases either for IBP relations or for generalized recurrence relations and explicit formula for tadpole integral: one can reduce any integral J3 (1 , 2 , 3 ) with integer , 2 , 3 to the set of basic integrals (d) (d) (d) (d) J3 (1, 1, 1), J3 (0, 1, 1) , J3 (1, 0, 1) , J3 (1, 1, 0). It turns out that Gr¨ obner bases for generalized recurrence relations is much more efficient than for IBP relations!!
(d)

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 11


Novel mathematical . . .
(d)

Example: reduction of the integral J3 (3, 5, 4) IBP relations: 72 sec Generalized recurrence relations: 9 sec For higher powers of propagators the difference in time is more than 20 times.

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 12


Novel mathematical . . .

There are even more efficient relations. For the considered example they are: (d - 2)1 2 1+ 2+ J (d - (d -
(d) 3 (d) 2)1 3 1+ 3+ J3 (d) 2)2 3 2+ 3+ J3

(1 , 2 , 3 ) = -2m2 3 3+ + (d - 2 - 23 ) J3 3

(d-2) (d-2) (d-2)

(1 , 2 , 3 ), (1 , 2 , 3 ), (1 , 2 , 3 ),

(1 , 2 , 3 ) = -2m2 1 2+ + (d - 2 - 22 ) J3 2

(1 , 2 , 3 ) = -2m2 1 1+ + (d - 2 - 21 ) J3 1

1 1+ + 2 2+ + 3 3+ - (d - 1 - 2 - 3 ) J

(d) 3

(1 , 2 , 3 ) = 0.
(d)

To find these relations we used Gr¨ obner bases. For the considered integral J3 (3, 5, 4) exploiting above relations only 3 seconds were needed to reduce it to basic integrals. The reason is that this optimal set of relations has no explicit dependence on kinematical Gram determinants! This was one of the criteria for finding those relations. Reduction of d d - 2 was very essential! Gram determinants disappear only in relations connecting integrals with different dimensions of the space-time! Similar recurrence relations were discovered for the one-loop multi-leg integrals. The calculations of the one-loop five gluon amplitude are now in progress. Depending on diagram evaluations are from 10 to 100 times faster than with Gr¨ obner bases for generalized recurrence relations.
O. V. Tarasov JINR, Dubna, Russia ­ Septemb er 25, 2011­ 13


Novel mathematical . . .

Dimensional recurrences Dimensional recurrences are particular case of generalized recurrence relations. They include integral with fixed powers of propagators but with different shifts of the space - time dimension and simpler integrals considered as inhomogeneous part of the equation. General solution of dimensional recurrences can be written in the form: Mk (d, {mj }, {pi pk }) =
s

s (d, {mj }, {pi pk }) ws (d, {mj }, {pi pk })

where s are functions from the fundamental set of solutions for and ws are the so-called `periodics` satisfying the following condition: ws (d + 2, {mj }, {pi pk }) = ws (d, {mj }, {pi pk }) They can be found, for example, from the comparison of the above solution with the asymptotic value of the integral at d . In some cases one can obtain simple differential equation with respect to kinematic variables for ws (d, {mj }, {pi pk }).

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 14


Novel mathematical . . .

With the help of these algorithms new analytic results were obtained:

· hypergeometric representation for the one - loop integrals corresponding to diagrams with three- and four external legs · analytic formula for the one-loop massless pentagon type integral · hypergeometric representation for the two-loop 'sunrise' propagator type integral Recently Li and Smirnov applied this method in calculating four loop massless propagator type integrals satisfying first order dimensional recurrence relation.

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 15


Novel mathematical . . .

Functional equations for Feynman integrals
Feynman integrals satisfy recurrence relations which we write in the form Qj Ij,n =
j k,r
Rk,r Ik

,r

where Qj , Rk are polynomials in masses, scalar products of external momenta, d, and powers of propagators. Ik,r - are integrals with r external lines. In recurrence relations some integrals are more complicated than the others: they have more arguments than the others.

General method for deriving functional equations:
By choosing kinematic variables, masses, indices of propagators remove most complicated integrals, i.e. impose conditions : Qj = 0 keeping at least some other coefficients different from zero Rk = 0

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 16


Novel mathematical . . .

Example: one-loop n-point integrals (d) Integrals In satisfy generalized recurrence relations O.T. in Phys.Rev.D54 (1996) p.6479
n

G

n-1 j

( j+ Ind+2) - (j n )I

(d) n

=
k=1

( (j k n )k- Ind) ,

where j± shifts indices j j ± 1, , j m2 j p1 p1 . . . p1 p Y11 . . . Y12 . . .
n n-1

G

n-1

= -2n

p1 p . . .

2

... .. .
1

p1 pn-1 . . .
n-1 pn-1

,

p2 pn-

... p

n =

. . . Y1n . .. . . . ... Y
nn

,

Yij = m2 + m2 - pij , i j

pij = (pi - pj )2 ,

Y1n Y2
O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 17


Novel mathematical . . .

At n = 3, j = 1 we get equation: G2 1+ I
(d+2) 3

(m2 , m2 , m2 , p23 , p13 , p12 ) 1 2 3
(d) 3

-(1 3 )I = 2(p
12 13

(m2 , m2 , m2 , p23 , p13 , p12 ) 1 2 3
(d) (d) 2

+p

23 23

- p13 )I2 (m2 , m2 , p12 ) 1 2 - p12 )I (m2 , m2 , p13 ) - 4p23 I2 (m2 , m2 , p23 ). 1 3 2 3
(d)

+2(p where

+p

3 =

2 2 2 12 + 2p13 + 2p23 - 4p13 p23 - 4p12 p13 - 4p23 p12 , 2(m2 - m2 )[(m2 - m2 )p13 - (m2 - m2 )p12 ] - 2m2 p2 - 2m2 p2 2 3 1 2 1 3 1 23 3 12 -2m2 p2 - 2(m2 - m2 )(m2 - m2 )p23 + 2(m2 + m2 )p12 p13 2 13 1 3 1 2 2 3 +2(m2 + m2 )p23 p12 + 2(m2 + m2 )p13 p23 - 2p12 p13 p23 3 1 2 1

G2 = 2p

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 18


Novel mathematical . . .

Coefficients in front of I3 depend on 6 variables p12 , p13 , p23 , m2 , m2 , m2 . To remove I 1 2 3 from the equation we must solve system of equations G 1
2 3

3

= 2p2 + 2p2 + 2p2 - 4p13 p23 - 4p12 p13 - 4p23 p12 = 0, 12 13 23 = -2p2 - 4m2 p 23 1 +2m2 p 2
13 23

+ 2m2 p23 + 2m2 p 2 3

23

+ 2p12 m
12

2 3 12

- 2m2 p13 + 2p13 p23 - 2m2 p 3 2

+ 2p23 p

=0

This system can be resolved w.r.t. p13 , p23 . There is a nontrivial solution + 2p12 (m2 + m2 ) - (p12 + m2 - m2 ) 1 3 1 2 p13 = s13 (m , , , p12 ) = , 2p12 12 + 2p12 (m2 + m2 ) + (p12 - m2 + m2 ) 2 2 2 2 3 1 2 p23 = s23 (m1 , m2 , m3 , p12 ) = . 2p12
2 1

m2 2

m2 3



12

where =±
12

+ 4p12 m2 . 3

ij = p2 + m4 + m4 - 2pij m2 - 2pij m2 - 2m2 m2 . ij i j i j ij

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 19


Novel mathematical . . .

This solution leads to the following functional equation
(d) I2

(

m2 1

,

m2 2

, p12 ) = +

p p

12

12

+ m2 - m2 - (d) 2 2 1 2 I2 (m1 , m3 , s13 (m2 , m2 , m2 , p12 )) 1 2 3 2p12 - m2 + m2 + (d) 2 2 1 2 I2 (m2 , m3 , s23 (m2 , m2 , m2 , p12 )). 1 2 3 2p12

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 20


Novel mathematical . . .

Substituting m2 = 0 into functional equation we have : 3
(d) I2

(

m2 1

,

m2 2

, p12 ) = +

p p

12

12

+ m2 - m2 - 1 2 2p12 - m2 + m2 + 1 2 2p12

12

I2 (m2 , 0, s13 ) 1 I2 (0, m2 , s23 ) 2
(d)

(d)

12

where s s
13

= =



12

12

23 12

+ 2p12 m2 - (p12 + m2 - m2 )12 1 1 2 , 2p12 + 2p12 m2 + (p12 - m2 + m2 )12 2 1 2 , 2p12



= ± 12 .

Integral with arbitrary mases and momentum can be expressed in terms of integrals with one propagator massless !!!

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 21


Novel mathematical . . .

Analytic result for I2 (0, m2 , p2 ) is known Bollini and Giambiagi (1972b), Boos and Davydychev (1990rg): 1, 2 - d ; q 2 (d) (d) 2 . I2 (0, m2 , p2 ) = I2 (0, m2 , 0) 2 F1 2 d m ;
2

(d)

where I2 (0, m2 , 0) = - 1 -
(d) (d)

d 2

md

-4

.

Substituting this expression for I2 (0, m2 , p2 ) into functional equation we get complete agreement with the known result for I2 (m2 , m2 , p12 ) 1 2
(d)

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 22


Novel mathematical . . .

Setting m2 = 0 into the previous functional equation we have :
(d) I2

(

m2 1

m4 m2 (d) 2 1 I2 m1 , 0, 1 , 0, p12 ) = p12 p12
(d) 2

(p12 - m2 ) (d 1 + I2 p12

)

(p12 - m2 )2 1 0, 0, p12
d 2

.

where I
(d)

2 - d 2 d - 1 2 2 (0, 0, p ) = (-p2 ) (d - 2)
2

-2

.

Integral I2 on the right hand side has inverse argument . In fact this equation corresponds to the well known formula for analytic continuation: 1, 2 - d ; 1, 2 - d ; 1 1 2 2 z = 2 F1 2 F1 d d z z 2; 2; +
d 2

d -1 2 (-z ) (d - 2)

d 2

-2

1 1- z

d-3

.

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 23


Novel mathematical . . .

Very similar functional equations do exist for the three-, four-, e.t.c integrals. Functional equations here play the same role as 24 Kummer relations for Gauss' hypergeometric function! Therefore functional equations can be used for analytic continuation of functions with several variables. It is not so easy to obtain formulae for analytic continuation for hypergeometric functions with several variables. For the rather simple Appell function F1 explicit representation in terms of Gauss hypergeometric function was used. For analytic continuation of Feynman integrals with the help of functional equations explicit representation is not needed! It will be interesting to obtain functional equations for some Green functions to all orders of perturbation theory. Probably one can use use Dyson-Schwinger equations and exploit functional equations for the kernels of this equations.

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 24


Novel mathematical . . .

As was realized many years ago in papers by Regge (1967), Feynman integrals are generalized hypergeometric functions. This conjecture was confirmed through the evaluations of specific Feynman integrals. From numerous results we know that Feynman integrals can be expressed in terms of generalized hypergeometric functions, Appell functions F1 , F2 ,F3 , F4 Laurichella, Laurichella-Saran e.t.c functions. . These results were obtained using rather different methods, e.g. · by directly evaluating the integrals from their Feynman parameter representations, · by applying Mellin-Barnes integral representations, · by solving recurrence relations, · by making use of the negative-dimension approach, · or by using spectral representations. As a method for finding relations between hypergeometric functions, Srivastava and Karlsson in their book advocated the evaluation of integrals reducible to hypergeometric functions by several different methods and the comparison of the results thus obtained. In this respect, the evaluation of Feynman integrals may be considered as a rich source for finding relations between hypergeometric functions.
O. V. Tarasov JINR, Dubna, Russia ­ Septemb er 25, 2011­ 25


Novel mathematical . . .

As an example we consider the evaluation of the one-loop propagator type integral with arbitrary masses and arbitrary powers of propagators: I
(d) 1 2

(m2 1

,

m2 2

; s12 ) =

dd q 1 . i d/2 [(q - p1 )2 - m2 ]1 [(q - p2 )2 - m2 ]2 2 1 1 + 2 - (1 )(2 )
d 2 0 1

This integral can be written as an integral over Feynman parameters: I
(d) 1
2

(

m2 1

,

m2 2

; s12 ) = (-1)

1 +

2

dx x [s12 x2 + x (m2 1

1

-1

(1 - x)

2

-1

-

m2 2

- s12 ) +

m2 ]1 +2 - 2

d 2

.

Representing the quadratic polynomial in the denominator as s12 x2 + x(m2 - m2 - s12 ) + m2 = m2 (1 - x+ x)(1 - x- x), 1 2 2 2 where x± = 1+x-y± x2 + y 2 + 1 - 2xy - 2x - 2y , 2 s12 x = 2, m2 m2 y = 1, m2 2

and then comparing our integral with the integral representation for the Appell function F1
O. V. Tarasov JINR, Dubna, Russia ­ Septemb er 25, 2011­ 26


Novel mathematical . . .

the following result follows: I
(d) 1
2

(

m2 1

,

m2 2

(-1)1 +2 1 + 2 - d 2 ; s12 ) = (1 + 2 )(m2 )1 +2 -d/2 2 d d âF1 1 , 1 + 2 - , 1 + 2 - ; 1 + 2 ; x- , x+ , 2 2

An analytic result for this integral in terms of two Appell functions F4 was derived by E. Boos and A. Davydychev. Comparing both results we can derive relation between F1 and F4 functions. Just for simplicity we consider the case m1 = m2 . By using Mellin-Barnes representation E. Boos and A. Davydychev obtained the following result
() Id2 (m2 , m2 ; s12 ) = (-1)1 +2 (m2 )d/2-1 -2 1 1 , 2 , 1 + 2 - d ; x1 1 + 2 - d 2 2 . â 3 F2 1 +2 1 +2 +1 (1 + 2 ) ; x2 2, 2

Comparing this formula with our result taken at m2 = m2 = m2 , we obtain: 1 2 , - , ; x 2 - 2x, x + 2 - 2x , F1 , , ; ; x - x x = 3 F2 +1 2 2, 2 ;
O. V. Tarasov JINR, Dubna, Russia ­ Septemb er 25, 2011­ 27


Novel mathematical . . .

which may be rewritten as: F1 x , , ; ; x, x-1 = 3 F2 , - , ;
+1 2, 2

;

x2 . 4(x - 1)



To the best of our knowledge, there is no such a relation in the mathematical literature. Another interesting relation can be obtained from the comparison of imaginary the two-loop "sunrise integral"calculated by two different methods: 12 11 22 2 , 2 ; (x - 3)(x + 1)3 3(x + 3)(x - 1)3 2 3 , 3 ; x (x - 9) = 2 F1 2 F1 3 2 + 3) (x (x2 + 3)3 1 ; (x + 3)(x - 1) 1;

part of .

The hypergeometric function 2 F1 on the left-hand side of this equation is proportional to the complete elliptic integral of the first kind. Relations between hypergeometric functions with parameters 1/2, 1/2, 1 and 1/3, 2/3, 1 but with arguments different from that in the above equation were first derived by Ramanujan. Several other relations one can find in the paper: B. A. Kniehl and O.V. T., (arXiv:1108.6019 [math-ph])
O. V. Tarasov JINR, Dubna, Russia ­ Septemb er 25, 2011­ 28


Novel mathematical . . .

Summary · generalized recurrence relations provide us a tool for efficient evaluation of Feynman integrals but further investigation concerning optimal sets (Gr¨ obner bases) of recurrence relations is needed · the method of dimensional recurrences can be used in calculation of multiscale integrals as well as multiloop integrals. Dimensional recurrences are simpler than differential equations because singularity structure of differential equations w.r.t kinematic variables is more complicated than w.r.t. d · functional equations represent a powerful instrument for analytic continuation of Feynman integrals. For a detailed classification of these equations group theoretical approach should be formulated. · computational machinery for Feynman integrals can be used to obtain relationships for hypergeometric functions that will be useful in other applications. From the already known results we can essentially extend lists of formulae given in the well known books by Baitmen-Erdely and Brychkov,Marichev,Prudnikov.

O. V. Tarasov

JINR, Dubna, Russia

­ Septemb er 25, 2011­ 29