In many cases, the results of analytical calculation of Feynman diagrams can be represented as combinations of hypergeometric functions. For practical applications, finding a hypergeometric representation is not enough. It is necessary to construct the socalled epsilonexpansion, which we may understand as the construction of the analytical coefficients of the Laurent expansion of hypergeometric functions around rational values of their parameters. In this direction, very limited results are available.
One of the classical tasks in mathematics is to find the full set of parameters and arguments for which hypergeometric functions are expressible in terms of algebraic functions. Quantum field theory makes a quantum generalisation of this classical task: to find the full set of parameters and arguments so that the allorder epsilonexpansion is expressible in terms of known functions or identify the full set of functions which must be invented in order to construct the allorder epsilonexpansion of generalized hypergeometric functions. The complete solution of this problem is still open.
There are three different ways to describe special functions: by a series whose coeffcients satisfy certain recurrence relations; as a solution of a system of differential and difference equations (holonomic approach); and as integrals of Euler or MellinBarnes type. For functions of a single variable all these representations are equivalent, but some properties of the function may be expressed more readily in one representation than another. These three different representations of special functions have led physicists to three different approaches to developing the epsilon expansion of hypergeometric functions.
In the Euler integral representation, the most important results are related to the construction of the allorder epsilonexpansion of Gauss hypergeometric functions with special values of parameters in terms of Nielsen polylogarithms [D00,DK00,DK01] . There are several important masterintegrals expressible in terms of Gauss (2F1) hypergeometric functions. This set of integrals includes oneloop propagator type diagram with arbitrary values of mass and momentum; twoloop bubble integral with an arbitrary values of masses, and oneloop massless vertex diagram with three nonzero external momenta [BD91,DT93,DT96] .
The series representation is an intensively studied approach. Particularly impressive results involving series representations were derived in the framework of the nested sum approach for hypergeometric functions with a balanced set of parameters in [MUW02,W04] , and in framework of the generating function approach for hypergeometric functions with one unbalanced set of parameters in [DK04,KWY07b,KK08a] .
An approach using the iterated solution of differential equations has been explored in [KWY07a,KWY07c,KK08a] . One of the advantage of the iterated solution approach over the series approach is that it provides a more efficient way to calculate each order of the epsilonexpansion, since it relates each new term to the previously derived terms, rather than having to work with an increasingly large collection of independent sums at each order. This technique includes two steps: (i) the differential reduction algorithm (to reduce a generalized hypergeometric function to basic functions); (ii) iterative solution of the proper differential equation for the basic functions (equivalent to iterative algorithms for calculating the analytical coefficients of the epsilonexpansion).
2F1(I1+a*ep, I2+b*ep; I3+p/q+c*ep;z) ,
2F1(I1+p/q+a*ep, I2+p/q+b*ep; I3+p/q + c*ep;z) ,
2F1(I1+p/q+a*ep, I2+b*ep; I3+c*ep;z) ,
2F1(I1+p/q+a*ep, I2+b*ep; I3+p/q + c*ep;z)
The first five coefficients of the epsilonexpansion for basis hypergeometric functions are available here.
For generalized hypergeometric functions with integer and /or halfinteger values of parameters, and special values of argument (z=1/4,1/2,3/4,1) the epsilon epxansion up to constant of weight 5 is constructed [FK99,DK01] . For discussion of proper basis we refer to Basis for Single scale massive Feynman diagrams In this case, the result of expansion can be written in terms of generalized logsine functions and calculated with high accuracy by the help of program LSJK
The application of our resuls for calculation of different masterintegrals are given in [JKV03,JK04,K06] .
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Gauss hypergeometric function :  

hepth/0602028 : Eqs. (4.1), (4.2) 
FORM  hypergeometric2F1 . 
Mathematica hypergeom.m , 

hepth/0612240 : Eqs. (3.3), (3.4) 
FORM  hypergeometric2F1  angle represenation . 

hepth/0612240 : Eqs. (2.19)(2.22) 
FORM  hypergeometric2F1  integer values of parameters . 

Generalized hypergeometric function:  
hepth/0708.0803 Eqs. (2.15), (3.1)(3.4), (3.5) 
FORM  hypergeometric pFp1  integer values of parameters . 
3F2 hypergeometric function:  

FORM  hypergeometric 3F2  with one halfinteger parameter .  
FORM  hypergeometric 3F2  with integer parameters . 
4F3 hypergeometric function:  

FORM  4F3  with one halfinteger parameter: one set of basis functions .  
FORM  hypergeometric 4F3  with integer parameters . 
5F4 hypergeometric function:  

FORM  5F4  with one halfinteger parameter .  
FORM  hypergeometric 5F4  with integer parameters . 