Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order epsilon-expansion of generalized hypergeometric functions with one half-integer value of parameter

We continue the study of the construction of analytical coefficients of the epsilon-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we show the following results: Theorem A: The multiple (inverse) binomial sums of arbitrary weight and depth (see Eq. (1.1)) are expressible in terms of Remiddi-Vermaseren functions. Theorem B: The epsilon expansion of a hypergeometric function with one half-integer value of parameter (see Eq. (1.2)) is expressible in terms of the harmonic polylogarithms of Remiddi and Vermaseren with coefficients that are ratios of polynomials. Some extra materials are available via the www at this http://theor.jinr.ru/~kalmykov/hypergeom/hyper.htmlComment: 24 pages, latex with amsmath and JHEP3.cls; v2: some typos corrected and a few references added; v3: few references added

Derivatives of Horn-type hypergeometric functions with respect to their parameters

We consider the derivatives of Horn hypergeometric functions of any number variables with respect to their parameters. The derivative of the function in $n$ variables is expressed as a Horn hypergeometric series of $n+1$ infinite summations depending on the same variables and with the same region of convergence as for original Horn function. The derivatives of Appell functions, generalized hypergeometric functions, confluent and non-confluent Lauricella series and generalized Lauricella series are explicitly presented. Applications to the calculation of Feynman diagrams are discussed, especially the series expansion in $\epsilon$ within dimensional regularization. Connections with other classes of special functions are discussed as well.Comment: 27 page

On one master integral for three-loop on-shell HQET propagator diagrams with mass

An exact expression for the master integral I_2 arising in three-loop on-shell HQET propagator diagrams with mass is derived and its analytical expansion in the dimensional regularization parameter epsilon is given.Comment: 6 pages, 1 figure; v3: completely re-written, 2 new authors, many new results, additional reference

ϵ\epsilon-Expansion of Multivariable Hypergeometric Functions Appearing in Feynman Integral Calculus

We present a new methodology to perform the $\epsilon$-expansion of hypergeometric functions with linear $\epsilon$ dependent Pochhammer parameters in any number of variables. Our approach allows one to perform Taylor as well as Laurent series expansion of multivariable hypergeometric functions. Each of the coefficients of $\epsilon$ in the series expansion is expressed as a linear combination of multivariable hypergeometric functions with the same domain of convergence as that of the original hypergeometric function thereby providing a closed system of expressions. We present illustrative examples of hypergeometric functions in one, two and three variables which are typical of Feynman integral calculus

A new approach to the epsilon expansion of generalized hypergeometric functions

Assumed that the parameters of a generalized hypergeometric function depend linearly on a small variable $\varepsilon$, the successive derivatives of the function with respect to that small variable are evaluated at $\varepsilon=0$ to obtain the coefficients of the $\varepsilon$-expansion of the function. The procedure, quite naive, benefits from simple explicit expressions of the derivatives, to any order, of the Pochhammer and reciprocal Pochhammer symbols with respect to their argument. The algorithm may be used algebraically, irrespective of the values of the parameters. It reproduces the exact results obtained by other authors in cases of especially simple parameters. Implemented numerically, the procedure improves considerably the numerical expansions given by other methods.Comment: Some formulae adde

MultiHypExp\texttt{MultiHypExp}: A Mathematica Package For Expanding Multivariate Hypergeometric Functions In Terms Of Multiple Polylogarithms

We present the Mathematica package $\texttt{MultiHypExp}$ that allows for the expansion of multivariate hypergeometric functions (MHFs), especially those likely to appear as solutions of multi-loop, multi-scale Feynman integrals, in the dimensional regularization parameter. The series expansion of MHFs can be carried out around integer values of parameters to express the series coefficients in terms of multiple polylogarithms. The package uses a modified version of the algorithm prescribed in arXiv:2208.01000v2. In the present work, we relate a given MHF to a Taylor series expandable MHF by a differential operator. The Taylor expansion of the latter MHF is found by first finding the associated partial differential equations (PDEs) from its series representation. We then bring the PDEs to the Pfaffian system and further to the canonical form, and solve them order by order in the expansion parameter using appropriate boundary conditions. The Taylor expansion so obtained and the differential operators are used to find the series expansion of the given MHF. We provide examples to demonstrate the algorithm and to describe the usage of the package, which can be found in https://github.com/souvik5151/MultiHypExp