A Table Errata is submitted concerning a formula in terms of a sum of two Gauss hypergeometric functions for the Ferrers function of the second kind (associated Legendre function of the second kind “on the cut”). This error occurs on p. 167 of Magnus, Oberhettinger & Soni (1966) Formulas and Theorems for the Special Functions of Mathematical Physics (third enlarged edition, Springer-Verlag, New...

In this work we derive a new integral of the hypergeometric differential equation, valid for arbitrary parameters α, β, γ and which is expressed in terms of indefinite integrals. This leads to new types of relations between F (α, β, γ;x) and its contiguous functions : F (α, β, γ;x) can be related to only one of the functions contiguous to it.These equations enable one to extend the formulae giv...

The second Appell’s hypergeometric function F2(a, b, b ′, c, c′;x, y) has a Mellin convolution integral representation in the region (x + y) < 1 and a > 0. We apply a recently introduced asymptotic method for Mellin convolution integrals to derive three asymptotic expansions of F2(a, b, b ′, c, c′;x, y) in decreasing powers of x and y with x/y bounded. For certain values of the real parameters ...

The subject of this article belongs to the general question Under which condition(s) suitably normalized transcendental functions take algebraic values at algebraic arguments? Already the classical examples of Weierstrass’ result concerning the exponential function and Theodor Schneider’s result about the elliptic modular function show that arguments and values in these cases are of particular ...

In [1] I.M.Gelfand introduced the conception of hypergeometric functions, associated with the Grassmanian Gk,n of k-dimensional subspaces in C . The class of these functions includes many classical hypergeometric functions as particular cases. Series of consequent papers of I.M.Gelfand and his coathors ([2]–[6]) are devoted to the theory of these functions. In particular, the paper [4] is devot...

We consider a point process on one-dimensional lattice originated from the harmonic analysis on the infinite symmetric group, and defined by the z-measures with the deformation (Jack) parameter 2. We derive an exact Pfaffian formula for the correlation function of this process. Namely, we prove that the correlation function is given as a Pfaffian with a 2×2 matrix kernel. The kernel is given in...

Throughout the paper, we denote this equation by E(a, b, c). Since Kummer we know that in general there are 24 hypergeometric series which express solutions of E(a, b, c). The subject of this paper is Gauss hypergeometric functions when some or all of the numbers a, b, c, c − a, c − b, a − b, c − a − b are integers. We refer to these functions and to corresponding hypergeometric equations as de...