Related papers: Equations of hypergeometric type in the degenerate…
In this paper we continue investigation of the hypergeometric function ${}_4F_3(1)$ as the function of its seven parameters. We deduce several reduction formulas for this function under additional conditions that one of the top parameters…
This is a study of terminating and ill-defined Gauss hypergeometric functions. Corresponding hypergeometric equations have a degenerate set of of 24 Kummer's solutions. We describe those solutions and relations between them.
The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients. These fundamental solutions are directly connected with multiple…
The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with three singular coefficients, which could be expressed in terms of a confluent hypergeometric function…
We give a systematic and unified discussion of various classes of hypergeometric type equations: the hypergeometric equation, the confluent equation, the F_1 equation (equivalent to the Bessel equation), the Gegenbauer equation and the…
When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find…
In this paper, using the similarity method, we construct particular solutions with singularities for degenerate high-order equations. The considered equations have singularities of the first and second kind. Particular solutions are…
The 15 Gauss contiguous relations for ${}_2F_1$ hypergeometric series imply that any three ${}_2F_1$ series whose corresponding parameters differ by integers are linearly related (over the field of rational functions in the parameters). We…
We present infinitely many solutions of the general Heun equation in terms of generalized hypergeometric functions. Each solution assumes that two restrictions are imposed on the involved parameters: a characteristic exponent of one of the…
We express explicitly the Heckman-Opdam hypergeometric function for the root system of type A with a certain degenerate parameter in terms of the Lauricella hypergeometric function.
In this paper, we introduce degenertae generalized hypergeometric functions and study degenerate hypergeometric numbers of order p. These numbers involving of lambda-binomial coefficients and lambda-falling sequence, and can be represented…
In this paper, we introduce a new class of confluent hypergeometric functions of many variables, study their properties, and determine a system of partial differential equations that this function satisfies. It turns out that all the…
For certain class of hypergeometric functions ${}_3F_2$ with rational parameters, we give a sufficient condition for the special value at $1$ to be expressed in terms of logarithms of algebraic numbers. We give two proofs, both of which are…
In first order formulation of pure gravity, we find a new class of solutions to the equations of motion represented by degenerate four-geometries. These configurations are described by non- invertible tetrads with two zero eigenvalues and…
Fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients were constructed recently. These fundamental solutions are directly connected with multiple Lauricella hypergeometric function and…
Hypergeometric functions and their generalizations play an important r\^{o}les in diverse applications. Many authors have been established generalizations of hypergeometric functions by a number ways. In this paper, we aim at establishing…
We study the linear Pfaffian systems satisfied by a certain class of hypergeometric functions, which includes Gau\ss's ${}_2 F_{1}$, Thomae's ${}_L F_{L-1}$ and Appell-Lauricella's $F_D$. In particular, we present a fundamental system of…
Any three basic hypergeometric series {}_{2}phi_{1} whose respective parameters (a, b, c) differ by integer powers of the base q satisfy a linear relation with coefficients which are rational functions of a, b, c, q and the variable x.…
A class of second-order differential equations commonly arising in physics applications are considered, and their explicit hypergeometric solutions are provided. Further, the relationship with the Generalized and Universal Associated…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…