Related papers: On computing some special values of hypergeometric…
In Slater's 1960 standard work on confluent hypergeometric functions, also called Kummer functions, a number of asymptotic expansions of these functions can be found. We summarize expansions derived from a differential equation for large…
Special functions have been used widely in many problems of applied sciences. However, there are considerable numbers of problems in which exact solutions could not be achieved because of undefined sums or integrals involving special…
In this paper, we will obtain new algebraic transformations of the $_2F_1$-hypergeometric functions. The main novelty in our approach is the interpretation of identities among $_2F_1$-hypergeometric functions as identities among automorphic…
In this paper, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and develop some transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have…
An algorithm is given for computing explicit formulas for the generators of relations among the invariant rational functions for vector-valued bilinear forms. These formulas have applications in the geometry of Riemannian submanifolds and…
We show how certain hypergeometric functions play an important role in finding fundamental solutions for a generalized Tricomi operator.
Generalized integral formulas involving the generalized Bessel-Maitland function are considered and it expressed in terms of generalized Wright hypergeometric functions. By assuming appropriate values of the parameters in the main results,…
We prove a variety of explicit formulas relating special values of generalized hypergeometric functions to lattice sums with four indices of summation. These results are related to Boyd's conjectured identities between Mahler measures and…
Recursive formulas extending some known $_{2}F_{1}$ and $_{3}F_{2}$ summation formulas by using contiguous relations have been obtained. On the one hand, these recursive equations are quite suitable for symbolic and numerical evaluation by…
In this paper, we use some standard numerical techniques to approximate the hypergeometric function $$ {}_2F_1[a,b;c;x]=1+\frac{ab}{c}x+\frac{a(a+1)b(b+1)}{c(c+1)}\frac{x^2}{2!}+\cdots $$ for a range of parameter triples $(a,b,c)$ on the…
HYPERDIRE is a project devoted to the creation of a set of Mathematica based programs for the differential reduction of hypergeometric functions. The current version includes two parts: the first one, FdFunction, for manipulations with…
Special values of the Lommel functions allow the calculation of Fresnel like integrals. These closed form expressions along with their asymptotic values are reported.
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…
Some integral properties of Jack polynomials, hypergeometric functions and invariant polynomials are studied for real normed division algebras.
In this article, we provide an application of hypergeometric evaluation identities, including a strange valuation of Gosper, to prove several supercongruences related to special valuations of truncated hypergeometric series. In particular,…
The differential-reduction algorithm, which allows one to express generalized hypergeometric functions with parameters of arbitrary values in terms of such functions with parameters whose values differ from the original ones by integers, is…
Series containing the digamma function arise when calculating the parametric derivatives of the hypergeometric functions and play a role in evaluation of Feynman diagrams. As these series are typically non-hypergeometric, a few instances…
Estimating the coefficient functionals on various classes of holomorphic functions traditionally forms an important field of geometric complex analysis and its mathematical and physical applications. These coefficients reflect fundamental…
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 article three expansion formulas for a generalized hypergeometric function $_4F_3$ are derived, when its upper parameters differ by integers. Though the results are special cases of a general continuation formula for $_pF_q$, they…