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In 1946, Magnus presented an addition theorem for the confluent hypergeometric function of the second kind $U$ with argument $x+y$ expressed as an integral of a product of two $U$'s, one with argument $x$ and another with argument $y$. We take advantage of recently obtained asymptotics for $U$ with large complex first parameter to determine a domain of convergence for Magnus' result. Using well-known specializations of $U$, we obtain corresponding integral addition theorems with precise domains of convergence for modified parabolic cylinder functions, and Hankel, Macdonald, and Bessel functions of the first and second kind with order zero and one.

In this paper, we use combinatorial group theory and a limiting process to connect various types of hypergeometric series, and of relations among such series. We begin with a set S of 56 distinct translates of a certain function M, which takes the form of a Barnes integral, and is expressible as a sum of two very-well-poised $$_9F_8$$ hypergeometric series of unit argument. We consider a known, transitive action of the Coxeter group $$W(E_7)$$ on this set. We show that, by removing from $$W(E_7)$$ a particular generator, we obtain a subgroup that is isomorphic to $$W(D_6)$$ , and that acts intransitively on S, partitioning it into three orbits, of sizes 32, 12, and 12, respectively. Taking certain limits of the M functions in the first orbit yields a set of 32 J functions, each of which is a sum of two Saalschutzian $$_4F_3$$ hypergeometric series of unit argument. The original action of $$W(D_6)$$ on the M functions in this orbit is then seen to correspond to a known action of this group on this set of J functions. In a similar way, the image of each of the size-12 orbits, under a similar limiting process, is a set of 12 L functions that have been investigated in earlier works. In fact, these two image sets are the same. The limiting process is seen to preserve distance, except on pairs consisting of one M function from each size-12 orbit. Finally, each known three-term relation among the J and L functions is seen to be obtainable as a limit of a known three-term relation among the M functions.

Recently, a large number of integral formulas involving Bessel functions and their extensions have been investigated. The objective of this paper is to establish four classes of integral formulas associated with the Struve functions, which are expressed in terms of the Fox-Wright function. Among a variety of special cases of the main results, we present only six integral formulas involving trigonometric and hyperbolic functions.

Mittag-Leffler functions has many applications in various areas of Physical, biological ,applied, earth Sciences and Engineering. It is used in solving problems of fractional order differential, integral and difference equations. In this paper, we aim to define the m-parameter Mittag-Leffler function, which can be reduced to various already known extensions of Mittag-Leffler function. We then, discuss its various properties like recurrence relations, differentiation formula and integral representations. We also represent the new m-parameter Mittag-Leffler function in terms of some known special functions such as Generalized hypergeometric function, Mellin Barnes integral, Wright hypergeometric function and Fox H-function. We also discuss its various integral transforms like Euler-Beta, Whittaker, Laplace and Mellin transforms. Further, fractional differential and integral operators are considered to discuss few properties of m-parameter Mittag-Leffler function. Also, we use the m-parameter Mittag-Leffler function to define a generalization of Prabhakar integral and discuss its properties. Further, relation of m-parameter Mittag Leffler function with various other functions such as exponential, trigonometric, hypergeometric and algebraic functions is obtained and represented graphically using MATHEMATICA 12.

The paper studies logarithmic convexity and concavity of the generalized hypergeometric function with respect to the simultaneous shift of several parameters. We use integral representations and properties of Meijer’s G function to prove log-convexity. When all parameters are shifted we use series manipulations to examine the power series coefficients of the generalized Turanian formed by the generalized hypergeometric function. In cases when all zeros of the generalized hypergeometric function are real, we further explore the consequences of the extended Laguerre inequalities and formulate a conjecture about reality of zeros.

We introduce the generalized hypergeometric function with matrix parameters. We also define two variable Appell matrix functions and find their regions of convergence as well as integral representations.

The aim of this paper is to establish certain integrals involving product of the Aleph function with Srivastava’s polynomials and Fox-Wright’s Generalized Hypergeometric function. Being unified and general in nature, these integrals yield a number of known and new results as special cases. For the sake of illustration, four corollaries are also recorded here as special case of our main results.

In this paper, the parameter derivatives to any order of the generalized hypergeometric function as well as for its various special cases are obtained.

In our previous work, we found sufficient conditions to be imposed on the parameters of the generalized hypergeometric function in order that it be completely monotonic or of Stieltjes class. In this paper we collect a number of consequences of these properties. In particular, we find new integral representations of the generalized hypergeometric functions, evaluate a number of integrals of their products, compute the jump and the average value of the generalized hypergeometric function over the branch cut [1,∞), and establish new inequalities for this function in the half-plane ℜz

We obtain a criterion for the convergence of the Mellin–Barnes integral representing the solution to a general system of algebraic equations. This yields a criterion for a nonnegative matrix to have positive principal minors. The proof rests on the Nilsson–Passare–Tsikh Theorem about the convergence domain of the general Mellin–Barnes integral, as well as some theorem of a linear algebra on a subdivision of the real space into polyhedral cones.

In this paper, we extend one of Erdelyi's classical integrals for the Gauss hypergeometric functions to a special class of generalized hypergeometric functions in which certain pairs of numerator and denomiator parameters differ by positive integers. Our main results are achieved by applying the fractional integration by parts and series manipulation technique. Some special cases are also pointed out which includes a new extension of a Thomae-type transformation.

In this paper we consider a master integral in four arbitrary parameters. The integrand involves the logarithmic function and the Gauss hypergeometric function, which in certain special cases the integral reduces to identities involving zeta functions. A relationship will also be created between the integral and Euler sums of arbitrary order and arbitrary argument. Many interesting new specific examples will be highlighted.

We employ some formulas on basic hypergeometric series and hypergeometric series and the p -adic Gamma function to establish several new supercongruences.

We employ some formulae on hypergeometric series and p-adic Gamma function to establish several congruences.

In the present paper, we study the mixed summation integral type operators having Baskakov and Szasz basis functions in summation and integration form, respectively. We also give here the alternate form of the operators in terms of hypergeometric functions and estimated moments of these operators using hypergeometric series. In the last section, we present some results for operators related to convergence.

Recently, Srivastava et al. [Integral Transforms Spec. Funct. 23 (2012), 659-683] introduced the incomplete Pochhammer symbols that led to a natural generalization and decomposition of a class of hypergeometric and other related functions as well as to certain potentially useful closed-form representations of definite and improper integrals of various special functions of applied mathematics and mathematical physics. In the present paper, our aim is to establish several formulas involving integral transforms and fractional derivatives of this family of incomplete hypergeometric functions. As corollaries and consequences, many interesting results are shown to follow from our main results.

In this paper, some new integral inequalities related to the bounded functions, involving hypergepmetric fractional integral operators, are established. Special cases of the main results are also pointed out.

We prove a new Bailey-type transformation relating WP-Bailey pairs. We then use this transformation to derive a number of new 3- and 4-term transformation formulae between basic hypergeometric series., Comment: 12 pages

We discuss recent work of the authors in which we study the translation of classical hypergeometric transformation and evaluation formulas to the finite field setting.

Integrals of logarithmic and hypergeometric functions are intrinsically connected with Euler sums. In this paper we explore many relations and explicitly derive closed form representations of integrals of logarithmic, hypergeometric functions and the Lerch phi transcendent in terms of zeta functions and sums of alternating harmonic numbers.

Summation, transformation and reduction formulas for various families of hypergeometric functions in one, two and more variables are potentially useful in many diverse areas of applications. The main object of this paper is to derive several substantially more general results on this subject than those considered recently by Neethu et al. [7] in connection with Bailey’s transformation involving the Gauss hypergeometrc function 2F1 (see [1]). The methodology used here is based essentially on some families of hypergeometric generating functions. Relevant connections of the results presented in this paper with those in the earlier works are also pointed out.

In the present paper, our aim is to establish several formulas involving integral transforms, fractional derivatives, and a certain family of extended generalized hypergeometric functions. As corollaries and consequences, many interesting results are shown to follow from our main results. A probability density function involving the extended generalized hypergeometric function is introduced, and its properties are studied. The corresponding properties of some of the classical probability distributions and their associated probability density functions are easily derivable as special cases of our general results. Copyright © 2016 John Wiley & Sons, Ltd.

We show that the Schrödinger and Klein–Gordon equations can both be derived from a hypergeometric differential equation. The same applies to non linear generalizations of these equations., Facultad de Ciencias Exactas, Instituto de Física La Plata