Decomposition formulas for some triple hypergeometric functions

With the help of some techniques based upon certain inverse pairs of symbolic operators, the authors investigate several decomposition formulas associated with Srivastava's hypergeometric functions HA, HB and HC in three variables. Many operator identities involving these pairs of symbolic operators are first constructed for this purpose. By means of these operator identities, as many as 15 decomposition formulas are then found, which express the aforementioned triple hypergeometric functions in terms of such simpler functions as the products of the Gauss and Appell hypergeometric functions. Other closely-related results are also considered briefly.     

On the hypergeometric matrix functions of two variables

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.     

A note on Turán type and mean inequalities for the Kummer function

Turán-type inequalities for combinations of Kummer functions involving Φ(a±ν,c±ν,x) and Φ(a,c±ν,x) have been recently investigated in [Á. Baricz, Functional inequalities involving Bessel and modified Bessel functions of the first kind, Expo. Math. 26 (3) (2008) 279-293; M.E.H. Ismail, A. Laforgia, Monotonicity properties of determinants of special functions, Constr. Approx. 26 (2007) 1-9]. In the current paper, we resolve the corresponding Turán-type and closely related mean inequalities for the additional case involving Φ(a±ν,c,x). The application to modeling credit risk is also summarized.     

Partial fraction expansion of the hypergeometric functions

In this paper, the partial fractions expansions for the generalized hypergeometric function will be presented. Our results generalized several well-known results in the literature.     

Some identities on Catalan numbers and hypergeometric functions via Catalan matrix power

In this paper we use the Catalan matrix power as a tool for deriving identities involving Catalan numbers and hypergeometric functions. For that purpose, we extend earlier investigated relations between the Catalan matrix and the Pascal matrix by inserting the Catalan matrix power and particulary the squared Catalan matrix in those relations. We also pay attention to some relations between Catalan matrix powers of different degrees, which allows us to derive the simplification formula for hypergeometric function ₃F₂, as well as the simplification formula for the product of the Catalan number and the hypergeometric function ₃F₂. Some identities involving Catalan numbers, proved by the non-matrix approach, are also given.     

Parameter derivatives of the generalized hypergeometric function

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

Refinements of transformation inequalities for zero-balanced hypergeometric functions

In the article, we present some refinements of three classes of transformation inequalities for zero-balanced hypergeometric functions by use of the updated monotonicity criterion for the quotient of power series.     

Jacobi matrices generated by ratios of hypergeometric functions

A problem of determining zeroes of the Gauss hypergeometric function goes back to Klein, Hurwitz, and Van Vleck. In this very short note we show how ratios of hypergeometric functions arise as m-functions of Jacobi matrices and we then revisit the problem based on the recent developments of the spectral theory of non-Hermitian Jacobi matrices.     

Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters a,b and z

In a recent paper (Temme, 2021) new asymptotic expansions are given for the Kummer functions $M(a,b,z)$ and $U(a,b+1,z)$ for large positive values of $a$ and $b$, with $z$ fixed and special attention for the case $a～b$. In this paper we extend the approach and also accept large values of $z$. The new expansions are valid when at least one of the parameters $a$, $b$, or $z$ is large. We provide numerical tables to show the performance of the expansions.     

Inequalities and monotonicity of ratios for generalized hypergeometric function

We find two-sided inequalities for the generalized hypergeometric function of the form _q+1F_q(−x) with positive parameters restricted by certain additional conditions. Both lower and upper bounds agree with the value of _q+1F_q(−x) at the endpoints of positive semi-axis and are asymptotically precise at one of the endpoints. The inequalities are derived from a theorem asserting the monotony of the quotient of two generalized hypergeometric functions with shifted parameters. The proofs hinge on a generalized Stieltjes representation of the generalized hypergeometric function. This representation also provides yet another method to deduce the second Thomae relation for ₃F₂(1) and leads to an integral representations of ₄F₃(x) in terms of the Appell function F₃. In the last section of the paper we list some open questions and conjectures.     

Certain bilateral generating relations for generalized hypergeometric functions

Recently, we introduced a class of generalized hypergeometric functionsI n:(b _q)/α:(a _p) (x, w) by using a difference operator Δ _x,w , where . In this paper an attempt has been made to obtain some bilateral generating relations associated withI _n ^ga (x, w). Each result is followed by its applications to the classical orthogonal polynomials.     

Elliptic hypergeometric functions and Calogero-Sutherland-type models

We consider an elliptic analogue of the Gauss hypergeometric function and two of its multivariate generalizations. We describe their relation to elliptic beta integrals, the exceptional Weyl group E₇, the elliptic hypergeometric equation, and Calogero-Sutherland-type models. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 2, pp. 311–324, February, 2007.     

Classes of series identities and associated hypergeometric reduction formulas

The main object of the present paper is to investigate some classes of series identities and their applications and consequences leading naturally to several (known or new) hypergeometric reduction formulas. We also indicate how some of these series identities and reduction formulas would yield several series identities which emerged recently in the context of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order).     

Automatic generation of hypergeometric identities by the beta integral method

In this article, hypergeometric identities (or transformations) for _p+1F_p-series and for Kampé de Fériet series of unit arguments are derived systematically from known transformations of hypergeometric series and products of hypergeometric series, respectively, using the beta integral method in an automated manner, based on the Mathematica package HYP. As a result, we obtain some known and some identities which seem to not have been recorded before in literature.     

Index transforms associated with generalized hypergeometric functions

We deal with a class of integral transformations whose kernels contain the Clausenian hypergeometric function ₃F₂(a₁,a₂,a₃;b₁,b₂;z). These transforms are defined in terms of integrals with respect to their parameters. It involves as particular cases the familiar Olevskii and generalized Mehler–Fock transforms which are key tools in the methods of the mathematical theory of elasticity. The main theorem of boundedness of these operators as a map of L₂(?₊)L₂(?₊;x^?1 dx) is proved. Some examples of the Olevskii and Mehler–Fock type integrals are given. Copyright © 2003 John Wiley & Sons, Ltd.