Inequalities for the Gaussian hypergeometric function

we study the monotonicity of certain combinations of the Gaussian hypergeometric functions F(-1/2,1/2;1;1- xc) and F(-1/2- δ,1/2 + δ;1;1- xd) on(0,1) for given 0 c 5d/6 ∞ andδ∈(-1/2,1/2),and find the largest value δ1 = δ1(c,d) such that inequality F(-1/2,1/2;1;1- xc) F(-1/2- δ,1/2 + δ;1;1- xd) holds for all x ∈(0,1). Besides,we also consider the Gaussian hypergeometric functions F(a- 1- δ,1- a + δ;1;1- x3) and F(a- 1,1- a;1;1- x2) for given a ∈ [1/29,1) and δ∈(a- 1,a),and obtain the analogous results.     

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.     

A connection between a generalized Pascal matrix and the hypergeometric function

The n × n generalized Pascal matrix P(t) whose elements are related to the hypergeometric function ₂F₁(a, b; c; x) is presented and the Cholesky decomposition of P(t) is obtained. As a result, it is shown that

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is the solution of the Gauss's hypergeometric differential equation,

x(1 − x)y″ + [1 + (a + b − 1)x]y′ − ABY = 0

. where a and b are any nonnegative integers. Moreover, a recurrence relation for generating the elements of P(t) is given.     

New class of generating functions associated with generalized hypergeometric polynomials

In this paper authors prove a general theorem on generating relations for a certain sequence of functions. Many formulas involving the families of generating functions for generalized hypergeometric polynomials are shown here to be special cases of a general class of generating functions involving generalized hypergeometric polynomials and multiple hypergeometric series of several variables. It is then shown how the main result can be applied to derive a large number of generating functions involving hypergeometric functions of Kampé de Fériet, Srivastava, Srivastava-Daoust, Chaundy, Fasenmyer, Cohen, Pasternack, Khandekar, Rainville and other multiple Gaussian hypergeometric polynomials scattered in the literature of special functions.     

Extension of gamma, beta and hypergeometric functions

The main object of this paper is to present generalizations of gamma, beta and hypergeometric functions. Some recurrence relations, transformation formulas, operation formulas and integral representations are obtained for these new generalizations.     

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.     

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.     

Sharp power mean bounds for the Gaussian hypergeometric function

Sharp inequalities are established between the Gaussian hypergeometric function and the power mean. These results extend known inequalities involving the complete elliptic integral and the hypergeometric mean.     

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.     

Implicit function theorems for generalized equations

We show that Lipschitz and differentiability properties of a solution to a parameterized generalized equation 0 f(x, y) + F(x), wheref is a function andF is a set-valued map acting in Banach spaces, are determined by the corresponding Lipschitz and differentiability properties of a solution toz g(x) + F(x), whereg strongly approximatesf in the sense of Robinson. In particular, the inverse map (f + F)^–1 has a local selection which is Lipschitz continuous nearx ₀ and Fréchet (Gateaux, Bouligand, directionally) differentiable atx ₀ if and only if the linearization inverse (f (x ₀) + f (x₀) (× – x₀) + F(×))^–1 has the same properties. As an application, we study directional differentiability of a solution to a variational inequality.This work was supported by National Science Foundation Grant Number DMS 9404431.     

Asymptotic expansions of integrals of two Bessel functions via the generalized hypergeometric and Meijer functions

Asymptotic expansions of certain finite and infinite integrals involving products of two Bessel functions of the first kind are obtained by using the generalized hypergeometric and Meijer functions. The Bessel functions involved are of arbitrary (generally different) orders, but of the same argument containing a parameter which tends to infinity. These types of integrals arise in various contexts, including wave scattering and crystallography, and are of general mathematical interest being related to the Riemann—Liouville and Hankel integrals. The results complete the asymptotic expansions derived previously by two different methods — a straightforward approach and the Mellin-transform technique. These asymptotic expansions supply practical algorithms for computing the integrals. The leading terms explicitly provide valuable analytical insight into the high-frequency behavior of the solutions to the wave-scattering problems.     

Inequalities for the generalized elliptic integrals and modular functions

In this paper, the authors study monotonicity and convexity of the generalized elliptic integrals and certain combinations of these special functions, such as ma(r) and μa(r). Making use of these results, the authors obtain some sharp inequalities for the so-called Ramanujan's generalized modular functions.     

On certain transformations of poly-basic bilateral hypergeometric series

In this paper, we have established certain transformations of basic hypergeometric series with more than one base. Some of these lead to the relationship between product of two q-series. These results, in turn, lead to very interesting transformations of bi-basic and poly-basic q-series. A few of the results which are representative of the many results obtained are presented in this article.     

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.     

Inequalities and bounds for generalized complete elliptic integrals

Computable bounds for the generalized complete elliptic integrals of the first and second kind are obtained. Also, bounds for some combinations and products for integrals under discussion are established. It has been proven that both families of integrals are logarithmically convex as functions of the first parameter. This property has been employed to obtain several inequalities involving integrals in question.     

Arithmetical properties of hypergeometric Bernoulli numbers

In a recent paper, Byrnes et al. (2014) have developed some recurrence relations for the hypergeometric zeta functions. Moreover, the authors made two conjectures for arithmetical properties of the denominators of the reduced fraction of the hypergeometric Bernoulli numbers. In this paper, we prove these conjectures using some recurrence relations. Furthermore, we assert that the above properties hold for both Carlitz and Howard numbers.     

Invariance properties of generalized monotonicity

Generalized monotone maps are studied under affine variable transformations. The results enable us to generate generalized monotone matrices of any size. Various necessary conditions and sufficient conditions for generalized monotone matrices are derived. Furthermore. admissible translations of generalized monotone linear maps are studied. Finally, the maximal domain of generalized monotonicity is characterized.     

Inequalities for the generalized trigonometric and hyperbolic functions

The generalized trigonometric functions occur as an eigenfunction of the Dirichlet problem for the one-dimensional p$p$-Laplacian. The generalized hyperbolic functions are defined similarly. Some classical inequalities for trigonometric and hyperbolic functions, such as Mitrinovi?–Adamovi?’s inequality, Lazarevi?’s inequality, Huygens-type inequalities, Wilker-type inequalities, and Cusa–Huygens-type inequalities, are generalized to the case of generalized functions.