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.     

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.     

A new approach to the theory of classical hypergeometric polynomials

In this paper we present a unified approach to the spectral analysis of a hypergeometric type operator whose eigenfunctions include the classical orthogonal polynomials. We write the eigenfunctions of this operator by means of a new Taylor formula for operators of Askey-Wilson type. This gives rise to some expressions for the eigenfunctions, which are unknown in such a general setting. Our methods also give a general Rodrigues formula from which several well-known formulas of Rodrigues-type can be obtained directly. Moreover, other new Rodrigues-type formulas come out when seeking for regular solutions of the associated functional equations. The main difference here is that, in contrast with the formulas appearing in the literature, we get non-ramified solutions which are useful for applications in combinatorics. Another fact, that becomes clear in this paper, is the role played by the theory of elliptic functions in the connection between ramified and non-ramified solutions.

     

Some inverse Laplace transforms that contain the Marcum Q function and an expanded property of the Marcum Q function

ABSTRACT

The Laplace transforms of the transition probability density and distribution functions of the Feller process contain products of a Kummer and a Tricomi confluent hypergeometric function. The intricacies caused by the singularity at 0 of the Feller process imply that ultimately seven new inverse Laplace transforms can be derived of which four contain the Marcum Q function. The results of this paper together with a scarcely used link between the Marcum and Nuttall Q functions also provide two alternative proofs for an existing identity involving two Marcum Q functions with reversed arguments. The paper also expands the existing expression for the Marcum Q function with identical arguments and order 1. In particular, the new formula applies to all integer and fractional values of the order and is expressed in terms of the generalized hypergeometric function.     

A partial differential equation related to a problem in atmospheric pollution

We investigate a generalized form of a partial differential equation governing the diffusion of heavy pollutants into the atmosphere. In earlier treatments of the equation, the vertical component of turbulent exchange coefficient was assumed to be linear. Our generalization takes into account the nonlinear case of this component. Furthermore, two general identities involving the confluent hypergeometric function of the second kind are derived in the course of solving the given PDE.     

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.     

A simplification of Laplace’s method: Applications to the Gamma function and Gauss hypergeometric function

The main difficulties in the Laplace’s method of asymptotic expansions of integrals are originated by a change of variables. We propose a variant of the method which avoids that change of variables and simplifies the computations. On the one hand, the calculation of the coefficients of the asymptotic expansion is remarkably simpler. On the other hand, the asymptotic sequence is as simple as in the standard Laplace’s method: inverse powers of the asymptotic variable. New asymptotic expansions of the Gamma function Γ(z) for large z and the Gauss hypergeometric function ₂F₁(a,b,c;z) for large b and c are given as illustrations. An explicit formula for the coefficients of the classical Stirling expansion of Γ(z) is also given.     

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.     

On a generalization of Mittag-Leffler function and its properties

Let s and z be complex variables, Γ(s) the Gamma function, and for any complex ν the generalized Pochhammer symbol. The principal aim of the paper is to investigate the function

     

Infinite summation formulas related to Riemann Zeta function from hypergeometric series

Applying Gauss and Watson’s famous hypergeometric summation theorems, the authors establish two pattern infinite summation formulas involving generalized harmonic numbers related to Riemann Zeta function.     

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.     

The Laplace transform of the digamma function: An integral due to Glasser, Manna and Oloa

The definite integral

is related to the Laplace transform of the digamma function

by when . Certain analytic expressions for in the complementary range, , are also provided.

     

A duality approach to minimax results for quasi-saddle functions in finite dimensions

In this paper we show how saddle point theorems for a quasiconvex—quasiconcave function can be derived from duality theory. A symmetric duality framework that provides the machinery for deriving saddle point theorems is presented. Generating the theorems,via the framework, provides a deeper understanding of assumptions employed in existing theorems which do not utilize duality theory.     

Apparent singular points of factors of reducible generalized hypergeometric equations

We consider a reducible generalized hypergeometric equation, whose sub‐equation possesses apparent singular points. We determine the polynomial whose roots are these points. We show that this polynomial is a generalized hypergeometric polynomial.     

Applications to give an analytical solution to the Black Scholes equation

A simple ?-function and its dynamic equation is presented. An application to give an analytical solution to the Black Scholes equation is presented.     

Generalizations of Rodrigues type formulas for hypergeometric difference equations on nonuniform lattices

ABSTRACT

By building a second-order adjoint difference equations on nonuniform lattices, the generalized Rodrigues type representation for the second kind solution of a second-order difference equation of hypergeometric type on nonuniform lattices is given. The general solution of the equation in the form of a combination of a standard Rodrigues formula and a ‘generalized’ Rodrigues formula is also established.     

On the order of starlikeness of the shifted Gauss hypergeometric function

We determine for several ranges of real parameters the order of starlikeness of the shifted Gauss hypergeometric function and we give some consequences of our results, in particular some mapping properties of the Carlson-Shaffer convolution operator.     

A note on property of the Mittag-Leffler function

Recently the authors have found in some publications that the following property (0.1) of Mittag-Leffler function is taken for granted and used to derive other properties.

(0.1)