On some properties of the bibasic Humbert hypergeometric functions Ξ_(1) and Ξ_(2)

The main object of this paper is to deduce the bibasic Humbert functions Ξ_(1) and Ξ_(2)Some interesting results and elementary summations technique that was successfully employed,q-recursion,q-derivatives relations,the q-differential recursion relations,the q-integral representations for Ξ_(1) and Ξ_(2)are given.The summation formula derives a list of p-analogues of transformation formulas for bibasic Humbert functions that have been studied,also some hypergeometric functions properties of some new interesting special cases have been given.

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.

A Subclass of Harmonic Functions Associated with Wright’s Hypergeometric Functions

We introduce a new class of complex valued harmonic functions associated with Wright hypergeometric functions which are orientation preserving and univalent in the open unit disc. Further we define, Wright generalized operator on harmonic function and investigate the coefficient bounds, distortion inequalities and extreme points for this generalized class of functions.

Wright Type Hypergeometric Function and Its Properties

Let s and z be complex variables, Γ(s) be the Gamma function, and for any complex v be the generalized Pochhammer symbol. Wright Type Hypergeometric Function is defined (Virchenko et al. [1]), as: where which is a direct generalization of classical Gauss Hypergeometric Function 2F1(a,b;c;z). The principal aim of this paper is to study the various properties of this Wright type hypergeometric function 2R1(a,b;c;τ;z);which includes differentiation and integration, representation in terms of pFq and in terms of Mellin-Barnes type integral. Euler (Beta) transforms, Laplace transform, Mellin transform, Whittaker transform have also been obtained;along with its relationship with Fox H-function and Wright hypergeometric function.

Decomposition Formulas of Kampé de Fériets Double Hypergeometric Functions

This paper is sequel to the authors paper [18]. By using the generalized Burchnall-Chaundy operator method, the authors are aiming at deriving certain decomposition formulas for some interesting special cases of Kampe de Feriets series of double hypergeometric series F;.

Solution of Some Integral Equations Involving Confluent <i>k</i>-Hypergeometric Functions

The principle aim of this research article is to investigate the properties of k-fractional integration introduced and defined by Mubeen and Habibullah [1],and secondly to solve the integral equation of the form , for?k > 0, β > 0, y > 0, 0 is the confluent k-hypergeometric functions, by using k-fractional integration.

Some Formulas Involving Hypergeometric Functions in Four Variables

Several(generalized)hypergeometric functions and a variety of their extensions have been presented and investigated in the literature by many authors.In the present paper,we investigate four new hypergeometric functions in four variables and then establish several recursion formulas for these new functions.Also,some interesting particular cases and consequences of our results are discussed.

Generalized Form of Hermite Matrix Polynomials via the Hypergeometric Matrix Function

The object of this paper is to present a new generalization of the Hermite matrix polynomials by means of the hypergeometric matrix function. An integral representation, differential recurrence relation and some other properties of these generalized forms are established here. Moreover, some new properties of the Hermite and Chebyshev matrix polynomials are obtained. In particular, the two-variable and two-index Chebyshev matrix polynomials of two matrices are presented.

APPLICATIONS OF HYPERGEOMETRIC SUMMATION THEOREMS OF KUMMER AND DIXON INVOLVING DOUBLE SERIES

Using series iteration techniques identities and apply each of these identities in we derive a number of general double series order to deduce several hypergeometric reduction formulas involving the Srivastava-Daoust double hypergeometric function. The results presented in this article are based essentially upon the hypergeometric summation theorems of Kummer and Dixon.

Kummer’s 24 Solutions of the Hypergeometric Differential Equation with the Aid of Fractional Calculus

We know that the hypergeometric function, which is a solution of the hypergeometric differential equation, is expressed in terms of the Riemann-Liouville fractional derivative (fD). The solution of the differential equation obtained by the Euler method takes the form of an integral, which is confirmed to be expressed in terms of the Riemann-Liouville fD of a function. We can rewrite this derivation such that we obtain the solution in the form of the Riemann-Liouville fD of a function. We present a derivation of Kummer’s 24 solutions of the hypergeometric differential equation by this method.

第二类勒让德函数的重规格化及其优化

椭球谐函数级数展开是地球重力场椭球谐建模的基础。然而,处理椭球谐函数级数的主要困难在于计算第二类勒让德函数。而Jekeli的重规格化方法简化了这一计算过程。本文在Jekeli重规格化的基础上,详细推导了两种基于高斯超几何函数变换的优化递归方法,同时利用这两种优化递归的方法计算了第二类勒让德函数,并将其展开到二阶导数。通过数值计算,证明了优化递归方法可以有效加快收敛,缩短计算时间,并且适用阶数更高,这使得椭球谐函数级数在实际应用中更加方便、可行。

Generalized Gr?tzsch ring function and generalized elliptic integrals

Abstract. In this paper, we study the quotient of hypergeometric functions μα （r） in the theory of Ramanujan＇s generalized modular equation for α ∈（0, 1/2]. Several new inequalities are given for this and related functions. Our main results complement and generalize some known results in the literature.

Analytical Evaluation of Non-Elementary Integrals Involving Some Exponential, Hyperbolic and Trigonometric Elementary Functions and Derivation of New Probability Measures Generalizing the Gamma-Type and Gaussian-Type Distributions

The non-elementary integrals involving elementary exponential, hyperbolic and trigonometric functions, and where α,η and β are real or complex constants are evaluated in terms of the confluent hypergeometric function 1F1 and the hypergeometric function 1F2. The hyperbolic and Euler identities are used to derive some identities involving exponential, hyperbolic, trigonometric functions and the hypergeometric functions 1F1 and 1F2. Having evaluated, these non-elementary integrals, some new probability measures generalizing the gamma-type and Gaussian distributions are also obtained. The obtained generalized probability distributions may, for example, allow to perform better statistical tests than those already known (e.g. chi-square (x2) statistical tests and other statistical tests constructed based on the central limit theorem (CLT)), while avoiding the use of computational approximations (or methods) which are in general expensive and associated with numerical errors.

GENERALIZED FRACTIONAL CALCULUS OF THE ALEPH-FUNCTION INVOLVING A GENERAL CLASS OF POLYNOMIALS

The object of this article is to study and develop the generalized fractional calcu- lus operators given by Saigo and Maeda in 1996. We establish generalized fractional calculus formulas involving the product of R-function, Appell function F3 and a general class of poly- nomials. The results obtained provide unification and extension of the results given by Saxena et al. [13], Srivastava and Grag [17], Srivastava et al. [20], and etc. The results are obtained in compact form and are useful in preparing some tables of operators of fractional calculus. On account of the general nature of the Saigo-Maeda operators, R-function, and a general class of polynomials a large number of new and known results involving Saigo fractional calculus operators and several special functions notably H-function, /-function, Mittag-Leffier function, generalized Wright hypergeometric function, generalized Bessel-Maitland function follow as special cases of our main findings.

THE SPHERICAL FUNCTIONS AND THE CENTRAL LIMIT THEOREM ON SU(2,2)/S(U(2)×U(2))

Let G = SU（2, 2）, K = S（U（2） × U（2））, and for l ∈ Z, let {Tl}l∈z be a one-dimensional K-type and let El be the line bundle over G/K associated to Tl. It is shown that the Tl-spherical function on G is given by the hypergeometric functions of several variables. By applying this result, a central limit theorem for the space G/K is obtained.

CERTAIN SUBCLASS OF p-VALENT MEROMORPHIC ANALYTIC FUNCTIONS INVOLVING CERTAIN INTEGRAL OPERATOR

The purpose of the present paper is to introduce a new subclass of p-valent meromorphic functions by using certain integral operator and to investigate various properties for this subclass.

Applications of Multivalent Functions Associated with Generalized Fractional Integral Operator

By using a method based upon the Briot-Bouquet differential subordination, we investigate some subordination properties of the generalized fractional integral operator which was defined by Owa, Saigo and Srivastava [1]. Some interesting further consequences are also considered.

Some Wgh Inequalities for Univalent Harmonic Analytic Functions

In this paper, some Wgh inequalities for univalent harmonic analytic functions defined by Wright's generalized hypergeometric (Wgh) functions to be in certain classes are observed and proved. Some consequent results are also discussed.

On <i>p</i>and <i>q</i>-Horn’s Matrix Function of Two Complex Variables

The main aim of this paper is to define and study of a new Horn’s matrix function, say, the p and q-Horn’s matrix function of two complex variables. The radius of regularity on this function is given when the positive integers p and q are greater than one, an integral representation of pHq 2 is obtained, recurrence relations are established. Finally, we obtain a higher order partial differential equation satisfied by the p and q-Horn’s matrix function.

Certain <i>pl</i>(<i>m</i>,<i>n</i>)-Kummer Matrix Function of Two Complex Variables under Differential Operator

The main aim of this paper is to define and study of a new matrix functions, say, the pl(m,n)-Kummer matrix function of two complex variables. The radius of regularity, recurrence relation and several new results on this function are established when the positive integers p is greater than one. Finally, we obtain a higher order partial differential equation satisfied by the pl(m,n)-Kummer matrix function and some special properties.