Some expansion formulae for G-function involving bessel functions

In this paper we have derived some expansions for G-function involving Bessel functions with the help of the orthogonality property of Bessel functions.     

Some properties of generalized associated Legendre functions of second kind

We investigate some new properties of generalized associated Legendre polynomials of the second kind, establish new relationships between these polynomials, construct differential operators with the functions P _k ^m,n (z), Q _k ^m,n (z), and describe some applications.Kiev Polytechnical Institute. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 34–45, 1989.     

Two definite integrals involving products of four Legendre functions

The definite integrals \( \int _{-1}^1x[P_\nu (x)]^4\mathrm{d}x\) and \( \int _{0}^1x[P_\nu (x)]^2\{[P_\nu (x)]^2-[P_\nu (-x)]^2\}\mathrm{d}x\) are evaluated in closed form, where \( P_\nu \) stands for the Legendre function of degree \( \nu \in \mathbb C\). Special cases of these integral formulae have appeared in arithmetic studies of automorphic Green’s functions and Epstein zeta functions.     

Some alternative principles and mixed minimax theorems involving two functions

Some alternative principles and quantitative-topological minimax theorems involving two functions are given. Consequently, Geraghty–Lin's minimax theorem and Terkelsen's minimax theorem are generalized. This revised version was published online in June 2006 with corrections to the Cover Date.     

New generating functions involving several complex variables

In the present paper the authors derive a number of new classes of generating functions for certain multiple sequences of functions of several complex variables. These generating-function relationships involve either Taylor or Laurent series and provide extensions of several known results given earlier by H.M. Srivastava, R.G. Buschman, and D. Zeitlin. The authors also apply their main results to obtain the corresponding generating functions for the generalized (Lauricella's) hypergeometric functions of several complex variables, which were introduced elsewhere by H.M. Srivastava and M.C. Daoust.     

Some estimates for the error in mixed Fourier-Bessel expansions of functions of two variables

Some issues concerning expansions of functions of two variables in mixed Fourier-Bessel series are considered. In particular, the rate of their convergence in the classes of functions characterized by generalized moduli of continuity are estimated, and estimates of the remainder terms are obtained.     

Some results involving Bessel function and Fox’s H-function of two variables

In this paper, we have presented two integrals and employed them to establish one Fourier Bessel expansion for Fox’s H-function of two variables.     

Some identities involving theta functions

Let (|q|<1). For k∈N it is shown that there exist k rational numbers A(k,0),…,A(k,k−1) such that

     

Hypergeometric functions of two variables

Summary The systematic investigation of contour integrals satisfying the system of partial differential equations associated with Appell's hypergeometric functionF ₁ leads to new solutions of that system. Fundamental sets of solutions are given for the vicinity of all singular points of the system of partial differential equations. The transformation theory of the solutions reveals connections between the system under consideration and other hypergeometric systems of partial differential equations. Presently it is discovered that any hypergeometric system of partial differential equations of the second order (with two independent variables) which has only three linearly independent solutions can be transformed into the system ofF ₁ or into a particular or limiting case of this system. There are also other hypergeometric systems (with four linearly independent solutions) the integration of which can be reduced to the integration of the system ofF ₁.     

Multiple integrals involving theH-function of two variables

In this paper, we establish two multiple integrals involving the products of generalised hypergeometric function,H-function of one and two variables. The integrals evaluated are very general in nature and generalise the well-known integrals due to Mittal and Gupta, Goyal, Olkha, Dahiya and others. Some interesting integrals involving the products of elementary special functions and orthogonal polynomials have also been obtained as particular cases of the main results.     

Jensen's operator inequality for functions of two variables

The operator convex functions of two variables are characterized in terms of a non-commutative generalization of Jensen's inequality.

     

On an integral transform with generalized associated Legendre functions

We study some new properties of generalized associated Legendre functions of first and second kind P _k ^m,n (z) and Q _k ^m,n (z). Applying these functions, we introduce an integral transform that can be used in solving boundary-value problems of mathematical physics.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 33–43.