Evaluation of Certain Integrals Involving the Product of Classical Hermite's Polynomials Using Laplace Transform Technique and Hypergeometric Approach

In this paper some novel integrals associated with the product of classical Hermite's polynomials ∫_(-∞)~(+∞)(x~2)~mexp(-x~2){Hr (x)}2dx, ∫_0~∞exp(-x~2)H_(2k)(x)H_(2s+1)(x)dx,∫_0~∞exp(-x~2)H_(2k)(x)H_(2s)(x)dx and ∫_0~∞exp(-x~2)H_(2k+1)(x)H_(2s+1)(x)dx,are evaluated using hypergeometric approach and Laplace transform method, which is a different approach from the approaches given by the other authors in the field of special functions. Also the results may be of significant nature, and may yield numerous other interesting integrals involving the product of classical Hermite's polynomials by suitable simplifications of arbitrary parameters.     

两类Kampéde Fériet级数的简化和求和公式

基于两个₃F₂-超几何级数求和公式,该文建立了两个一般的双变量级数变换公式.在经典超几何级数求和公式的帮助下,这两个变换公式变形出一系列形如F_q:1;0^p:2;1的Kampéde Fériet级数求和公式.另外,利用四个Saalschützian₄F₃ [1]-求和公式,一些形如F_q:1;1^p:2;2的Kampéde Fériet级数的简化和变换公式也被推导出来.     

A certain class of incomplete elliptic integrals and associated definite integrals

In many seemingly diverse physical contexts (including, for example, certain radiation field problems, studies of crystallographic minimal surfaces, the theory of scattering of acoustic or electromagnetic waves by means of an elliptic disk, studies of elliptical crack problems in fracture mechanics, and so on), a remarkably large number of general families of elliptic-type integrals, and indeed also many definite integrals of such families with respect to their modulus (or complementary modulus), are known to arise naturally. Motivated essentially by these and many other potential avenues of their applications, we present here a systematic account of the theory of a certain family of incomplete elliptic integrals in a unified and generalized manner. By means of the familiar Riemann–Liouville fractional differintegral operators, we obtain several explicit hypergeometric representations and apply these representations with a view to deriving various associated definite integrals, not only with respect to the modulus (or complementary modulus), but also with respect to the amplitude of the incomplete elliptic integrals involved therein.     

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.     

Some families of generating functions for a class of bivariate polynomials

The main purpose of this paper is to present various families of generating functions for a class of polynomials in two variables. Furthermore, several general classes of bilinear, bilateral or mixed multilateral generating functions are obtained for these polynomials.     

Reduction and transformation formulae for bivariate basic hypergeometric series

The main object of this paper is to establish several bivariate basic hypergeometric series identities by means of elementary series manipulation. Some of them can be applied to yield transformation and reduction formulae for q-Kampé de Fériet functions.