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 ∫-∞＋∞（x2）mexp（-x2）{Hr（x）}2dx,∫0∞exp（-x2）H2k（x）H2s＋1（x）dx,∫0∞exp（-x2）H2k（x）H2s（x）dx and ∫0∞exp（-x2）H2k＋1（x）H2s＋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 spe- cial 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.

Some Summation Theorems for Truncated Clausen Series and Applications

The main aim of this paper is to derive some new summation theorems for terminating and truncated Clausen’s hypergeometric series with unit argument,when one numerator parameter and one denominator parameter are negative integers.Further,using our truncated summation theorems,we obtain the Mellin transforms of the product of exponential function and Goursat’s truncated hypergeometric function.