TY - JOUR
T1 - Evaluation of Certain Integrals Involving the Product of Classical Hermite's Polynomials Using Laplace Transform Technique and Hypergeometric Approach
AU - M. I. Qureshi &  Saima Jabee
JO - Analysis in Theory and Applications
VL - 4
SP - 355
EP - 365
PY - 2017
DA - 2017/11
SN - 33
DO - http://doi.org/10.4208/ata.2017.v33.n4.5
UR - https://global-sci.org/intro/article_detail/ata/10702.html
KW - Gauss’s summation theorem, classical Hermite’s polynomials, generalized hypergeometric function, generalized Laguerre’s polynomials.
AB - <p style="text-align: justify;">In this paper some novel integrals associated with the product of classical
Hermite’s polynomials $$\int^{+∞}_{−∞} (x^2)^m exp(−x^2)\{H_r(x)\}^2dx,\ \int^∞_0
exp(−x^2)H_{2k} (x)H_{2s+1}(x)dx,$$ $$\int^∞_0
exp(−x^2
)H_{2k}(x)H_{2s}(x)dx&nbsp; \ \text{and}\ \int^∞_0
exp(−x^2)H_{2k+1}(x)H_{2s+1}(x)dx,$$ 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.</p>