International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 4, Pages 737-740
doi:10.1155/S0161171291000996
Some formulas of L. Carlitz on Hermite polynomials
Department of Mathematics, University of Calcutta, Calcutta 700 019, India
Received 31 May 1991
Copyright © 1991 S. K. Chatterjea and S. M. Eaqub Ali. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We have used the idea of ‘quasi inner product’ introduced by L. R.
Bragg in 1986 to consider generating series
∑n=0∞Hn2(x)Hn2(y)tn22n(n!)2
studied by L. Carlitz in 1963. The pecularity of the series is that there is (n!)2
in the denominator, which has a striking deviation from the usuaI generating series
containing n! in the denominator. Our generating function for the said generating
series is quite different from that of Carlitz, but somewhat analogous to generating
integrals derived by G. N. Watson (Higher Transcendental function Vol.III, P 271-272
for the case of Legendre, Gegenbauer and Jacobi polynomials.