Some formulas of L. Carlitz on Hermite polynomials

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.