On the square integrability of the q-Hermite functions

Abstract

Overlap integrals over the full real line -infinity < x < infinity for a family of the q-Hermite functions H-n(sin kx \ q)e(-x2/2), 0 < q = e(-2k2) < 1 are evaluated. In particular, an explicit form of the squared norms for these q-extensions of the Hermite functions (or the wave functions of the linear harmonic oscillator in quantum mechanics) is obtained. The classical Fourier-Gauss transform connects the q-Hermite functions with different values 0 < q < 1 and q > 1 of the parameter q. An explicit expansion of the q-Hermite polynomials H-n(sin kx \ q) in terms of the Hermite polynomials H-n(x) emerges as a by-product. (C) 1998 Elsevier Science B.V. All rights reserved.