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On the square integrability of the q-Hermite functions

Authors
Journal
Journal of Computational and Applied Mathematics
0377-0427
Publisher
Elsevier
Publication Date
Volume
99
Identifiers
DOI: 10.1016/s0377-0427(98)00142-3
Disciplines
  • Physics

Abstract

Abstract Overlap integrals over the full real line −∞< x<∞ for a family of the q-Hermite functions H n( sinκx¦q) e −x 2 2 , 0< q=e −2 κ 2<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( sinκx¦q) in terms of the Hermite polynomials H n ( x) emerges as a by-product.

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