q-Analogues of the Fourier and Hankel Transforms

摘要

The purpose of the paper is to state two types of orthogonality relations for the q-Bessel functions, to show that the first type is immediately implied by the generating function, and to rewrite the first type as the second type by use of a simple, but possibly new symmetry for the functions. It then shows that the second type of orthogonality is, on the one hand, a limit case of the orthogonality for the little q-Jacobi polynomials, and, on the other hand, allows the Hankel transform inversion formula as a limit case for q approaching 1. It generalizes the two orthogonality relations to two equivalent formulas, which are respectively the q-analogues of Graf's addition formula and the Weber-Schafheitlin discontinuous integral. The special cases of the q-Fourier-cosine, and sine transforms are covered. Appendix A contains rigorous proofs of two limit results.