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On the zeros of the Hahn-Exton \(q\)-Bessel function and associated \(q\)- Lommel polynomials. (English) Zbl 0811.33013

The Hahn-Exton \(q\)-Bessel function was in its general form originally introduced by H. Exton [Jñānabha, Sect. A 8, 49-56 (1978; Zbl 0468.33002)], and one of its more interesting properties is that of \(q\)- Fourier Bessel orthogonality. This study is devoted to results appertaining to the zeros of the functions under consideration. The proofs concerned rest upon an explicit evaluation of the \(q\)-Fourier Bessel integral and other formulae.
Associated \(q\)-Lommel polynomials connected with the Hahn-Exton \(q\)- Bessel function are then discussed. Comparisons with associated \(q\)- Lommel polynomials connected with the Jackson \(q\)-Bessel functions show that, in some respects, the latter function seems to be more closely related to the classical Bessel function. It is suggested that both of the \(q\)-analogues of the Bessel function are interesting each in its own way, and both possess elegant properties. Koelink’s study [Report W 91-26, University of Leiden (1991)] shows that a common generalization exists of the Jackson and Hahn-Exton \(q\)-Bessel functions.

MSC:

33D70 Other basic hypergeometric functions and integrals in several variables

Citations:

Zbl 0468.33002
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