Positivity of the generalized translation associated with the q-Hankel transform and applications

Abstract

A new formulation of the Graf-type addition formula related to the third Jackson q-Bessel function gives a solution for the problem of the positivity of the generalized q-translation operator associated with the q-Hankel transform. Next some applications in q-theory are treated, for instance the relationship between the q-Bessel- positive and negative-definite functions. We also show how the positivity of the q-Bessel translation operator plays a central role in q-Fourier analysis, namely in the study of Markov operators in the q-context. The paper concludes with the nonnegative product linearization of the q⁻²-Lommel polynomials.

Keywords:

• q-Bessel Fourier transform
• positivity of the generalized translation
• q-Bessel positive and negative definite
• Markov operators
• q-Lommel polynomials

Mathematics Subject Classification:

• 33D15
• 47A05

Acknowledgements

The author would like to thank the referees for their comments that help to improve the manuscript.