This paper deals with firstly the analogue of the Bochner theorem related to q-Bessel function of the third kind and secondly we give a new proof of the analogue of Bernstein's theorem via the q-Taylor formula. Finally we show that the q-Bessel positive definite and D q -completely monotonic functions are linked.
In this paper we study the positivity of the generalized q-translation associated with the q-Bessel Hahn Exton function which is deduced by a new formulation of the Graf's addition formula related to this function.
Spectral theory from the second-orderq-difference operatorΔqis developed. We give an integral representation of its inverse, and the resolvent operator is obtained. As application, we give an analogue of the Poincare inequality. We introduce the Zeta function for the operatorΔqand we formulate some of its properties. In the end, we obtain the spectral measure.
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