Geodesic
Polynomial Expansions for Solutions of Higher-Order Bessel Heat Equation in Quantum Calculus
Fractional calculus and applied analysis, Tome 10 (2007) no. 1, pp. 39-58.

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In this paper we give the q-analogue of the higher-order Bessel operators studied by I. Dimovski [3],[4], I. Dimovski and V. Kiryakova [5],[6], M. I. Klyuchantsev [17], V. Kiryakova [15], [16], A. Fitouhi, N. H. Mahmoud and S. A. Ould Ahmed Mahmoud [8], and recently by many other authors. Our objective is twofold. First, using the q-Jackson integral and the q-derivative, we aim at establishing some properties of this function with proofs similar to the classical case. Second, our goal is to construct the associated q-Fourier transform and the q-analogue of the theory of the heat polynomials introduced by P. C. Rosenbloom and D. V. Widder [22]. For some value of the vector index, our operator generalizes the q-jα Bessel operator of the second order in [9] and a q-Third operator in [12].
Keywords: q-Analysis, q-Fourier Transform, q-Heat Equation, q-Laguerre Polynomials, q-Heat Polynomials, 33C10, 33D60, 26D15, 33D05, 33D15, 33D90 
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Ben Hammouda, M.S.; Nemri, Akram. Polynomial Expansions for Solutions of Higher-Order Bessel Heat Equation in Quantum Calculus. Fractional calculus and applied analysis, Tome 10 (2007) no. 1, pp. 39-58. http://geodesic.mathdoc.fr/item/FCAA_2007_10_1_a2/