A Novel Family of q-Mittag-Leffler-Based Bessel and Tricomi Functions via Umbral Approach

Many properties of special polynomials, such as recurrence relations, sum formulas, integral transforms and symmetric identities, have been studied in the literature with the help of generating functions and their functional equations. In this paper, we introduce hybrid forms of q-Mittag-Leffler functions. The q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are constructed using a q-symbolic operator. The generating functions, series definitions, q-derivative formulas and q-recurrence formulas for q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are obtained. The Nq-transforms and Nq-transforms of q-Mittag-Leffler–Bessel and q-Mittag-Leffler–Tricomi functions are obtained. These hybrid q-special functions are also studied by plotting their graphs for specific values of the indices and parameters.