Generalized \(q\)-Bernoulli polynomials generated by Jackson \(q\)-Bessel functions (Q2136152)

The purpose of this article is to introduce new generalized \(q\)-Bernoulli polynomials. The \(g_{\alpha}\) functions are products of a \(\Gamma_q\) function and a \(q\)-Bessel function. These generalized \(q\)-Bernoulli polynomials, some of them \(q\)-Appell polynomials, are generated by the \(g_{\alpha}\) functions in the denominators. Explicit sum expressions for these polynomials are proved. Of the recurrence formulas, (2.16,17) follow because of properties of \(q\)-Appell polynomials [\textit{T. Ernst}, A comprehensive treatment of \(q\)-calculus. Basel: Birkhäuser (2012; Zbl 1256.33001), 4.107]. Connection relations between \(q\)-Bernoulli polynomials with Al-Salam polynomials as coefficients are proved. Several complicated formulas are proved by using the Cauchy product formula. Theorem 2.17 expresses powers of \(x\) as finite sums with \(q\)-binomial coefficients and \(q\)-Bernoulli polynomials, similar to the formula [loc. cit., 4.149]. Asymptotic formulas for \(\alpha\to\infty\) are given. Finally, connection relations between one of these generalized \(q\)-Bernoulli polynomials and the \(q\)-Laguerre and the little \(q\)-Legendre polynomials are given.