Some formulae for the \(q\)-Bernoulli and Euler polynomials of higher order
Let \(q\in \mathbb C_p\), \(|q-1|_p<p^{-1/(p-1)}\). The author defines higher \(q\)-Bernoulli numbers via a \(q\)-analog of the Volkenborn integral [see also the author, Arch. Math. 76, 190--195 (2001; Zbl 0986.11010)], by analogy with a well-known representation for classical Bernoulli numbers. In a similar way, the \(q\)-Euler polynomials and \(q\)-Euler numbers are defined. Some formulas for sums of their products are proved. For the higher \(q\)-Bernoulli numbers, a Kummer-type congruence is obtained.
scientific article; zbMATH DE number 1866610