Journal of Integer Sequences, Vol. 26 (2023), Article 23.1.1

Two-Parameter Identities for q-Appell Polynomials

Emanuele Munarini
Dipartimento di Matematica
Politecnico di Milano
Piazza Leonardo da Vinci 32
20133 Milano


In this paper, by using the techniques of the q-exponential generating series, we extend a well-known two-parameter identity for the Appell polynomials to the q-Appell polynomials of type I and II. More precisely, we obtain two different q-analogues of such an identity. Then, we specialize these identities for some q-polynomials arising in combinatorics, in q-calculus or in the theory of orthogonal polynomials. In particular, we consider the generalized q-Bernoulli and q-Euler polynomials and then we deduce some further identities involving the Bernoulli and Euler numbers. In this way, as a byproduct, we derive the symmetric Carlitz identity for the Bernoulli numbers. Finally, we find a (non-symmetric) q-analogue of Carlitz's identity involving the q-Bernoulli numbers of type I and II.

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(Concerned with sequences A000166 A000364 A000522 A001586 A028296 A122045.)

Received December 3 2021; revised version received March 7 2023. Published in Journal of Integer Sequences, March 8 2023.

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