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Two-Parameter Identities for ***q*-Appell Polynomials

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Emanuele Munarini

Dipartimento di Matematica

Politecnico di Milano

Piazza Leonardo da Vinci 32

20133 Milano

Italy

**Abstract:**

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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