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About Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size (enumerative combinatorics), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems (algebraic combinatorics).
In \cite{Roman} page 25 we read that, a sequence $s_n(x)$ is Sheffer for $(g(t), f(t))$, for some invertible $g(t)$, if and only if
$$ s_n(x+y)=\sum\limits_{k=0}^{\infty}\binom nk p_k(y) s_{nk}(x),$$
for all $y$ in complex numbers, where $p_n(x)$ is associated to $f(t)$.
Noting to the fact that $e_q(x+y) \neq e_q(x)e_q(y)$, leads to conclude that $ s_{n,q}(x+y) \neq e_q(yt)s_{n,q}(x)$, and ,therefore, we do not have the $q$analogue of the identity above directly. Is it possible to express the $q$analogue of the above mentioned identity in any other way, or should we neglect such an identity for $q$Sheffer sequences at all?
Any contribution is appreciated in advance.
\bibitem{Roman}Roman S., Rota G. The umbral calculus. Advances Math. 1978;27:95–188.