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\vskip 1cm{\LARGE\bf 
Shifting Property for Riordan, Sheffer and \\
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Connection Constants Matrices
}
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\large
Emanuele Munarini\footnote{The work is partially supported by MIUR (Ministero dell'Istruzione, dell'Universit\`a e della Ricerca).}\\
Politecnico di Milano\\
Dipartimento di Matematica\\
Piazza Leonardo da Vinci 32 \\
20133 Milano \\
Italy\\
\href{mailto:emanuele.munarini@polimi.it}{\tt emanuele.munarini@polimi.it}
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\begin{abstract}
 We study the \emph{shifting property} of
 a matrix $ R = [r_{n,k}]_{n,k\geq0} $ and a sequence $ (h_n)_{n\in\NN} $,
 i.e., the identity
 $$ \sum_{k=0}^n r_{n,k} h_{k+1} = \sum_{k=0}^n r_{n+1,k+1} h_k \, , $$
 when $ R $ is a \emph{Riordan matrix}, a \emph{Sheffer matrix} (exponential Riordan matrix),
 or a \emph{connection constants matrix} (involving symmetric functions and continuants).
 Moreover, we consider the shifting identity for several sequences of combinatorial interest,
 such as the binomial coefficients, the polynomial coefficients,
 the Stirling numbers (and their $q$-analogues), the Lah numbers, the De Morgan numbers,
 the generalized Fibonacci numbers, the Bell numbers, the involutions numbers,
 the Chebyshev polynomials, the Stirling polynomials, the Hermite polynomials,
 the Gaussian coefficients, and the $q$-Fibonacci numbers.
\end{abstract}


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