Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2

Shifting Property for Riordan, Sheffer and Connection Constants Matrices

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


We study the shifting property of a matrix $ R = [r_{n,k}]_{n,k\geq0} $ and a sequence $ (h_n)_{n\in\mathbb{N} } $, i.e., the identity

\begin{displaymath}\sum_{k=0}^n r_{n,k} h_{k+1} = \sum_{k=0}^n r_{n+1,k+1} h_k \, , \end{displaymath}

when R is a Riordan matrix, a Sheffer matrix (exponential Riordan matrix), or a 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.

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(Concerned with sequences A000045 A000073 A000078 A000085 A000108 A000110 A000255 A000262 A000932 A000984 A001591 A001592 A001764 A002720 A002793 A004319 A006629 A006630 A006631 A007318 A007405 A008277 A008287 A025174 A027907 A035343 A040027 A049425 A063260 A063265 A066178 A079262 A102594 A102893 A105287 A122189 A131689 A132393 A171890 A213651 A213652 A230547 A233657 A236194.)

Received September 28 2016; revised versions received October 3 2016; July 6 2017; August 24 2017; August 25 2017. Published in Journal of Integer Sequences, August 31 2017.

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