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The Sheffer group and the Riordan group. (English) Zbl 1123.05007
Summary: We define the Sheffer group of all Sheffer-type polynomials and prove the isomorphism between the Sheffer group and the Riordan group. An equivalence of the Riordan array pair and generalized Stirling number pair is also presented. Finally, we discuss a higher dimensional extension of Riordan array pairs.

MSC:
05A15 Exact enumeration problems, generating functions
05E15 Combinatorial aspects of groups and algebras (MSC2010)
11B73 Bell and Stirling numbers
11B83 Special sequences and polynomials
13F25 Formal power series rings
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
05A40 Umbral calculus
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