Characterization of $(c)$-Riordan Arrays, Gegenbauer-Humbert-Type Polynomial Sequences, and $(c)$-Bell Polynomials |
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Authors: | Henry W GOULD and Tianxiao HE |
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Institution: | Department of Mathematics, West Virginia University, Morgantown, WV $26505$, USA;Department of Mathematics, Illinois Wesleyan University, Bloomington, IL 61702, USA |
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Abstract: | Here presented are the definitions of $(c)$-Riordan arrays and $(c)$-Bell polynomials which are extensions of the classical Riordan arrays and Bell polynomials. The characterization of $(c)$-Riordan arrays by means of the $A$- and $Z$-sequences is given, which corresponds to a horizontal construction of a $(c)$-Riordan array rather than its definition approach through column generating functions. There exists a one-to-one correspondence between Gegenbauer-Humbert-type polynomial sequences and the set of $(c)$-Riordan arrays, which generates the sequence characterization of Gegenbauer-Humbert-type polynomial sequences. The sequence characterization is applied to construct readily a $(c)$-Riordan array. In addition, subgrouping of $(c)$-Riordan arrays by using the characterizations is discussed. The $(c)$-Bell polynomials and its identities by means of convolution families are also studied. Finally, the characterization of $(c)$-Riordan arrays in terms of the convolution families and $(c)$-Bell polynomials is presented. |
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Keywords: | Riordan arrays $(c)$-Riordan arrays $A$-sequence $Z$-sequence $(c)$-Bell polynomials $(c)$-hitting-time subgroup |
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