| Characterization of $(c)$-Riordan Arrays, Gegenbauer-Humbert-Type Polynomial Sequences, and $(c)$-Bell Polynomials |
| Received:December 07, 2012 Revised:February 18, 2013 |
| Key Words:
Riordan arrays $(c)$-Riordan arrays $A$-sequence $Z$-sequence $(c)$-Bell polynomials $(c)$-hitting-time subgroup.
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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. |
| Citation: |
| DOI:10.3770/j.issn:2095-2651.2013.05.001 |
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