Characterization of $(c)$-Riordan Arrays, Gegenbauer-Humbert-Type Polynomial Sequences, and $(c)$-Bell Polynomials
Received:December 07, 2012  Revised:February 18, 2013
Key Word: Riordan arrays   $(c)$-Riordan arrays   $A$-sequence   $Z$-sequence   $(c)$-Bell polynomials   $(c)$-hitting-time subgroup.  
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Author NameAffiliation
Henry W. GOULD Department of Mathematics, West Virginia University, Morgantown, WV $26505$, USA 
Tianxiao HE Department of Mathematics, Illinois Wesleyan University, Bloomington, IL 61702, USA 
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      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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