Characterization of (c)-Riordan arrays, Gegenbauer-Humbert-type Polynomial Sequences, and (c)-Bell Polynomials

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