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On An Extension Of Riordan Array And Its Application In The Construction Of Convolution-Type And Abel-Type Identities, Tian-Xiao He, Leetsch Hsu, Xing Ron Ma
On An Extension Of Riordan Array And Its Application In The Construction Of Convolution-Type And Abel-Type Identities, Tian-Xiao He, Leetsch Hsu, Xing Ron Ma
Tian-Xiao He
Using the basic fact that any formal power series over the real or complex number field can always be expressed in terms of given polynomials {pn(t)}{pn(t)}, where pn(t)pn(t) is of degree nn, we extend the ordinary Riordan array (resp. Riordan group) to a generalized Riordan array (resp. generalized Riordan group) associated with {pn(t)}{pn(t)}. As new application of the latter, a rather general Vandermonde-type convolution formula and certain of its particular forms are presented. The construction of the Abel type identities using the generalized Riordan arrays is also discussed.
The Sheffer Group And The Riordan Group, Tian-Xiao He, Peter J.S. Shiue, Leetsch C. Hsu
The Sheffer Group And The Riordan Group, Tian-Xiao He, Peter J.S. Shiue, Leetsch C. Hsu
Tian-Xiao He
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.