On an extension of Riordan array and its application in the construction of convolution-type and Abel-type identities

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 { p n ( t ) } , where p n ( t ) is of degree n , we extend the ordinary Riordan array (resp. Riordan group) to a generalized Riordan array (resp. generalized Riordan group) associated with { p n ( 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.