The Umbral Calculus

Generating functions for unification of the multidimensional Bernstein polynomials and their applications

“…(cf Roman 29 ). Combining (32) and (34), we obtain the following combinatorial sum including the Stirling numbers of the second kind and the first kind Cauchy numbers:…”

Representations of Sheffer Polynomials

“…Various expansion formulas associated with this delta operator yield a powerful way to obtain identities for this class of polynomials (see [4;7,Chap. 1;8]). If (qn)n E N is of convolution type and deg qn = n, then there exists a unique delta operator Q for (qn)n E Nand (qn)n E N is said to be the basic set of Q. Sheffer polynomials, which are a generalization of polynomials of convolution type, are also considered in [4].…”

“…k=O A third definition (due to Roman [8]) is in terms of linear functionals on the space of polynomials: (sn)n E N is Sheffer if deg sn = n and there exist an invertible linear functional A (i.e., A 1 =1= 0) and a delta operator Q such that AQksn = 0kn' where 0kn is the Kronecker delta.…”

“…We now show that linear functionals can be represented by integrals. Roman strongly advocates the use of functionals instead of operators in the umbral calculus (see [8]). The above theorem can be used to prove another characterization theorem for Sheffer polynomials.…”

A note on the discrete Cauchy‐Kovalevskaya extension

“…In the present paper, we reformulate the approaches considered on both papers. The construction stated in Section 3 makes use of Roman's umbral calculus approach 9 and of the discrete Fourier theory considered by Gürlebeck and Sprößig on their former book, 10 following up author's recent contribution. 11 In Section 4, it was obtained an explicit space-time integral representation for the Cauchy-Kovalevskaya extension on the momentum space, based on Cauchy principal value representations for 1.1, and on a Laplace transform identity associated to the generalized Mittag-Leffler function (cf Samko et al 12, p. 21 )…”