An Algebraic Exposition of Umbral Calculus with Application to General Linear Interpolation Problem – a Survey

A systematic exposition of Sheffer polynomial sequences via determinantal form is given. A general linear interpolation problem related to Sheffer sequences is considered. It generalizes many known cases of linear interpolation. Numerical examples and conclusions close the paper. 1. The modern umbral calculus In the 1970s Rota and his collaborators [17,19,20] began to construct a completely rigorous foundation for the classical umbral calculus, consisted primarily of a symbolic technique for the manipulation of numerical and polynomial sequences. The theory of Rota et al. was based on the relatively modern ideas of linear functional, linear operator and adjoint. This theory followed that less efficient of generating function methods; in fact, Appell [1], Sheffer [22] and Steffensen [23] based their theories on formal power series. These theories can be criticized both for their formalism not suitable for nonspecialists and for insufficient computational tools. The umbral calculus, because of its numerous applications in many branches of mathematics, physics, chemistry and engineering [24], has received many attentions from researchers. Recently, Di Bucchianico and Loeb [14] summarized and documented more than five hundred old and new findings related to umbral calculus. In last years attention has centered on finding novel approaches. For example, in [21] the connection between Sheffer polynomials and Riordan array is sketched and in [16] the isomorphism between the Sheffer group and the Riordan Group is proved. In [5,27] two different matrix approaches to Appell polynomials are given, in [9,26], these methods have been extended to Sheffer polynomials, and in [11], to binomial polynomial sequences. Recently, the relation between the umbral calculus and the general linear interpolation problem has been highlighted [6–8,10,11]. In this survey we give a unitary matrix approach to Sheffer polynomials, including Appell and binomial type polynomial sequences. Our theory of Sheffer sequences assumes binomial type polynomial sequences, therefore the paper is organized as 2010 Mathematics Subject Classification: 11B83; 65F40.