Leetsch C. Hsu

The Sheffer group and the Riordan group

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.

A symbolic operator approach to several summation formulas for power series II

Here expounded is a kind of symbolic operator method that can be used to construct many transformation formulas and summation formulas for various types of power series including some old ones and more new ones.

On certain summation problems and generalizations of Eulerian polynomials and numbers

Abstract The main purpose of this paper is to introduce two kinds of generalized Eulerian polynomials and numbers which are associated with the constructions of certain summation formulas for extended arithmetic-geometric series. Also we present some basic formulas related to certain generalized Eulerian numbers, Carlitz-Howards degenerated Stirling numbers and Dickson-Stirling numbers.

Cycle indicators and Special Functions

Abstract. It is shown that a class of special functions (involving Sheffer-type polynomials and Gegenbauer-Humbert-type polynomials) could have cycle indicator representations and some recurrence relations. This implies the conclusion that classical special functions with simple logarithms of generating functions can be classified this way. This paper is mainly devoted to establish various relations and identities for special functions and remarkable number sequences.

Symbolization of generating functions; an application of the Mullin-Rota theory of binomial enumeration

We have found that there are more than a dozen classical generating functions that could be suitably symbolized to yield various symbolic sum formulas by employing the Mullin-Rota theory of binomial enumeration. Various special formulas and identities involving well-known number sequences or polynomial sequences are presented as illustrative examples. The convergence of the symbolic summations is discussed.

A pair of operator summation formulas and their applications

Two types of symbolic summation formulas are reformulated using an extension of Mullin-Rotas substitution rule in [R. Mullin, G.-C. Rota, On the foundations of combinatorial theory: III. Theory of binomial enumeration, in: B. Harris (Ed.), Graph Theory and its Applications, Academic Press, New York, London, 1970, pp. 167-213], and several applications involving various special formulas and identities are presented as illustrative examples.

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 eld can always be expressed in terms of given polynomials fpn(t)g, where pn(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 fpn(t)g. 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.

Convergence of the summation formulas constructed by using a symbolic operator approach

This paper deals with the convergence of the summation of power series of the form @S@a@?@k(@k)@g^@k, where 0 @? @a @? b @? ~, and { (@k)} is a given sequence of numbers with @k @e (@a, b) or(t) a differentiable function defined on (a, b). Here, the summation is found by using the symbolic operator approach shown in [1]. We will give a different type of the remainder of the summation formulas. The convergence of the corresponding power series will be determined consequently. Several examples such as the generalized Eulers transformation series will also be given. In addition, we will compare the convergence of the given series transforms.

On a Class of Combinatorial Sums Involving Generalized Factorials

The object of this paper is to show that generalized Stirling numbers can be effectively used to evaluate a class of combinatorial sums involving generalized factorials.

On an Extension of Abel-Gontscharoff's Expansion Formula

We present a constructive generalization of Abel-Gontscharoff’s series expansion to higher dimensions. A constructive application to a problem of multivariate interpolation is also investigated. In addition, two algorithms for constructing the basis functions of the interpolants are given.