Tian-Xiao He

Sequence characterization of Riordan arrays

In the realm of the Riordan group, we consider the characterization of Riordan arrays by means of the A- and Z-sequences. It corresponds to a horizontal construction of a Riordan array, whereas the traditional approach is through column generating functions. We show how the A- and Z-sequences of the product of two Riordan arrays are derived from those of the two factors; similar results are obtained for the inverse. We also show how the sequence characterization is applied to construct easily a Riordan array. Finally, we give the characterizations relative to some subgroups of the Riordan group, in particular, of the hitting-time subgroup.

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.

Bivariate C 1 Quadratic Finite Elements and Vertex Splines

Following work of Heindl and of Powell and Sabin, each triangle of an arbitrary (regular) triangulation A of a polygonal region ii in R is subdi- vided into twelve triangles, using the three medians, yielding the refinement A of A , so that C quadratic finite elements can be constructed. In this paper, we derive the Bezier nets of these elements in terms of the parameters that de- scribe function and first partial derivative values at the vertices and values of the normal derivatives at the midpoints of the edges of A. Consequently, bivariate C quadratic (generalized) vertex splines on A have an explicit formulation. Here, a generalized vertex spline is one which is a piecewise polynomial on the refined grid partition A and has support that contains at most one vertex of the original partition A in its interior. The collection of all C quadratic general- ized vertex splines on A so constructed is shown to form a basis of S2 (A), the vector space of all functions on C (I2) whose restrictions to each triangular cell of the partition A are quadratic polynomials. A subspace with the basis given by appropriately chosen generalized vertex splines with exactly one vertex of A in the interior of their supports, that reproduces all quadratic polynomials, is identified, and hence, has approximation order three. Quasi-interpolation for- mulas using this subspace are obtained. In addition, a constructive procedure that yields a locally supported basis of yet another subspace with dimension given by the number of vertices of A, that has approximation order three, is given.

On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2

Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a general method to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.

Eulerian polynomials and B-splines

An interrelationship between Eulerian polynomials, Eulerian fractions and Euler-Frobe nius polynomials, Euler-Frobenius fractions, and B-splines is presented. The properties of Eulerian polynomials and Eulerian fractions and their applications in B-spline interpolation and evaluation of Riemann zeta function values at odd integers are given. The relation between Eulerian numbers and B-spline values at knot points are also discussed.

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.

Construction of boundary quadrature formulas using wavelets

This paper gives a refinement of a general method for the construction of multivariate numerical integration formulas that use merely boundary points as evaluation points. Boundary quadrature formulas are constructed by using the optimal dimensionality-reducing expansion and quadrature formulas for the integrals of periodic functions with wavelet weights. Boundary quadrature formulas are also used to solve boundary value problems of partial differential equations.

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.