Physical Sciences and Mathematics Commons^™

Composite Dilation Wavelets With High Degrees, Tian-Xiao He

Tian-Xiao He

No abstract provided.

Asymptotic Expansions And Computation Of Generalized Stirling Numbers And Generalized Stirling Functions, Tian-Xiao He

Tian-Xiao He

Here presented is a unified approach to Stirling numbers and their generalizations as well as generalized Stirling functions by using generalized factorial functions, k-Gamma functions, and generalized divided difference. Previous well-known extensions of Stirling numbers due to Riordan, Carlitz, Howard, Charalambides-Koutras, Gould-Hopper, Hsu-Shiue, Tsylova Todorov, Ahuja-Enneking, and Stirling functions introduced by Butzer and Hauss, Butzer, Kilbas, and Trujilloet and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations and generating functions are discussed. Some asymptotic expansions for the generalized Stirling functions and generalized Stirling numbers are established. …

Construction Of Spline Type Orthogonal Scaling Functions And Wavelets, Tian-Xiao He, Tung Nguyen, '15

Tian-Xiao He

No abstract provided.

Enumeration Problems For A Linear Congruence Equation, Tian-Xiao He, Wun-Seng Chou, Peter Shiue

Tian-Xiao He

Let m ≥ 2 and r ≥ 1 be integers and let c Є Zm = {0, 1, …,m ─ 1}. In this paper, we give an upper bound and a lower bound for the number of unordered solutions x1, …, xn Є Zm of the congruence x1 + x2 + ••• + xr ≡ c mod m. Exact formulae are also given when m or r is prime. This solution number involves the Catalan number or generalized Catalan number in some special cases. Moreover, the enumeration problem has interrelationship with the restricted integer partition.

Polynomials That Have Golden Ratio Zeros, Tian-Xiao He, Jack Maier, Kurt Vanness

Tian-Xiao He

When the golden ratio and its conjugate are zeros to a polynomial, two of the coefficients are functions of the Fibonacci sequence in terms of the other coefficients, which characterize the polynomial completely. These functions are used to derive some Fn, Ln, and golden ratio identities. In many cases, this is generalized to the Lucas sequences Un and Vn, with an associated quadratic root pair. Horadam sequences are produced in the series of linear and constant coefficients of the series of polynomials Having ra and rb zeros when all of the other coefficients are equal.

On An Extension Of Riordan Array And Its Application In The Construction Of Convolution-Type And Abel-Type Identities, Tian-Xiao He, Leetsch Hsu, Xing Ron Ma

Tian-Xiao He

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

Hyperbolic Expressions Of Polynomial Sequences And Parametric Number Sequences Defined By Linear Recurrence Relations Of Order 2, Tian-Xiao He, Peter J.-S. Shiue, Tsui-Wei Weng

Tian-Xiao He