Physical Sciences and Mathematics Commons^™

Application Of Polynomial Interpolation In The Chinese Remainder Problem, Tian-Xiao He, S. Macdonald, P. J.-S. Shiue

Tian-Xiao He

Applications Of Riordan Matrix Functions To Bernoulli And Euler Polynomials, Tian-Xiao He

Tian-Xiao He

Shift Operators Defined In The Riordan Group And Their Applications, Tian-Xiao He

Tian-Xiao He

Construction Of Nonlinear Expression For Recursive Number Sequences, Tian-Xiao He

Tian-Xiao He

The Pascal Matrix Function And Its Applications To Bernoulli Numbers And Bernoulli Polynomials And Euler Numbers And Euler Polynomials, Tian-Xiao He, Jeff Liao, Peter Shiue

Tian-Xiao He

Wavelet Analysis And Applications In Economics And Finance, Tian-Xiao He, Tung Nguyen, '15

Tian-Xiao He

Convexity Of Spherical Bernstein-B´Ezier Patches And Circular Bernstein-B´Ezier Curves, Tian-Xiao He, Ram Mohapatray

Tian-Xiao He

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

Tian-Xiao He

No abstract provided.

Frames And Spline Framelets, Tian-Xiao He, Tung Nguyen, '15, Nahee Kim, '15

Tian-Xiao He

No abstract provided.

Parametric Catalan Numbers And Catalan Triangles, Tian-Xiao He

Tian-Xiao He

Here presented a generalization of Catalan numbers and Catalan triangles associated with two parameters based on the sequence characterization of Bell-type Riordan arrays. Among the generalized Catalan numbers, a class of large generalized Catalan numbers and a class of small generalized Catalan numbers are defined, which can be considered as an extension of large Schroder numbers and small Schroder numbers, respectively. Using the characterization sequences of Bell-type Riordan arrays, some properties and expressions including the Taylor expansions of generalized Catalan numbers are given. A few combinatorial interpretations of the generalized Catalan numbers are also provided. Finally, a generalized Motzkin numbers …

A Unified Approach To Generalized Stirling Functions, Tian-Xiao He

Tian-Xiao He

Here presented is a unified approach to generalized Stirling functions by using generalized factorial functions, $k$-Gamma functions, generalized divided difference, and the unified expression of Stirling numbers defined in \cite{He11}. Previous well-known Stirling functions introduced by Butzer and Hauss \cite{BH93}, Butzer, Kilbas, and Trujilloet \cite{BKT03} and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations, generating functions, and asymptotic properties are discussed, which extend the corresponding results about the Stirling numbers shown in \cite{HS98} to the defined Stirling functions.

Eulerian Polynomials And B-Splines, Tian-Xiao He

Tian-Xiao He

Here presented is the interrelationship between Eulerian polynomials, Eulerian fractions and Euler-Frobenius polynomials, Euler-Frobenius fractions, B-splines, respectively. 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.

Sequences Of Numbers Meet The Generalized Gegenbauer-Humbert Polynomials, Tian-Xiao He, Peter J.-S. Shiue, Tsui-Wei Weng

Tian-Xiao He

Here we present a connection between a sequence of numbers generated by a linear recurrence relation of order 2 and sequences of the generalized Gegenbauer-Humbert polynomials. Many new and known formulas of the Fibonacci, the Lucas, the Pell, and the Jacobsthal numbers in terms of the generalized Gegenbauer-Humbert polynomial values are given. The applications of the relationship to the construction of identities of number and polynomial value sequences defined by linear recurrence relations are also discussed.

Generalized Exponential Euler Polynomials And Exponential Splines, Tian-Xiao He

Tian-Xiao He

Here presented is constructive generalization of exponential Euler polynomial and exponential splines based on the interrelationship between the set of concepts of Eulerian polynomials, Eulerian numbers, and Eulerian fractions and the set of concepts related to spline functions. The applications of generalized exponential Euler polynomials in series transformations and expansions are also given.

Characterizations Of Orthogonal Generalized Gegenbauer-Humbert Polynomials And Orthogonal Sheffer-Type Polynomials, Tian-Xiao He

Tian-Xiao He

We present characterizations of the orthogonal generalized Gegen-bauer-Humbert polynomial sequences and the orthogonal Sheffer-type polynomial sequences. Using a new polynomial sequence transformation technique presented in [12], we give a method to evaluate the measures and their supports of some orthogonal generalized Gegenbauer-Humbert polynomial sequences.

Sequences Of Non-Gegenbauer-Humbert Polynomials Meet The Generalized Gegenbauer-Humbert Polynomials, Tian-Xiao He, Peter Shiue

Tian-Xiao He

Here,we present a connection between a sequence of polynomials generated by a linear recurrence relation of order 2 and sequences of the generalized Gegenbauer Humbert polynomials. Many new and known transfer formulas between non-Gegenbauer-Humbert polynomials and generalized Gegenbauer-Humbert polynomials are given. The applications of the relationship to the construction of identities of polynomial sequences defined by linear recurrence relations are also discussed.

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. Three algorithms for calculating the Stirling numbers based on our generalization are also …

Generalized Zeta Functions, Tian-Xiao He

Tian-Xiao He

We present here a wide class of generalized zeta function in terms of the generalized Mobius functions and its properties.

Riordan Arrays Associated With Laurent Series And Generalized Sheffer-Type Groups, Tian-Xiao He

Tian-Xiao He

A relationship between a pair of Laurent series and Riordan arrays is formulated. In addition, a type of generalized Sheffer groups is defined using Riordan arrays with respect to power series with non-zero coefficients. The isomorphism between a generalized Sheffer group and the group of the Riordan arrays associated with Laurent series is established. Furthermore, Appell, associated, Bell, and hitting-time subgroups of the groups are defined and discussed. A relationship between the generalized Sheffer groups with respect to different power series is presented. The equivalence of the defined Riordan array pairs and generalized Stirling number pairs is given. A type …

Boundary Type Quadrature Formulas Over Axially Symmetric Regions, Tian-Xiao He

Tian-Xiao He

A boundary type quadrature formula (BTQF) is an approximate integration formula with all its of evaluation points lying on the Boundary of the integration domain. This type formulas are particularly useful for the cases when the values of the integrand functions and their derivatives inside the domain are not given or are not easily determined. In this paper, we will establish the BTQFs over sonic axially symmetric regions. We will discuss time following three questions in the construction of BTQFs: (i) What is the highest possible degree of algebraic precision of the BTQF if it exists? (ii) What is the …

A Pair Of Operator Summation Formulas And Their Applications, Tian-Xiao He, Leetsch C. Hsu, Dongsheng Yin

Tian-Xiao He

Two types of symbolic summation formulas are reformulated using an extension of Mullin–Rota’s 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 Sequences Of Numbers And Polynomials Defined By Linear Recurrence Relations Of Order 2, Tian-Xiao He, Peter J.-S. Shiue

Tian-Xiao He

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 generalmethod 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.

Sequence Characterization Of Riordan Arrays, Tian-Xiao He, Renzo Sprugnoli

Tian-Xiao He

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 …

Characterization Of Compactly Supported Renable Splines With Integer Matrix, Tian-Xiao He, Yujing Guana

Tian-Xiao He

Let M be an integer matrix with absolute values of all its eigenvalues being greater than 1. We give a characterization of compactly supported M-refinable splines f and the conditions that the shifts of f form a Riesz basis.

A Symbolic Operator Approach To Power Series Transformation-Expansion Formulas, Tian-Xiao He

Tian-Xiao He

In this paper we discuss a kind of symbolic operator method by making use of the defined Sheffer-type polynomial sequences and their generalizations, which can be used to construct many power series transformation and expansion formulas. The convergence of the expansions are also discussed.

Padé Spline Functions, Tian-Xiao He

Tian-Xiao He

We present here the definition of Pad´e spline functions, their expressions, and the estimate of the remainders of pad´e spline expansions. Some algorithms are also given.

Symbolization Of Generating Functions; An Application Of The Mullin–Rota Theory Of Binomial Enumeration, Tian-Xiao He, Peter J.S. S, Leetsch C. Hsu

Tian-Xiao He

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.

Fourier Transform Of Bernstein–Bézier Polynomials, Tian-Xiao He, Charles K. Chui, Qingtang Jiang

Tian-Xiao He

Explicit formulae, in terms of Bernstein–Bézier coefficients, of the Fourier transform of bivariate polynomials on a triangle and univariate polynomials on an interval are derived in this paper. Examples are given and discussed to illustrate the general theory. Finally, this consideration is related to the study of refinement masks of spline function vectors.

Construction Of Biorthogonal B-Spline Type Wavelet Sequences With Certain Regularities, Tian-Xiao He

Tian-Xiao He

No abstract provided.

The Sheffer Group And The Riordan Group, Tian-Xiao He, Peter J.S. Shiue, Leetsch C. Hsu

Tian-Xiao He

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.