SelectedWorks - Tian-Xiao He

Contact

Selected Works of Tian-Xiao He

Earl and Marion A. Beling Professorship of Natural Sciences and Mathematics

Dr. Tian-Xiao He received a Ph.D. from the Dalian University of Technology and a Ph.D. from the Texas A&M University. His research interests include spline functions, approximation theory, wavelet analysis, numerical analysis, enumerative combinatorics, and some topics in computational mathematics.

Mentor

Works
About

×

About Tian-Xiao He

Dr. Tian-Xiao He received a Ph.D. from the Dalian University of Technology and a Ph.D. from the Texas A&M University. His research interests include spline functions, approximation theory, wavelet analysis, numerical analysis, enumerative combinatorics, and some topics in computational mathematics.

Present	Earl and Marion A. Beling Professorship of Natural Sciences, Illinois Wesleyan University
Institution Organization Position Type Position Title Time Period to Options Current Position Affiliate my profile with this Institution
Present	Professor of Mathematics, Illinois Wesleyan University
Institution Organization Position Type Position Title Time Period to Options Current Position Affiliate my profile with this Institution

Applied Mathematics and Mathematics

Articles (93)

Centralizers of the Riordan Group

Indian Journal of Pure and Applied Mathematics (2022)

Tian-Xiao He and Yuanziyi Zhang

In this paper, we discuss centralizers in the Riordan group. We will see that Faà di Bruno’s formula is an ...

Some identities of Gaussian binomial coefficients

Annales Mathematicae et Informaticae (2022)

Tian-Xiao He, Anthony G. Shannon and Peter J.-S. Shiue

In this paper, we present some identities of Gaussian binomial coefficients with respect to recursive sequences, Fibonomial coefficients, and complete ...

The Dual of Number Sequences, Riordan Polynomials, and Sheffer Polynomials

Special Matrices (2021)

Tian-Xiao He and José l Ramírez

In this paper we introduce different families of numerical and polynomial sequences by using Riordan pseudo involutions and Sheffer polynomial ...

On the Solutions of Three-Variable Frobenius-Related Problems Using Order Reduction ...

International Journal of Mathematics and Mathematical Sciences (2021)

Tian-Xiao He, Rama Venkat and Peter J.-S. Shiue

This paper presents a new approach to determine the number of solutions of three-variable Frobenius-related problems and to find their ...

One-pth Riordan Arrays in the Construction of Identities

Journal of Mathematical Research with Applications (JMRA) (2021)

For an integer p ≥ 2 we construct vertical and horizontal one-pth Riordan arrays from a Riordan array. When p ...

Half Riordan Array Sequences

Linear Algebra and Its Applications (2020)

We construct sequences of vertical and horizontal halves of various Riordan arrays by using the vertical and horizontal halve Riordan ...

On the Applications of the Girard-Waring Identities

Journal of Computational Analysis and Applications (2020)

Tian-Xiao He and Peter J.-S. Shiue

We present here some applications of Girard-Waring identities. Many various identities for things like elementary mathematics and other advanced mathematics ...

Recursive Sequences and Girard-Waring Identities with Applications in Sequence Transformation

Electronic Research Archive (2020)

Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie and Minghao Chen

We present here a generalized Girard-Waring identity constructed from recursive sequences. We also present the construction of Binet Girard-Waring identity ...

A Note on Eulerian Numbers and Toeplitz Matrices

Special Matrices (2020)

Tian-Xiao He and Peter J.-S. Shiue

This note presents a new formula of Eulerian numbers derived from Toeplitz matrices via Riordan array approach.

Palindromes and Pseudo-Involution Multiplication

Linear Algebra and Its Applications (2020)

Tian-Xiao He and Louis Shapiro

A Riordan array (g,f) is called a pseudo-involution if (g,f)M (or equivalently, M (g,f)), where M = (1, - z) ...

A-sequences, Z-sequence, and B-sequences of Riordan matrices

Discrete Mathematics (2020)

We define two type B-sequences of Riordan arrays and present the A-sequence characterization and Z-sequence characterization of the Riordan matrices ...

Computation of Fermat's Pseudoprimes

Journal of Discrete Mathematical Sciences and Cryptography (2020)

Tian-Xiao He, Peter J.-S. Shiue and Yaotsu Chang

This note discussed the computation of Fermat’s pseudoprimes (Fermat’s Psp).

Generalized Schröder Matrices Arising from Enumeration of Lattice Paths

Czechoslovak Mathematical Journal (2019)

Tian-Xiao He, Sheng-liang Yang and Lin Yang

We introduce a new family of generalized Schröder matrices from the Riordan arrays which are obtained by counting of the ...

On Applications of the Linear Recurrence Relationships to Pseudoprimes

Journal of Mathematical Research with Applications (2019)

Tian-Xiao He and P. J.-S. Shiue

We present here some results on the applications of linear recursive sequences of order 2 to the Fermat pseudoprimes, Fibonacci ...

Divisibility of Generalized Catalan Numbers and Raney Numbers

Journal of Computational Analysis and Applications (2019)

Tian-Xiao He, Jacob Bobrowski and Peter J.-S. Shiue

The Raney numbers, also called Fuss-Catalan numbers, are defined by Rk(n, r) = r(...)/(kn + r). A generalized Lobb numbers ...

A Unified Approach for the Catalan Matrices by Using Riordan ...

Linear Algebra and its Applications (2018)

Tian-Xiao He, Yan-Ni Dong, Sheng-liang Yang and Yan-Xue Xu

We study generalized Catalan matrices based on the Riordan array and Fuss–Catalan numbers. A unified combinatorial interpretation for the entries ...

Permanents of Tridiagonal and Hessenberg Matrices Representing Recursive Number and ...

Journal of Combinatorial Mathematics and Combinatorial Computing (2018)

Tian-Xiao He and Ximing Dong

In this article, we present a generalized approach to construct tridiagonal matrices and Hessenberg matrices representing recursive number sequences and ...

Sequence Characterizations of Double Riordan Arrays and Their Compressions

Linear Algebra and its Applications (2018)

Inspired by the Fibonacci tree shown in the recent book, Catalan Numbers, by Richard Stanley, we present a combinatorial way ...

Matrix Representation of Recursive Sequences of Order 3 and Its ...

Journal of Mathematical Research with Applications (JMRA) (2018)

Tian-Xiao He, Jeff H.C. Liao and Peter J.S. Shiue

Here presented is a matrix representation of recursive number sequences of order 3 defined by an = pan-1 + qan-2 ...

On the Primality of the Generalized Fuss-Catalan Numbers

Journal of Integer Sequences (2018)

Tian-Xiao He, Wun-Seng Chou and Peter J.-S. Shiue

In this note, we determine all primes among the Fuss-Catalan numbers the generalized Fuss-Catalan numbers, the Lobb numbers, and the ...

Some Matrix Identities on Colored Motzkin Paths

Discrete Mathematics (2017)

Tian-Xiao He, Sheng-liang Yang and Yan-Ni Dong

Merlini and Sprugnoli (2017) give both an algebraic and a combinatorial proof for an identity proposed by Louis Shapiro by ...

Fuss–Catalan Matrices, Their Weighted Sums, and Stabilizer Subgroups of ...

Linear Algebra and its Applications (2017)

Tian-Xiao He and Louis W. Shapiro

In this paper, we present the Riordan arrays called Fuss–Catalan matrices which are constructed by the convolutions of the generating ...

(m, r)-CENTRAL RIORDAN ARRAYS AND THEIR APPLICATIONS

Czechoslovak Mathematical Journal (2017)

Tian-Xiao He, Sheng-liang Yang and Yan-Xue Xu

Excerpt from the abstract:It is known that the (m, r)-central coefficient triangles of any Riordan array are also Riordan arrays. ...

A Uniﬁed Approach to Construct a Class of Daubechies Orthogonal ...

Journal of Mathematical Research with Applications (2017)

Tian-Xiao He and Tung Nguyen

We use Lorentz polynomials to give an efficient way to prove Daubechies’results on the existence of spline type orthogonal scaling ...

An Approach to the Construction of Linear Divisibility Sequences of ...

Journal of Integer Sequences (2017)

Tian-Xiao He and Peter J.-S. Shiue

We present a unified approach to construct divisibility sequences of higher orders by using divisibility sequences of order 2.