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Half Inverse Riordan Arrays and Their Related Vertical Recurrence Relation

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Journal of Mathematical Research with Applications 2023年第4期43卷 379-388页

作者： tianxiao he Department of Mathematics Illinois Wesleyan University

We discuss two different procedures to study the half Riordan arrays and their inverses. One of the procedures shows that every Riordan array is the half Riordan array of a unique Riordan array. It is well known that every Riordan array has its half Riordan ***, this paper answers the converse question: Is every Riordan array the half Riordan array of some Riordan arrays? In addition, this paper shows that the vertical recurrence relation of the column entries of the half Riordan array is equivalent to the horizontal recurrence relation of the original Riordan array’s row entries.

关键词： Riordan array Riordan group A-sequence Appell subgroup Lagrange (associated) subgroup Bell subgroup hitting-time subgroup derivative subgroup

Construction of Nonlinear Expression for Recursive Number Sequences

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Journal of Mathematical Research with Applications 2015年第5期35卷 473-483页

作者： tianxiao he Department of Mathematics Illinois Wesleyan University

A type of nonlinear expressions of Lucas sequences are established inspired by Hsu [A nonlinear expression for Fibonacci numbers and its ***.,2012,32(6):654–658].Using the relationships between the Lucas sequence and other linear recurring sequences satisfying the same recurrence relation of order 2,i.e.,the Horadam sequences,we may transfer the identities of Lucas sequences to the latter.

关键词： Fibonacci numbers Pell numbers Jacobsthal numbers Lucas numbers linear recurrence relation Horadam sequence impulse response sequence

A Unified Approach to Construct a Class of Daubechies Orthogonal Scaling Functions

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Journal of Mathematical Research with Applications 2017年第1期37卷 29-39页

作者： tianxiao he Tung NGUYEN Department of Mathematics Illinois Wesleyan University Department of Mathematics University of Wisconsin-Madison

We use Lorentz polynomials to give an efficient way to prove Daubechies’ results on the existence of spline type orthogonal scaling functions and to evaluate a class of Daubechies scaling functions in a unified appro... 详细信息

关键词： Daubechies scaling functions MRA scaling function B-spline Lorentz polynomial

Characterization of (c)-Riordan Arrays, Gegenbauer-Humbert-Type Polynomial Sequences, and (c)-Bell Polynomials

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Journal of Mathematical Research with Applications 2013年第5期33卷 505-527页

作者： henry W. GOULD tianxiao he Department of Mathematics West Virginia University Department of Mathematics Illinois Wesleyan University

here presented are the definitions of(c)-Riordan arrays and(c)-Bell polynomials which are extensions of the classical Riordan arrays and Bell *** characterization of(c)-Riordan arrays by means of the A-and Z-sequences is given,which corresponds to a horizontal construction of a(c)-Riordan array rather than its definition approach through column generating *** exists a one-to-one correspondence between GegenbauerHumbert-type polynomial sequences and the set of(c)-Riordan arrays,which generates the sequence characterization of Gegenbauer-Humbert-type polynomial *** sequence characterization is applied to construct readily a(c)-Riordan *** addition,subgrouping of(c)-Riordan arrays by using the characterizations is ***(c)-Bell polynomials and its identities by means of convolution families are also ***,the characterization of(c)-Riordan arrays in terms of the convolution families and(c)-Bell polynomials is presented.

关键词： Riordan arrays (c)-Riordan arrays A-sequence Z-sequence (c)-Bell polynomials (c)-hitting-time subgroup.

A BRIEF BIOGRAPHY OF PROFESSOR L. C. HSU In celabration of Professor Hsu for his 80's birthday

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Analysis in Theory and Applications 2000年第3期16卷 1-9页

作者： he tianxiao Illinois Wesleyan University USA

L.C. Hsu (Lizhi Xu), advisor, Institute of Mathematics, Academia Sinica ； Honorary Director, Institute of Applied Mathematics, Dalian University of Technology; Honorary Chairman, Department of Mathematics, Huazhong University of Sicence and Technology;Editor-in-Chief, Approximation Theory and Its Applications； Editor-in-Chief, Journal of Mathematical Research ＆- Exposition; Edotor-in-Chief, Mathematical Methodology Series, Jiangsu Education Press; and Learned Advisor, Liaoning University. Prof. Hsu devotes himself to the study of mathematics and has obtained various noteworthy results in the fields of asymptotics, approximation, numerical analysis, combinatorics, and mathematical methodology, etc.

关键词： A BRIEF BIOGRAPHY OF PROFESSOR L HSU In celabration of Professor Hsu for his 80’s birthday

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On the Applications of the Linear Recurrence Relationships to Pseudoprimes

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Journal of Mathematical Research with Applications 2019年第6期39卷 563-574页

作者： tianxiao he Peter J.-S.SHIUE Department of Mathematics Illinois Wesleyan University Department of Mathematical Sciences University of Nevada Las Vegas

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

关键词： Girard-Waring identities Recursive sequences Fermat’s pseudoprimes Fibonacci numbers Lucas numbers Fibonacci pseudoprimes Dickson numbers Dickson pseudoprimes

A Unified Approach to Generalized Stirling Functions

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Journal of Mathematical Research with Applications 2012年第6期32卷 631-646页

作者： tianxiao he Department of Mathematics and Computer Science, Illinois Wesleyan University,Bloomington, IL 61702-2900, USA Department of Mathematics and Computer Science Illinois Wesleyan University USA Bloomington 61702-2900

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 [16]. Previous well-known Stirling functions introduced by Butzer and Hauss [4], Butzer, Kilbas, and Trujilloet [6] 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 [21] to the defined Stirling functions.

关键词： Stirling numbers Stirling functions factorial polynomials generalized factorial divided difference k-Gamma functions Pochhammer symbol and k-Pochhammer symbol.

ON AN EXTENSION OF ABEL-GONTSCHAROFF'S EXPANSION FORMULA

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Analysis in Theory and Applications 2005年第4期21卷 359-369页

作者： tianxiao he Leetsch C. Hsu Peter J. S. Shiue Illinois Wesleyan University USA Dalian University of Technology China University of Nevada Las Vegas USA

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 algo... 详细信息

关键词： Abel-Gontscharoff's Expansion Formula Abel-Gontscharoff-Gould polynomial multivariate Abel-Gontscharoff interpolation higher dimensional dot product annihilation coefficients

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Application of Polynomial Interpolation in the Chinese Remainder Problem

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Journal of Mathematical Research with Applications 2017年第1期37卷 90-96页

作者： tianxiao he Scott MACDONALD Peter J.-S.SHIUE Department of Mathematics Illinois Wesleyan University Math Learning Center University of Nevada Las Vegas Department of Mathematical Sciences University of Nevada

This paper presents an application of polynomial interpolation in the solution of the Chinese Remainder Problem for bother integers and polynomials.

关键词： Chinese Remainder problem Chinese Remainder theorem Lagrange interpolation Newton interpolation