Physical Sciences and Mathematics Commons^™

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

Tian-Xiao He

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.

Characterization Of (C)-Riordan Arrays, Gegenbauer-Humbert-Type Polynomial Sequences, And (C)-Bell Polynomials, Tian-Xiao He, Henry Gould

Tian-Xiao He

Here presented are the definitions of (c)-Riordan arrays and (c)-Bell polynomials which are extensions of the classical Riordan arrays and Bell polynomials. The 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 functions. There exists a one-to-one correspondence between Gegenbauer-Humbert-type polynomial sequences and the set of (c)-Riordan arrays, which generates the sequence characterization of Gegenbauer-Humbert-type polynomial sequences. The sequence characterization is applied to construct readily a (c)-Riordan array. In addition, subgrouping of (c)-Riordan arrays by using the characterizations …

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

Tian-Xiao He

No abstract provided.

Impulse Response Sequences And Construction Of Number Sequence Identities, Tian-Xiao He

Tian-Xiao He

In this paper, we investigate impulse response sequences ov er the integers by pre-senting their generating functions and expressions. We also establish some of the corre-sponding identities. In addition, we give the relationship between an impulse response sequence and all linear recurring sequences satisfying the same linear recurrence rela- tion, which can be used to transfer the identities among different sequences. Finally, we discuss some applications of impulse response sequences to the structure of Stirling numbers of the second kind, the Wythoff array, and the Boustro phedon transform.

Adding It Up: In His Teaching And Research, Math Professor Tian-Xiao He Embraces The Joy Of Exploring An Oft-Feared Subject, Kim Hill

Tian-Xiao He

Professor of Mathematics Tian-Xiao He says reaching the number “100” is not significant. Colleagues and former students beg to differ.

It’s not the numeral following “99” under debate, but rather the number of papers published in peer-reviewed journals that He has written or co-authored. To be precise (after all, this is mathematics), He has published 111 papers and five books since his graduate school days in the 1980s.

Q-Analogues Of Symbolic Operators, Tian-Xiao He, Michael Dancs

Tian-Xiao He

Here presented are 𝑞-extensions of several linear operators including a novel 𝑞-analogue of the derivative operator 𝐷. Some 𝑞-analogues of the symbolic substitution rules given by He et al., 2007, are obtained. As sample applications, we show how these 𝑞-substitution rules may be used to construct symbolic summation and series transformation formulas, including 𝑞-analogues of the classical Euler transformations for accelerating the convergence of alternating series.

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.

Some Dense Subsets Of Real Numbers And Their Applications, Tian-Xiao He, Peter Shiue, Xiaoya Zha

Tian-Xiao He

We give a collection of subsets which are dense in the set of real numbers. Several applications of the dense sets are also presented.

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.

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.

A Pair Of General Series-Transformation Formulas, Tian-Xiao He, Leetsch Hsu, Peter Shiue

Tian-Xiao He

No abstract provided.