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

Row Sums And Alternating Sums Of Riordan Arrays, Tian-Xiao He, Louis W. Shapiro

Tian-Xiao He

Application Of Fa´A Di Bruno’S Formula In The Construction Of Combinatorial Identities, Tian-Xiao He

Tian-Xiao He

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

Tian-Xiao He

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. …

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

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 …

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 …

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.

On The Construction Of Number Sequence Identities, Tian-Xiao He, Wun-Seng Chou

Tian-Xiao He

To construct a class of identities for number sequences generated by linear recurrence relations. An alternative method based on the generating functions of the sequences is given. The equivalence between two methods for linear recurring sequences are also shown. However, the second method is not limited to the linear recurring sequences, which can be used for a wide class of sequences possessing rational generating functions. As examples, Many new and known identities of Stirling numbers of the second kind, Pell numbers, Jacobsthal numbers, etc., are constructed by using our approach. Finally, we discuss the hyperbolic expression of the identities of …

Expression And Computation Of Generalized Stirling Numbers, Tian-Xiao He

Tian-Xiao He

Schroder Matrix As Inverse Of Delannoy Matrix, Tian-Xiao He, Sheng-Liang Yang, Sai-Nan Zheng, Shao-Peng Yuan

Tian-Xiao He

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.

The Characterization Of Riordan Arrays And Sheffer-Type Polynomial Sequences, Tian-Xiao He

Tian-Xiao He

Here we present a characterization of Sheffer-type polynomial sequences based on the isomorphism between the Riordan group and Sheffer group and the sequence characterization of Riordan arrays. We also give several alternative forms of the characterization of the Riordan group, Sheffer group and their subgroups. Formulas for the computation of the generating functions of Riordan arrays and Sheffer-type polynomial sequences from the characteristics are shown. Furthermore, the applications of the characteristics to lattice walks and recursive construction of Sheffer-type polynomial sequences are also given.

A Note On Horner's Method, Tian-Xiao He, P. J.-S. Shiue

Tian-Xiao He

Here we present an application of Horner's method in evaluating the sequence of Stirling numbers of the second kind. Based on the method, we also give an e_cient way to calculate the diference sequence and divided diference sequence of a polynomial, which can be applied in the Newton interpolation. Finally, we survey all of the results in Proposition 1.4.

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.

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.

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 …