Physical Sciences and Mathematics Commons^™

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.