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Articles 1 - 4 of 4
Full-Text Articles in Physical Sciences and Mathematics
A Pair Of Operator Summation Formulas And Their Applications, Tian-Xiao He, Leetsch C. Hsu, Dongsheng Yin
A Pair Of Operator Summation Formulas And Their Applications, Tian-Xiao He, Leetsch C. Hsu, Dongsheng Yin
Tian-Xiao He
Two types of symbolic summation formulas are reformulated using an extension of Mullin–Rota’s substitution rule in [R. Mullin, G.-C. Rota, On the foundations of combinatorial theory: III. Theory of binomial enumeration, in: B. Harris (Ed.), Graph Theory and its Applications, Academic Press, New York, London, 1970, pp. 167–213], and several applications involving various special formulas and identities are presented as illustrative examples.
On Sequences Of Numbers And Polynomials Defined By Linear Recurrence Relations Of Order 2, Tian-Xiao He, Peter J.-S. Shiue
On Sequences Of Numbers And Polynomials Defined By Linear Recurrence Relations Of Order 2, Tian-Xiao He, Peter J.-S. Shiue
Tian-Xiao He
Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a generalmethod to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.
Sequence Characterization Of Riordan Arrays, Tian-Xiao He, Renzo Sprugnoli
Sequence Characterization Of Riordan Arrays, Tian-Xiao He, Renzo Sprugnoli
Tian-Xiao He
In the realm of the Riordan group, we consider the characterization of Riordan arrays by means of the A- and Z-sequences. It corresponds to a horizontal construction of a Riordan array, whereas the traditional approach is through column generating functions. We show how the A- and Z-sequences of the product of two Riordan arrays are derived from those of the two factors; similar results are obtained for the inverse. We also show how the sequence characterization is applied to construct easily a Riordan array. Finally, we give the characterizations relative to some subgroups of the Riordan group, in particular, of …
Characterization Of Compactly Supported Renable Splines With Integer Matrix, Tian-Xiao He, Yujing Guana
Characterization Of Compactly Supported Renable Splines With Integer Matrix, Tian-Xiao He, Yujing Guana
Tian-Xiao He
Let M be an integer matrix with absolute values of all its eigenvalues being greater than 1. We give a characterization of compactly supported M-refinable splines f and the conditions that the shifts of f form a Riesz basis.