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- Tian-Xiao He, Leetsch C. Hsu, Peter Jau-Shyong Shiue
- Discrete Mathematics
- 2008

This paper deals with the summation problem of power series of the form S b a (f ; x) = a≤k≤b f (k)x k , where 0 ≤ a < b ≤ ∞, and {f (k)} is a given sequence of numbers with k ∈ [a, b) or f (t) is a differentiable function defined on [a, b). We present a symbolic summation operator with its various expansions, and construct several summation formulas with… (More)

- Tian-Xiao He
- J. Computational Applied Mathematics
- 2012

Here presented is the interrelationship between Eulerian polynomials, Eule-rian 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.… (More)

- Tian-Xiao He, Leetsch C. Hsu, Peter Jau-Shyong Shiue
- Discrete Applied Mathematics
- 2007

We define the Sheffer group of all Sheffer-type polynomials and prove the isomorphism between the Sheffer group and the Rior-dan group. An equivalence of the Riordan array pair and generalized Stirling number pair is also presented. Finally, we discuss a higher dimensional extension of Riordan array pairs. of the paper was generated while preparing for a… (More)

- Tian-Xiao He, Renzo Sprugnoli
- Discrete Mathematics
- 2009

Keywords: Riordan arrays A-sequence Z-sequence Hitting-time subgroup a b s t r a c t 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… (More)

- Tian-Xiao He, Leetsch C. Hsu, Peter Jau-Shyong Shiue
- Computers & Mathematics with Applications
- 2007

We have found that there are more than a dozen classical generating functions that could be suitably symbolized to yield various symbolic sum formulas by employing Mullin-Rota's theory of binomial enumeration. Various special formulas and identities involving well-known number sequences or polynomial sequences are presented as illustrative examples. The… (More)

- Tian-Xiao He, Leetsch C. Hsu, Peter Jau-Shyong Shiue
- Computers & Mathematics with Applications
- 2006

This paper deals with the convergence of the summation of power series of the form S b a (f ; x) = a≤k≤b f (k)x k , where 0 ≤ a < b ≤ ∞, and {f (k)} is a given sequence of numbers with k ∈ [a, b) or f (t) a differentiable function defined on [a, b). Here the summation is found by using the symbolic operator approach shown in [4]. We will give a different… (More)

With the aid of multivariate Sheffer-type polynomials and differential operators, this paper provides two kinds of general expansion formulas, called respectively the first expansion formula and the second expansion formula, that yield a constructive solution to the problem of the expansion of A(ˆ t)f (g(t)) (a composition of any given formal power series)… (More)

- Tian-Xiao He, Leetsch C. Hsu, Dongsheng Yin
- Computers & Mathematics with Applications
- 2009

Two types of symbolic summation formulas are reformulated using an extension of Mullin-Rota's substitution rule in [1], and several applications involving various special formulas and identities are presented as illustrative examples.

- Tian-Xiao He
- 2004

—Let φ be an orthonormal scaling function with approximation degree p − 1, and let Bn be the B-spline of order n. Then, spline type scaling functions defined by ¯ fn = f * Bn (n = 1, 2,. . .) possess higher approximation order, p + n − 1, and compact support. The corresponding biorthogonal wavelet functions are also constructed. This technique is extended… (More)

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… (More)