Tian-Xiao He

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This paper deals with the summation problem of power series of the form Sb 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)
In the realm of the Riordan group, we consider the characterization of Riordan arrays by means of the Aand 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 Aand Z-sequences of the product of two Riordan arrays are derived from those of the(More)
Here presented is the interrelationship between Eulerian polynomials, Eulerian fractions and Euler-Frobenius polynomials, Euler-Frobenius fractions, Bsplines, 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)
We define the Sheffer group of all Sheffer-type polynomials and prove the isomorphism between the Sheffer group and the Riordan 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. AMS Subject Classification: 05A15, 11B73, 11B83,(More)
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)
This paper deals with the convergence of the summation of power series of the form Sb 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 type(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(t̂)f(ĝ(t)) (a composition of any given formal power series)(More)
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 f̄n = 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 to the(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 Aand Z-sequences is given, which corresponds to a horizontal construction of a (c)Riordan array rather than its definition approach through(More)