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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)

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)

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)

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)

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)

- Michael J. Dancs, Tian-Xiao He, T. X. He
- 2005

In this short paper, we give several new formulas for ζ(n) when n is an odd positive integer. The method is based on a recent proof, due to H. Tsumura, of Euler's classical result for even n. Our results illuminate the similarities between the even and odd cases, and may give some insight into why the odd case is much more difficult.

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)

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 general method… (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
- 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)