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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)

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)

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 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 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)

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)

- Tian-Xiao He
- 2013

In this paper, we investigate impulse response sequences over the integers by presenting their generating functions and expressions. We also establish some of the corresponding identities. In addition, we give the relationship between an impulse response sequence and all linear recurring sequences satisfying the same linear recurrence relation , which can… (More)

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.