Tian-Xiao He

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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 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)
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