A Symbolic Operator Approach to Several Summation Formulas for Power Series

Tian-Xiao He; Leetsch C. Hsu; Peter J.-S. Shiue; D. C. Torney

A Symbolic Operator Approach to Several Summation Formulas for Power Series

Tian-Xiao He, Leetsch C. Hsu, Peter J.-S. Shiue, D. C. Torney

Illinois Wesleyan University

Research output: Journal Article › Article › peer-review

Abstract

<p> This paper deals with the summation problem of power series of the form Sba (f; x) = ∑a ≤ k ≤ b f(k) xk, where 0≤ a < b ≤ ∞, and {f(k)} is a given sequence of numbers with k &Jukcy; [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 estimable remainders for Sba (f; x), by the aid of some classical interpolation series due to Newton, Gauss and Everett, respectively.</p>

Original language	American English
Journal	Journal of Computational and Applied Mathematics
Volume	177
State	Published - 2005

Keywords

Eulerian fraction
Eulerian numbers
Eulerian polynomial
Evertt’s interpolation
Gauss interpolation.
Newton’s interpolation
Symbolic summation operator
generating function
power series

Disciplines

Applied Mathematics
Mathematics

Access to Document

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Cite this

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title = "A Symbolic Operator Approach to Several Summation Formulas for Power Series",

abstract = " This paper deals with the summation problem of power series of the form Sba (f; x) = ∑a ≤ k ≤ b f(k) xk, where 0≤ a < b ≤ ∞, and {f(k)} is a given sequence of numbers with k &Jukcy; [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 estimable remainders for Sba (f; x), by the aid of some classical interpolation series due to Newton, Gauss and Everett, respectively.",

keywords = "Eulerian fraction, Eulerian numbers, Eulerian polynomial, Evertt{\textquoteright}s interpolation, Gauss interpolation., Newton{\textquoteright}s interpolation, Symbolic summation operator, generating function, power series",

author = "Tian-Xiao He and Hsu, {Leetsch C.} and Shiue, {Peter J.-S.} and Torney, {D. C.}",

year = "2005",

language = "American English",

volume = "177",

journal = "Journal of Computational and Applied Mathematics",

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T1 - A Symbolic Operator Approach to Several Summation Formulas for Power Series

AU - He, Tian-Xiao

AU - Hsu, Leetsch C.

AU - Shiue, Peter J.-S.

AU - Torney, D. C.

PY - 2005

Y1 - 2005

N2 - This paper deals with the summation problem of power series of the form Sba (f; x) = ∑a ≤ k ≤ b f(k) xk, where 0≤ a < b ≤ ∞, and {f(k)} is a given sequence of numbers with k &Jukcy; [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 estimable remainders for Sba (f; x), by the aid of some classical interpolation series due to Newton, Gauss and Everett, respectively.

AB - This paper deals with the summation problem of power series of the form Sba (f; x) = ∑a ≤ k ≤ b f(k) xk, where 0≤ a < b ≤ ∞, and {f(k)} is a given sequence of numbers with k &Jukcy; [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 estimable remainders for Sba (f; x), by the aid of some classical interpolation series due to Newton, Gauss and Everett, respectively.

KW - Eulerian fraction

KW - Eulerian numbers

KW - Eulerian polynomial

KW - Evertt’s interpolation

KW - Gauss interpolation.

KW - Newton’s interpolation

KW - Symbolic summation operator

KW - generating function

KW - power series

M3 - Article

VL - 177

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

ER -