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Full-Text Articles in Physical Sciences and Mathematics
Eulerian Polynomials And B-Splines, Tian-Xiao He
Eulerian Polynomials And B-Splines, Tian-Xiao He
Tian-Xiao He
Here presented is the interrelationship between Eulerian polynomials, Eulerian 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. The relation between Eulerian numbers and B-spline values at knot points are also discussed.
A Symbolic Operator Approach To Several Summation Formulas For Power Series, Tian-Xiao He, Leetsch Hsu, Peter Shiue, D. Torney
A Symbolic Operator Approach To Several Summation Formulas For Power Series, Tian-Xiao He, Leetsch Hsu, Peter Shiue, D. Torney
Tian-Xiao He
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 Є [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.