Rapidly Convergent Summation Formulas involving Stirling Series

Abstract

This paper presents a family of rapidly convergent summation formulas for various finite sums of analytic functions. These summation formulas are obtained by applying a series acceleration transformation involving Stirling numbers of the first kind to the asymptotic, but divergent, expressions for the corresponding sums coming from the Euler-Maclaurin summation formula. While it is well-known that the expressions obtained from the Euler-Maclaurin summation formula diverge, our summation formulas are all very rapidly convergent and thus computationally efficient.

The Generalization of Faulhaber's Formula to Sums of Arbitrary Complex Powers

“…We believe that they will also have applications in physics [18] such as the extended version of Faulhaber's formula [19,20]. With the universal technique, explained in this paper, one can obtain other summation formulas of this type [21,22], as for example with Theorem 10 and equation (28) we obtain: (68)…”

“…. }, we apply the Weniger transformation formula (21) directly to the function f 1 (z) with n := 1 − s − a and O n+a (z) = O 1−s (z) = 0. Similarly from equation (34), we obtain the formula (30) by applying Theorem 10 with R n (z) := z a U n+a (z) to the analytic function f 2 (z) defined by…”