Euler-Boole Summation Revisited

Some series involving the Euler zeta function

“…where E n (t) is the n -th quasi-periodic Euler functions defended by (6) and (8). Using the Boole summation formula (see Lemma 2.1 above), we obtain the following formula.…”

Extension of Summation Formulas involving Stirling series

“…Definition 4. (Euler numbers) [3] We define the sequence of Euler numbers {E k } ∞ k=0 by the generating function identity 2e…”

“…Definition 5. (Bernoulli polynomials) [2,3] We define for n ∈ N 0 the n-th Bernoulli polynomial B n (x) via the following exponential generating function as…”

“…For this, we need the following Lemma 7. (Extended Euler-Maclaurin summation formula) [3,5] Let f be an analytic function. Then for all x ∈ R + , we have that…”

“…In this paper we will use the Euler-Maclaurin summation formula [3,5] to obtain rapidly convergent series expansions for finite sums involving Stirling series [1]. Our key tool will be the so called Weniger transformation [1].…”