New series representations of {\pi}, {\pi}3 and {\pi}5 in terms of Euler numbers and {\pi}2, {\pi}4 and {\pi}6 in terms of Bernoulli numbers

TL;DR

This paper introduces new series formulas expressing odd and even powers of pi using Euler and Bernoulli numbers, respectively, providing novel mathematical representations.

Contribution

It presents the first empirical derivation of series representations of pi powers in terms of Euler and Bernoulli numbers.

Findings

Series for {}, {}^3, {}^5 using Euler numbers

Series for {}^2, {}^4, {}^6 using Bernoulli numbers

Empirical derivation of these series representations

Abstract

New series representations for odd powers of {\pi} i.e. {\pi}, {\pi}3 and {\pi}5 in terms of Euler numbers and even powers of {\pi} i.e. {\pi}2, {\pi}4 and {\pi}6 in terms of Bernoulli numbers have been obtained empirically.