New interesting Euler sums

Abstract

. We present here some new and interesting Euler sums obtained by means of related integrals and elementary approach. We supplement Euler’s general recurrence formula with two general formulas of the form ∑ n (cid:2) 1 O ( m ) n (cid:2) 1 ( 2 n − 1 ) p + 1 ( 2 n ) p (cid:3) and ∑ n (cid:2) 1 O n ( 2 n − 1 ) p ( 2 n + 1 ) q , where O ( m ) n = n ∑ j = 1 1 ( 2 j − 1 ) m . Two formulas for ζ ( 5 ) are also derived.