id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-4152	Corcino, Cristina Bordaje; Corcino, Roberto Bagsarsa; Casquejo, Jeremar	Fourier Expansion, Integral Representation and Explicit Formula at Rational Arguments of the Tangent Polynomials of Higher-Order	2021	10	.pdf	application/pdf	2648	132	83	Writing ( e2t + 1 )r as ( e2t + 1 )r = (−1)r ( e2t(−1)− 1 )r = (−1)r ( e2t · e−2tk − 1 )r = (−1)r ( e2(t−tk) − 1 )r , and since e−2tk = e−2(2k+1)π 2 i = e−(2k+1)πi = −1, we have (t− tk) r ( 2 e2t + 1 )r ext tn+1 = 2r(t− tk) r (−1)r ( e2(t−tk) − 1 )r · ext tn+1 = (−1)r (2(t− tk)) r (e2(t−tk) − 1)r · ext tn+1 = (−1)r ( ∞∑ n=0 Br n (2(t− tk)) n n! ) ext t−(n+1), where Br n denotes the Bernoulli numbers of order r defined by the generating function C. Corcino, R. Corcino, J. Casquejo / Eur. = dr−1 dtr−1 { (−1)r ( ∞∑ n=0 Br n (2(t− tk)) n n! ) ext t−(n+1) } = (−1)r dr−1 dtr−1 {( ext ∞∑ n=0 Br n (2(t− tk)) n n! ) t−(n+1) } = (−1)r r−1∑ j=0 ( r − 1 j ) dr−1−j dtr−1−j t−(n+1) ·	cache/ejpam-4152.pdf	txt/ejpam-4152.txt