Some integral and asymptotic formulas associated with the Hurwitz Zeta function

Estimating the digamma and trigamma functions by completely monotonicity arguments

“…In this paper, we study the complete monotonicity of the functions F a , then we apply these results to obtain new sharp bounds for the digamma and trigamma functions. The problem of estimating the gamma and polygamma functions has attracted the attention of many researchers, since they are close related to the theory of zeta functions [1,8,12,16,27,39], multiple gamma and related functions [7,[9][10][11]13,[24][25][26]30,33,35,38,40], gamma type distributions [15,29], or harmonic sums [23,34]. There are also many recent investigations dealing with one-sided and two-sided inequalities involving the digamma, trigamma, polygamma and other related functions, see e.g., [5,6,14,31,32].…”

Character analogue of the Boole summation formula with applications

“…The method used here have already been employed by Kanemitsu et al [19] to the Euler-MacLaurin summation formula to obtain integral representations for Hurwitz zeta function and its partial sum. For α = 0 and β = x, let f (t) = (t + a) s in Theorem 1.3.…”

“…Observe that the integrals in (19) and (20) can be emerged from Theorem 1.3 by taking the logarithm function. So, we may establish a connection between generalized Euler functions and some identities for logarithmic means.…”

New improvements of the Stirling formula

“…In consequence, it has been deeply studied by many authors due to its basic role in the theory of the gamma function and related functions. See, e.g., [2][3][4][5][6][7][8][9][10][11][12][13][17][18][19][20][21][22][23][24][25][26][27][28][29][31][32][33][34][35][36][37] and all references therein. We use here the discrete representation The superiority of these formulas over the Stirling formula follows by a subsequent inequality (2.2).…”