A unified presentation of the Gamma-type functions occurring in diffraction theory and associated probability distributions

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].…”

The Fox–Wright functions and Laguerre multiplier sequences

“…In the special case when F (x) is an entire function, open Problem 1.1 is intimately connected to a classical theorem of Laguerre (see Theorem 2.4). The specific entire functions we will consider in the sequel are members of the class of generalized Fox-Wright functions which recently have played an increasingly significant role in various types of applications (see [12,13,36,37]). …”

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).…”