Log-Convex Solutions to the Functional Equation f xq1.sg x.f x.: G-Type Functions

In this paper we discuss log-convex solutions f : RqªRq to the functional equation with initial condition given by f xq1.sg x.f x. for x)0 and f 1.s1, ). where g: RqªRq. Our main result, a generalization of the Bohr]Mollerup]Artin classical characterization of the gamma function, is that if g is eventually log- concave with the property that, for each w)0, g xqw.rg x.ª1 as xª`, then ). has a unique eventually log-convex solution, determined by the formula g n.. . . g 1.g x n. f x.s lim for x)0. nª` g nqx.. . . g x. A function f arising thus is called a G-type function, two examples of which are the gamma function G and the q-gamma function Gq 0-q-1.generated, respec- tively, by the functions g x.sx and g x.s 1yqx.r 1yq.. We establish for G-type functions analogues of Legendre’s Duplication Formula, Gauss’ Multiplication Formula, Stirling’s Formula, Euler’s constant, and Weierstrass’ infinite product for the gamma function, and we use the theory of G-type functions to find log-convex solutions f : RqªRq to certain functional equations of the type 1 my1 f x.f xq /. . . f xq /h x.s1. m m The backdrop of G-type functions serves to place classical theory into perspective. Throughout, the concepts of log-convexity and log-concavity play central roles.