Extension theorems for integral representations of solutions of a functional equation (Q916921)

Consider: \((1)\quad F(x+1)=g(x)F(x),\) \((2)\quad F(x)=\int^{\infty}_{0}e^{-t}\Phi (t,x-1)dt,\) \((3)\quad \Phi_ t(t,x)=g(x)\Phi (t,x-1)\) and \((4)\quad \Phi (t,-x)=1/\Phi (t,x).\) The classical example where \(F(x)=\Gamma (x),\) \(g(x)=x\) and \(\Phi (t,x)=t^ x\) suggests that solutions of the functional equation (1) may be found among functions of the form (2), where the kernel \(\Phi\) (t,x) satisfies the functional differential equation (3) and the reflection equation (4). The principal result of this paper is the following extension theorem. Let \(g: {\mathbb{R}}_+\to {\mathbb{R}}_+\) be given, and let \(\Phi^*\) be a positive valued function defined on \({\mathbb{R}}_+\times ((-1,0)\cup (0,1)).\) Assume that \((\Phi^*)_ t\) exists and that \(\Phi^*\) satisfies (3) and (4) for \(t>0\) and \(0<x<1.\) Assume that, for each x in (0,1), \((i)\quad \Phi^*(t,x)={\mathcal O}(e^{pt})\) as \(t\to \infty\) for some p depending on x, and \((ii)\quad \Phi^*(t,x)\to 0\) as \(t\to 0^+\). Then there is an extension \(\Phi\) : \({\mathbb{R}}_+\times (- 1,\infty)\) of \(\Phi^*\) which is of the form \(\Phi (t,x)=G(x)t^{\nu (x)}\) and which satisfies (3) for \(t>0\), \(x>0\). Moreover, the function F defined by (2) satisfies (1) for \(x>0\).