An exactly solvable self-convolutive recurrence

TL;DR

This paper presents an explicit, non-recursive solution to a self-convolutive recurrence related to hypergeometric functions, with applications in quantum field theory, Brownian motion, and combinatorics.

Contribution

It introduces a novel explicit solution for a specific recurrence using the Hilbert transform, linking it to moments and operator traces.

Findings

Explicit solution expressed as moments and operator traces

Applications demonstrated in quantum field theory and combinatorics

Connections established with asymptotic expansions of hypergeometric functions

Abstract

We consider a self-convolutive recurrence whose solution is the sequence of coefficients in the asymptotic expansion of the logarithmic derivative of the confluent hypergeometic function $U(a,b,z)$. By application of the Hilbert transform we convert this expression into an explicit, non-recursive solution in which the $n$th coefficient is expressed as the $(n-1)$th moment of a measure, and also as the trace of the $(n-1)$th iterate of a linear operator. Applications of these sequences, and hence of the explicit solution provided, are found in quantum field theory as the number of Feynman diagrams of a certain type and order, in Brownian motion theory, and in combinatorics.