An exactly solvable self-convolutive recurrence

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.

References in corpus (3)

• Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas
• The first-passage area for drifted Brownian motion and the moments of the Airy distribution
• Hedin's equations and enumeration of Feynman's diagrams

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