Lindel\"of Representations and (Non-)Holonomic Sequences

TL;DR

This paper explores how Lindel"of integral representations can analyze sequences with explicit formulas, revealing their transcendental nature and the absence of linear recurrences or differential equations, with applications to asymptotic estimates.

Contribution

It demonstrates the use of Lindel"of integral representations to establish non-existence results for linear recurrences and differential equations for certain sequences.

Findings

Sequences analyzed are transcendental.

No linear recurrences with polynomial coefficients exist for these sequences.

Asymptotic estimates are derived using Lindel"of methods.

Abstract

Various sequences that possess explicit analytic expressions can be analysed asymptotically through integral representations due to Lindel\"of, which belong to an attractive but somewhat neglected chapter of complex analysis. One of the outcomes of such analyses concerns the non-existence of linear recurrences with polynomial coefficients annihilating these sequences, and, accordingly, the non-existence of linear differential equations with polynomial coefficients annihilating their generating functions. In particular, the corresponding generating functions are transcendental. Asymptotic estimates of certain finite difference sequences come out as a byproduct of the Lindel\"of approach.