On an arithmetic triangle of numbers arising from inverses of analytic functions

Armen G Bagdasaryan

doi:10.1016/j.endm.2018.11.003

Abstract

The Lagrange inversion formula is a fundamental tool in combinatorics. In this work, we investigate an inversion formula for analytic functions, which does not require taking limits. By applying this formula to certain functions we have found an interesting arithmetic triangle for which we give a recurrence formula. We then explore the links between these numbers, Pascal's triangle, and Bernoulli's numbers, for which we obtain a new explicit formula. Furthermore, we present power series and asymptotic expansions of some elementary and special functions, and some links to the Online Encyclopedia of Integer Sequences (OEIS).

Highlights

The Lagrange inversion formula is a fundamental tool in combinatorics, with numerous applications and many generalizations [14], [6], [7] and [10]
We introduce a new inversion formula for analytic functions, based on the ordinary one-variable Lagrange inversion
Applying this formula to certain functions we find an arithmetic triangle linked to Pascal’s triangle, for which we give a recurrence formula. This continues work presented by the authors in [1]. We show that these numbers are closely related to Bernoulli numbers, which can be obtained by a new explicit formula

Summary

Introduction

The Lagrange inversion formula is a fundamental tool in combinatorics, with numerous applications and many generalizations [14], [6], [7] and [10]. We introduce a new inversion formula for analytic functions, based on the ordinary one-variable Lagrange inversion. This seems to be both simpler to use, and easier to apply than the classical Lagrange formula, which requires taking limits. Applying this formula to certain functions we find an arithmetic triangle linked to Pascal’s triangle, for which we give a recurrence formula. This continues work presented by the authors in [1]. We present power series and asymptotic expansions of some elementary and special functions [4], involving numbers An, and highlight links to existing entries in the Online Encyclopedia of Integer Sequences (OEIS) [13]

Inversion formula and applications

Recurrence formula

Series expansions involving the numbers Ak