Various considerations on hypergeometric series

TL;DR

This paper analyzes Euler's 1776 work on hypergeometric series, focusing on asymptotic behaviors, infinite products, and their relation to integrals, using the Euler-Maclaurin formula to explore foundational aspects of special functions.

Contribution

It provides a detailed examination of Euler's original methods for hypergeometric series and their asymptotic properties, highlighting historical and mathematical insights.

Findings

Euler's approach to asymptotic behavior of infinite products

Relations between infinite products and integrals

Use of Euler-Maclaurin formula in series analysis

Abstract

E661 in the Enestrom index. This was originally published as "Variae considerationes circa series hypergeometricas" (1776). In this paper Euler is looking at the asymptotic behavior of infinite products that are similar to the Gamma function. He looks at the relations between some infinite products and integrals. He takes the logarithm of these infinite products, and expands these using the Euler-Maclaurin summation formula. In section 14, Euler seems to be rederiving some of the results he already proved in the paper. However I do not see how these derivations are different. If any readers think they understand please I would appreciate it if you could email me. I am presently examining Euler's work on analytic number theory. The two main topics I want to understand are the analytic continuation of analytic functions and the connection to divergent series, and the asymptotic behavior of the Gamma function.