On highly transcendental quantities which cannot be expressed by integral formulas

TL;DR

This paper discusses certain transcendental quantities that cannot be expressed through integral formulas, exploring their properties and implications in the context of special functions and number theory, with historical references to Euler's work.

Contribution

It provides a historical perspective on transcendental quantities and discusses their non-representability by integral formulas, highlighting early ideas related to special functions and number theory.

Findings

Identification of transcendental quantities not expressible by integrals

Discussion of Euler's ideas on special functions and number representations

Historical insights into the development of transcendence theory

Abstract

E565 in the Enestrom index. Translated from the Latin original, "De plurimis quantitatibus transcendentibus quas nullo modo per formulas integrales exprimere licet" (1775). Euler does not prove any results in this paper. It seems to me like he is trying to develop some general ideas about special functions. He gives some examples of numbers he claims but does not prove cannot be represented by definite integrals of algebraic functions. Euler has the idea that if we knew more about the function with the power series $\sum x^{t_n}$ where $t_n$ is the $n$th triangular number, this could lead to a proof of Fermat's theorem that every positive integer is the sum of three triangular numbers. This doesn't end of being fruitful for Euler, but in fact later Jacobi proves a lot of results like this with his theta functions. The last paragraph (\S 9) is not clear to me. My best reading is that there are infinitely many "levels" of transcendental numbers and that this is unexpected or remarkable.