## Mathematics, Department of

#### Date of this Version

1943

#### Abstract

In this paper we shall be concerned with the functions φ^{k}_{a}(*z*) defined by the relation

(1) φ^{k}_{a}(*z*) ≡{*d*/(*dz*)logℓ_{α} (*z*,*q*)}^{k} = {[ℓά(*z*,*q*)] ÷ [ℓα(*z*,*q*)]^{k}, α + 0, 1, 2, 3,

where ℓ_{α}(*z, q*) is a Jacobi theta function and *k* is a positive integer. In the first place, we shall derive the Fourier developments which represent these functions in a certain strip of the complex plane; it will be seen that the Fourier coefficients of φ^{k}_{a}(*z*) depend on those of φ^{s}_{a}(*z*), *s* = 1, 2, 3, …, *k* - 1, through a recurrence relation of order *k*. Secondly, these developments, in conjunction with certain obvious identities, yield, through the method of paraphrase, some general arithmetical formulae of a type first given by Liouville. Indeed, we recover, in a simple manner, some results given without proof by Liouville, which were later proved by Bell through the use of somewhat complex identities involving a certain set of doubly periodic functions of the second kind. One of these results has recently been proved in a strictly elementary, but very ingenious way, by Uspensky. Finally, we indicate some applications of these formulae to the derivation of a certain type of arithmetic and algebraic identities.

## Comments

Published in

Bull. Amer. Math. Soc.49 (1943) 299-306. Used by permission.