On the Fourier Developments of a Certain Class of Theta Quotients

Authors

M. A. Basoco

Date of this Version

1943

Comments

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

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.