Infinite products involving Dirichlet characters and cyclotomic polynomials

Abstract

Using some basic properties of the gamma function, we evaluate a simple class of infinite products involving Dirichlet characters as a finite product of gamma functions and, in the case of odd characters, as a finite product of sines. As a consequence we obtain evaluations of certain multiple L-series. In the final part of this paper we derive expressions for infinite products of cyclotomic polynomials, again as finite products of gamma or of sine functions.

Special values of q-gamma products

“…k , where f (2) = f (4) = 0, f (1) = 1, f (3) = −1, and f (k) = f (k + 4), i.e., f is the unique non-principal Dirichlet character modulo 4. This observation is the key feature of the manuscript [3], and will be extensively exploited here. 1…”

“…By specializing these at values of q besides 1 we obtain several new values of infinite products in terms of e −π and gamma functions. This presents an extension of the program begun in [3], which generalized many classical product identities to identities involving Dirichlet characters. We also encounter in a natural way a variant of the cyclotomic polynomials defined by d) .…”

$ϕ$-Thue-Morse sequences and infinite products

“…Similarly, from Lemma 1 in [DV18], for each integer n ≥ 2, taking f (j) = −2ψ ¡φ(a) j − T 2q (a) j , a = 1 q2 n+2 and z j = 2j+1 q2 n+2 , we have q2 n+2 −1 j=0 f (j) = 0, so that…”

“…Proof From Lemma 1 in [DV18], for each integer n ≥ 2, taking f (j) = −φ(a) j − T q (a) j , a = 1 q2 n+1…”

“…) an , where (a n ) n∈N is the Thue-Morse sequence beginning with a 0 = 1, a 1 = −1. The author states that, rather than calculate directly n≥1 ( 2n 2n+1 ) an , one may work on the infinite product n≥1 (1 + ( an 2n+1 )).The last infinite product, inspired by [DV18], can be calculated as the limit of a sequence n≥1 (1 + (…”