Eisenstein Series and the Theta Functions of the borweins

摘要

Let q ∈ C satisfy |q| < 1, B_n (n = 0,1,2, ...) denote the nth Bernoulli number, G_(2n) (n = 0,1,2, ...) denote the (modified) Glaisher numbers defined by 3/(2+4 cosh t) =∑_(k=0)~∞ G_(2k) (t~(2k))/((2k)!), t ∈ C, |t| < (2π)/3, and for d ∈ Z ((-3)/d) = {1 if d ≡ 1 (mod 3), -1 id ≡ 2 (mod 3), 0 if d ≡ 0 (mod 3). Borwein and Borwein [5, pp. 695-696] defined the two-dimensional theta functions a(q) and b(q) by a(q):= ∑ q~(x~2+xy+y~2), b(q) : = ∑ ω~(x-y) q~(x~2+xy+y~2), where ω = e~(2πi/3), and (with Garvan) developed their properties in. In this paper we use a theorem of Alaca, Alaca, McAfee and Williams, which expresses a product of Lambert series as a Lambert series, to show that the Eisenstein series R_k(q):= (G_(2k)/3) + ∑_(n=1)~∞ (∑_(d|n)((-3)/d) d~2k)q~n, k = 0,1, 2,..., can be expressed in the form R_k(q):= ∑_(j=1)~k r(k,j)a~(2k+1-3j)(q)b~(3j)(q), r(k,j) ∈ Q, for k ≥ 1, and we give an explicit recurrence relation for the rational numbers r(k,j), see Theorem 1.5. This theorem extends and makes explicit a result of Cooper [8] proved by a different method. The key to this result is the identity ∑_(k=0)~n (2n/2k)R_k(q)R_(n-k)(q)+R_0(q)R_n(q) = 1/6 (E_(n+1)(q) - 3~(2n+2)E_(n+1)(q~3)), valid for n = 1, 2, 3, ...; see Theorem 1.4. This identity gives a new convolution sum identity for the arithmetic function ∑_(d|n)((-3)/d)d~(2k).