Fourier Coefficients of a Class of Eta Quotients of Weight 12 with Level 12

Baris Kendirli

Baris Kendirli Istanbul Kultur University, Turkey

Keywords: Dedekind eta function, eta quotients, Fourier series.

Abstract

Recently, Williams and then Yao, Xia and Jin discovered explicit formulas for the coefficients of the Fourier series expansions of a class of eta quotients. Williams expressed all coefficients of 126 eta quotients in terms of σ(n),σ(n/2),σ(n/3) and σ(n/6) and Yao, Xia and Jin, following the method of proof of Williams, expressed only even coefficients of 104 eta quotients in terms of σ₃(n),σ₃(n/2),σ₃(n/3) and σ₃(n/6). Here, we will express the even Fourier coefficients of 2 eta quotients i.e., the Fourier coefficients of the sum, f(q)+f(-q), of 2 eta quotients in terms of σ₅(n),σ₅(n/2),σ₅(n/3),σ₅(n/4),σ₅(n/6),σ₅(n/12),σ₁₁(n),σ₁₁(n/2),
σ₁₁(n/3),σ₁₁(n/4),σ₁₁(n/6),σ₁₁(n/12),τ(n)(tau function),τ(n/2),τ(n/3),τ(n/4),τ(n/6),τ(n/12)
and the odd Fourier coefficients of 393 eta quotients in terms of σ₅(n),σ₅(n/2),σ₅(n/3),σ₅(n/4),σ₅(n/6),σ₅(n/12),σ₁₁(n),σ₁₁(n/2),
σ₁₁(n/3),σ₁₁(n/4),σ₁₁(n/6),σ₁₁(n/12),τ(n),τ(n/2),τ(n/3),τ(n/4),τ(n/6),τ(n/12) and f₁₃,...,f₁₉.

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