Fourier Coefficients of a Class of Eta Quotients of Weight 12
Barış Kendirli *
Civil Engineering Department, Istanbul Kultur University, Turkey
*Author to whom correspondence should be addressed.
Abstract
Recently, Williams [1] and then Yao, Xia and Jin [2] 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 odd Fourier coefficients of 334 eta quotients in terms of σ11 (2n-1) and σ11 ((2n-1)/3)), i.e., the Fourier coefficients of the difference, f(q)-f(-q), of 334 eta quotients and we will
express the even Fourier coefficients of 198 eta quotients i.e., the Fourier coefficients of the sum, f(q)+f(-q), of 198 eta quotients in terms of σ11(n), σ11(n/2), σ11(n/3), σ11(n/4), σ11(n/6) and σ11(n/12)
Keywords: Dedekind eta function, eta quotients, Fourier series