FOURIER COEFFICIENTS OF A CLASS OF ETA QUOTIENTS

摘要

Recently, Williams, and then Yao, Xia and Jin, discovered explicit formulas for the coecients of the Fourier series expansions of a class of eta quotients. Williams expressed all coecients of one hundred and twenty-six eta quotients in terms of (n), ( n 2 ), ( n 3 ) and ( n 6 ), and Yao, Xia and Jin, following the method of Williams’ proof, expressed only the even coecients of one hundred and four eta quotients in terms of 3(n), 3( n 2 ), 3( n 3 ) and 3( n 6 ). Here, by using the method of Williams’ proof, we will express the odd Fourier coecients of seventy-four eta quotients f(q) in terms of 5 (2n 1) and 5 2n1 3 , i.e., the Fourier coecients of the di?erence f(q) f(q) of seventy-four eta quotients; and we will express the even Fourier coecients of sixty eta quotients, i.e., the Fourier coecients of the sum f(q) + f(q) of sixty eta quotients, in terms of 5 (n), 5 n 2 , 5 n 3 , 5 n 4 , 5 n 6 and 5 n 12 .