A Fourier analysis approach to disprove the weak Shanks conjecture

We derive formulas for the Fourier coefficients of $\mathrm{\mid }f{\mathrm{\mid }}^2$, where $f(z_1,z_2)=(1-\frac{z_1+z_2}{r}{)}^{-\alpha }$, in terms of hypergeometric functions. Using these formulas we provide additional counterexamples to the weak Shanks conjecture, which was recently disproven by Bénéteau, Khavinson and Seco. The obtained formulas allow for (numerical) optimization over the parameters $\alpha$ and $r$.