The autoregressive filter problem for multivariable degree one symmetric polynomials

The multivariable autoregressive filter problem asks for a polynomial $$p(z)=p(z_1, \ldots , z_d)$$ without roots in the closed d-disk based on prescribed Fourier coefficients of its spectral density function $$1/|p(z)|^2$$ . conditions derived this paper construction degree one symmetric reveal major divide between case at most two variables vs. three or more variables. latter involves elliptic functions, while former (due to [Geronimo and Woerdeman (Ann Math 160(3):839-906, 2004)]) only involve polynomials. variable is treated with detail, entails hypergeometric functions. Along way, we identify seemingly new relation $${}_2 F_{1}\left( {\frac{1}{3},\frac{2}{3}\atop 1}; \ z\right) $$ {\frac{1}{2},\frac{1}{2}\atop \widetilde{z}\right)