The autoregressive filter problem for multivariable degree one symmetric polynomials

Abstract

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 $$1/|p(z)|^2$$ . The conditions derived in this paper for the construction of a degree one symmetric polynomial reveal a major divide between the case of at most two variables vs. the the case of three or more variables. The latter involves multivariable elliptic functions, while the former (due to [Geronimo and Woerdeman (Ann Math 160(3):839-906, 2004)]) only involve polynomials. The three variable case is treated with more detail, and entails hypergeometric functions. Along the way, we identify a seemingly new relation between $${}_2 F_{1}\left( {\frac{1}{3},\frac{2}{3}\atop 1}; \ z\right) $$ and $${}_2 F_{1}\left( {\frac{1}{2},\frac{1}{2}\atop 1}; \ \widetilde{z}\right) $$ .