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 $1/|p(z)|^2$. 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 [J. S. Geronimo and H. J. Woerdeman, Ann. of Math. (2), 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 $_2F_1(\frac13,\frac23;1;z)$ and $_2F_1(\frac12,\frac12;1;\widetilde{z})$.