  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
$_2F_1(\frac12,\frac12;1;\widetilde{z})$.
