Sudden symmetry in simultaneous approximation (Q913011)

Let w be a primitive n-th root of unity and let \(f_ j(z)=f(w^{j- 1}z)\), \(j=1,2,...,n\). Consider the problem of finding polynomials \(P_ 0\) (the denominator) and \(P_ 1,...,P_ n\) with degrees p and s-k respectively where \(s=p+nk\) and such that \(P_ 0f_ j-P_ j=O(z^{s+1})\). This is a symmetric version of the so called simultaneous Padé approximation problem. If the problem has a unique solution \(P_ j/P_ 0\) then \(P_ 0\) has to be a polynomial in \(z^ n\). Since uniqueness doesn't always hold the author derives determinant conditions for \(P_ 0\) to be a polynomial in \(z^ n\) and shows that these are satisfied when f belongs to certain classes of hypergeometric functions.