Polynomial sequences with prescribed power sums of zeros (Q1318428)

Let \(V_ n(z)\) be a complex polynomial with zeros \(z_ 1,\dots,z_ n\). The value \(\sigma_ k=z_ 1^ k+\cdots+z_ n^ k\) is called a power sum of zeros (psz). In this paper, the author studies polynomials having a prescribed odd sequence \(\sigma_ 2,\sigma_ 3,\sigma_ 5,\dots\) of psz. These polynomials are given by \(V_ n(z)=z^ nU_ n(z^{-1})\) where \(U_ n(z)\) is the denominator of the \((n,n)\) Padé approximant for \(f(z)=\exp(g(z))\) with \(g(z)=2\sum_{m=1}^ \infty\sigma_{2m- 1}z^{1-2m}/(2m-1)\) (the factor \(1/(2m-1)\) is at the wrong place in the definition 1.1 on p. 318), therefore they form an orthogonal polynomial system (OPS) with respect to the sequence of MacLaurin coefficients of \(f(z)\). The origin of this problem goes back to Laguerre. Bessel polynomials are an example of such polynomials with \(\sigma_ 1=-1\) and \(\sigma_{2m-1}=0\) for \(m\geq 2\). More general properties concerning the differential equation and multiplicity of the zeros of the OPS are derived for the case where only a finite number of \(\sigma_{2m-1}\) are nonzero. Similar, somewhat weaker results hold for an infinite number of nonzero \(\sigma_{2m-1}\). Finally the totally positive polynomials, as discussed by Arms and Edrei [these are reversed denominators of diagonal Padé approximants for \(f(z)=e^ z\prod_{j=1}^ \infty(1+\alpha_ jz)/(a-\alpha_ jz)]\) are identified as a solution of the problem considered here. Results about the location and accumulation points of their zeros are derived. This includes a new generalization of Bessel polynomials.