Asymptotic properties of a family of orthogonal polynomial sequences
The authors consider the real J-fraction \[ 1+\frac{b_ 1}{z+b_ 2}- \frac{b_ 2b_ 3}{z+(b_ 3+b_ 4)}-\frac{b_ 4b_ 5}{z+(b_ 5+b_ 6)}-...\quad, \] with \(b_ n=1/\{4[4(n+1)^ 2-1]\}\), \(a\in {\mathcal R}\setminus \{-1/2,-3/2,-5/2,...\}\) which is the even contraction of the corresponding fraction (C-fraction) \[ 1+\frac{b_ 1z}{1}+\frac{b_ 2z}{1}+\frac{b_ 3z}{1}+...\quad. \] It is well known that the sequences of inverted denominators \(z^ nQ_ n(z^{-1})\) (orthogonal polynomials) and inverted numerators \(z^ nP_ n)z^{-1})\) (associated polynomials) are both convergent (so called `separate convergence' following from \(\sum^{\infty}_{n=1}| b_ n| <\infty)\). The authors give these limits explicitly \[ \lim_{n\to \infty}z^ nP_ n(z^{-1})=_ 0F_ 1(a+1/2;(1/16)z),\quad \lim_{n\to \infty}z^ nQ_ n(z^{-1})=_ 0F_ 1(a+3/2;(1/16)z). \] Moreover, the orthogonal polynomials are in the case \(a>-1/2\) connected with a determinate Stieltjes moment problem. The distribution function reduces for \(a=0\) to a step-function with infinitely many points of increase and is exhibited explicitly in the paper (its Stieltjes transform is 1/2\(\sqrt{z} \coth 1/2\sqrt{z}-2\sqrt{z}).\) The proofs are straightforward, using formulae for the coefficients in \(P_ n\), \(Q_ n\) with error term O(1/n). A nicely written, interesting paper.
scientific article; zbMATH DE number 4175379
scientific article