On a family of orthogonal polynomials related to elliptic functions
For fixed \(a\in\mathbb R\), the authors introduce the family \(\{G_n(x;a)\}_{n=0}^\infty\) of orthogonal polynomials generated by \[ -xG_n=2(n+1)(2n+1)G_{n+1}+2n(2n+1)G_{n-1}-2a(2n+1)^2G_n,\quad n>0 \] with \[ G_0(x;a):=1\quad\text{and}\quad G_1(x;a):=a-x/2, \] and the corresponding numerator polynomials \(\{G_n^*(x;a)\}\) which are solutions of the same recurrence relation, but has \(G_0^*(x;a):=0\) and \(G_1^*(x;a):=-1/2\). For these polynomials the generating functions are established, from which their asymptotics are worked out. Thereby it follows that the corresponding Hamburger moment problem is determinate if and only if \(| a| \geq 1\). For the determinate case the discrete orthogonality measure is derived, and for the indeterminate case an infinite family of weight functions is derived. When \(a=1\) and \(x\) is replaced by \(x-4\), the \(G_n\)'s are special continuous Hahn polynomials studied in \textit{G. E. Andrews} and \textit{R. A. Askey} Lect. Notes Math. 1171, 36-63 (1984; Zbl 0596.33016); 2. \textit{M. E. H. Ismail}, \textit{J. Letessier} and \textit{G. Valent}, SIAM J. Math. Anal. 20, No. 3, 727-737 (1989; Zbl 0692.33009). \(\{G_n(x;a)\}\) is also a sequence of orthogonal polynomials in \(a\) for fixed \(x\in\mathbb R\). This sequence is connected with certain elliptic functions. The absolutely continuous part of its orthogonality measure is found. It is supported on \([-1,1]\). The choice \(x:=0\) gives the Legendre polynomials.
scientific article; zbMATH DE number 1191163
scientific article