On a nonlinear recurrence related to Nevai polynomials (Q2736545)

Orthogonal polynomials on the real line corresponding to the weight function \(w(x) = \exp(-x^4)\) have been studied by Freud and, most notably, Nevai. These ``Nevai'' polynomials \(\{p_n(x)\}\) have been shown to satisfy the recurrence relation NEWLINE\[NEWLINEx\cdot p_n(x)=a_{n+1}p_{n+1}(x)+a_np_{n-1}(x)NEWLINE\]NEWLINE where the coefficients \(\{a_n\}\) in the recursion in turn can be generated from the nonlinear recurrence NEWLINE\[NEWLINEn=4a_n^2(a_{n+1}^2+a_n^2+a_{n-1}^2)NEWLINE\]NEWLINE for \(n = 1,2,3,\dots,\) with starting values for the sequence of \(a_0 = 0\) and \(a_1^2=\Gamma(\tfrac34)/\Gamma(\tfrac14)\). Efficient iterative schemes have been developed for computing the sequence \(\{a_n\}\) numerically, but a closed form representation of the \(\{a_n\}\) suitable for use in developing a generating function has proven elusive, although uniform asymptotic approximations for the Nevai polynomials have recently been obtained by \textit{B. Rui} and \textit{R. Wong} [J. Approx. Theory 98, No. 1, 146--166 (1999; Zbl 0952.41018)]. This paper describes work underway towards the construction of a generating function for the \(\{a_n\}\) and related nonlinear recurrences. In particular, a simple transformation of the preceding three-term recurrence can yield a nonlinear recurrence involving just two terms, which may ultimately be solvable in closed form.NEWLINENEWLINEFor the entire collection see [Zbl 0954.00030].