Nonlinear equations for the recurrence coefficients of discrete orthogonal polynomials (Q624580)

Let \(a\) and \(h>0\) be fixed and let \(a_{i}=a+ih\) for all integers \(i\). Let \( \left\{ P_{n}\right\} _{n=0}^{\infty }\) be monic polynomials that satisfy the orthogonality relation \[ \sum_{i=s}^{t}P_{m}(a_{i})P_{n}(a_{i})w(a_{i})=\zeta _{m}\delta _{m,n}, \] where \(t\in (0,+\infty ]\) and \(s\in [ -\infty ,t),\) and \(w\) is a positive weight function with finite moments and satisfies \( w(a_{s-1})=w(a_{t+1})=0.\) It is shown that \(\left\{ P_{n}\right\} \) satisfies the three-term recurrence relation \[ P_{n+1}(\lambda )+(\alpha _{n}-\lambda )P_{n}(\lambda )+\beta _{n}P_{n-1}(\lambda )=0, \] where \(P_{0}(\lambda )=1\) and \(P_{1}(\lambda )=\lambda -\alpha _{0},\) while the coefficients \(\left\{ \alpha _{n}\right\} \) and \(\left\{ \beta _{n}\right\} \) can be determined by several recurrence relations. Several illustrative examples are included by taking \(w(x)=1/\Gamma (x),1/\Gamma (2x),1/\Gamma (3x),\Gamma (x+b)\mu ^{x}/\Gamma (x)\) and \(\Gamma (x+b)\mu ^{x}/\Gamma (2x)\) where \(\mu \in (0,1).\) For example, when \(w(x)=1/\Gamma (x),\) \(\beta _{1}=1,\beta _{n}=n\) and \(\alpha _{n}=n+2\) can be computed from the recurrence relations and hence the governing recurrence for \(\left\{ P_{n}\right\} \) is \[ P_{n+1}(\lambda )+(n+2-\lambda )P_{n}(\lambda )+nP_{n-1}(\lambda )=0. \]