Analog of Favard's theorem for polynomials connected with difference equation of 4-th order (Q2761691)

The author studies systems of polynomials \(\{p_n(\lambda)\}_{n=0}^{\infty}\) that satisfy a recurrence relation of the form NEWLINE\[NEWLINE\alpha_{n-2} p_{n-2} (\lambda) + \overline{\beta_{n-1}} p_{n-1} (\lambda)+ \gamma_n p_n (\lambda) + \beta_n p_{n+1} (\lambda) + \alpha_n p_{n+2} (\lambda)= \lambda^2 p_n (\lambda),\;n=0,1,2,\dots,\tag{1}NEWLINE\]NEWLINE with \(\alpha_n>0\), \(\beta_n \in C\), \(\gamma_n \in R\), \(n=0,1,2,\dots\), \(\alpha_{-1}=\alpha_{-2}=\beta_{-1}=0\); \(p_{-1}=p_{-2}=0\), \(p_0 (\lambda) =1\), \(p_1 (\lambda) = c\lambda+b\), \(c>0\), \(b \in C\), \(\lambda \in C\). NEWLINENEWLINENEWLINEThe same relation can be written also for the vector of polynomials \(p\) in a matrix form \ \(J_5 p = \lambda^2 p\), with a five-diagonal, semi-infinite matrix \(J_5\). NEWLINENEWLINENEWLINEThis article is based on the earlier definitions and results by the same author [\textit{S. Zagorodniuk}, Serdica Math. J. 24, No. 3-4, 257-282 (1998; Zbl 0946.47010)]. Here, an analog of Favard's theorem (see Theorem 1.5 in [\textit{G. Freud}, ``Orthogonale Polynome'' (1969; Zbl 0169.08002)]), asserts that the system of polynomials \(\{p_n (\lambda)\}_{n=0}^{\infty}\) satisfying (1) is orthonormal on the real and imaginary axes in the complex plane: NEWLINE\[NEWLINE \int_{R \cup T} (p_n(\lambda),p_n(-\lambda)) d\sigma (\lambda) \overline{\binom {p_m (\lambda)}{p_m(-\lambda)}} = \delta_{n,m}, \quad n,m = \overline{0,\infty}; \;T = (-\infty,\infty), NEWLINE\]NEWLINE where \(\sigma (\lambda)\) is a symmetric, non-decreasing matrix-function with infinite number of points of increase. NEWLINENEWLINENEWLINEAlso, a Green formula for the difference equation of 4-th order, corresponding to (1), is proposed.