A note on orthogonal polynomials and oscillation criteria for second order linear difference equations
The paper is devoted to the study of the oscillation properties of solutions of the second-order linear difference equation (1) \(c_ nx_{n+1}+c_{n-1}x_{n-1}=b_ nx_ n\), \(n=1,2,...\), with \(x_ 0=0\), \(x_ 1=1\), where \(b_ n\) is real and \(c_ n>0\), the purpose being to show that the question of whether a solution \(x_ n\) of (1) is oscillatory may be phrased as a problem in the context of orthogonal polynomials. Define the sequence of polynomials \(P_ n(x)\) by the relations \(P_ n(x)=(x-d_ n)P_{n-1}(x)-\lambda_ nP_{n-2}(x)\), \(n=1,2,...\), \(P_{-1}(x)=0\), \(P_ 0(x)=1\), where \(d_ n\equiv -b_ n/c_ n\), \(\lambda_ n\equiv c_{n-1}/c_ n\), \(n=1,2,... \). If \({\mathfrak u}=\{u_ 0,u_ 1,...\}\) is an infinite sequence of real numbers, then S(\({\mathfrak u})\) denotes the number of sign changes in the sequence \({\mathfrak u}\) when we delete all zero terms. The main result of this paper refers to the equivalence of the following statements: (a) \(\{x_ n\}\) is oscillatory; (b) \(S({\mathbb{P}}(0))=\infty\), where \({\mathbb{P}}(0)=\{P_ 0(0),P_ 1(0),...\}\); (c) \(\eta_ j=\lim_{n\to \infty}x_{n.n-j+1}>0\) for all \(j=1,2,...\), where \(x_{n,i}\), \(i=1,...,n\), are the zeros of \(P_ n(x)\). On the other hand, the author shows that if \(b_ n\leq 0\) for at most finitely many n, then \(\{x_ n\}\) is oscillatory if and only if for every \(N\geq 0\) the sequence \(\{c^ 2_{N+n}/(b_{N+n}b_{N+n+1})\}_ 1^{\infty}\) is not a chain sequence.
scientific article; zbMATH DE number 3973473