On the Zeros of a Class of Polynomials Defined by a Three Term Recurrence Relation

✍ Scribed by E.K. Ifantis; P.N. Panagopoulos

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 316 KB
• Volume: 182
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.1994.1091

✦ Synopsis

Let (P_{N+1}(x)) be the polynomial which is defined recursively by (P_{0}(x)=0), (P_{1}(x)=1, \quad) and (\alpha_{n} P_{n+1}(x)+\alpha_{n-1} P_{n-1}(x)+b_{n} P_{n}(x)=x d_{n} P_{n}(x), \quad n=1, \quad 2, \ldots, N), where (\alpha_{n}, b_{n}, d_{n}) are real sequences with (\alpha_{n} \neq 0), for every (n=1,2, \ldots, N), and (k) terms of the sequence (\left{d_{n}\right}_{n=1}^{\infty}, 0 \leqslant k0) and the sequence (\left{\alpha_{n}^{2} / b_{n} b_{n+1}\right}_{n=1}^{x}) is a chain sequence then the polynomial (P_{N+1}(x)) is of degree (N-k) and has real and simple zeros different from zero. This result generalizes and simplifies previously known results with respect to the class of polynomials whose zeros are real and simple. 1994 Academic Press, Inc.