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Chain Sequences, Orthogonal Polynomials, and Jacobi Matrices

✍ Scribed by Ryszard Szwarc

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
337 KB
Volume
92
Category
Article
ISSN
0021-9045

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✦ Synopsis

Chain sequences are positive sequences [a n ] of the form a n = g n (1& g n&1 ) for a nonnegative sequence [g n ]. This concept was introduced by Wall in connection with continued fractions. In his monograph on orthogonal polynomials, Chihara conjectured that if a n 1 4 for each n then (a n & 1 4 ) 1 4 . We prove this conjecture and give other precise estimates for a n . We also characterize the chain sequences [a n ] whose terms are greater than 1 4 . We show connections to Jacobi matrices and orthogonal polynomials. In particular, we characterize the maximal chain sequences in terms of integrability properties of the spectral measure of the associated Jacobi matrix.

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