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Transactions of the American Mathematical Society

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Orthogonal polynomials defined by a recurrence relation


Author: Paul G. Nevai
Journal: Trans. Amer. Math. Soc. 250 (1979), 369-384
MSC: Primary 42C05
DOI: https://doi.org/10.1090/S0002-9947-1979-0530062-6
MathSciNet review: 530062
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Abstract: R. Askey has conjectured that if a system of orthogonal polynomials is defined by the three term recurrence relation

$\displaystyle x{p_{n\, - \,1}}\left( x \right)\, = \,\frac{{{\gamma _{n\, - \,1... ...amma _{n\, - \,2}}}} {{{\gamma _{n\, - \,1}}}}\,{p_{n\, - \,2}}\left( x \right)$

and

$\displaystyle {\alpha _n}\, = \,\frac{{{{( - 1)}^n}}} {n}\,{\text{const}}\,{\text{ + }}O\left( {\frac{1} {{{n^2}}}} \right),$

$\displaystyle \frac{{{\gamma _n}}} {{{\gamma _{n + 1}}}}\, = \,\frac{1} {2}\, +... ...1)}^n}}} {n}\,{\text{const}}\,{\text{ + }}O\left( {\frac{1} {{{n^2}}}} \right),$

then the logarithm of the absolutely continuous portion of the corresponding weight function is integrable. The purpose of this paper is to prove R. Askey's conjecture and solve related problems.

References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/S0002-9947-1979-0530062-6
Article copyright: © Copyright 1979 American Mathematical Society

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