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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Orthogonal polynomials defined by a recurrence relation

Author: Paul G. Nevai
Journal: Trans. Amer. Math. Soc. 250 (1979), 369-384
MSC: Primary 42C05
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

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


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

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

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?)

  • [1] K. M. Case, Orthogonal polynomials revisited, Theory and Application of Special Functions, (R. A. Askey, ed.), Academic Press, New York, 1975, pp. 289-304. MR 0390322 (52:11148)
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