Orthogonal polynomials defined by a recurrence relation
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- by Paul G. Nevai PDF
- Trans. Amer. Math. Soc. 250 (1979), 369-384 Request permission
Abstract:
R. Askey has conjectured that if a system of orthogonal polynomials is defined by the three term recurrence relation \[ x{p_{n - 1}}\left ( x \right ) = \frac {{{\gamma _{n - 1}}}} {{{\gamma _{n }}}} {p_n}\left ( x \right ) + {\alpha _{n - 1}}{p_{n - 1}}\left ( x \right ) + \frac {{{\gamma _{n - 2}}}} {{{\gamma _{n - 1}}}} {p_{n - 2}}\left ( x \right )\] and \[ {\alpha _n} = \frac {{{{( - 1)}^n}}} {n} {\text {const}} {\text { + }}O\left ( {\frac {1} {{{n^2}}}} \right ),\] \[ \frac {{{\gamma _n}}} {{{\gamma _{n + 1}}}} = \frac {1} {2} + \frac {{{{( - 1)}^n}}} {n} {\text {const}} {\text { + }}O\left ( {\frac {1} {{{n^2}}}} \right ),\] 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
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Additional Information
- © Copyright 1979 American Mathematical Society
- 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