Orthogonal polynomials defined by a recurrence relation

Nevai, Paul G.

doi:10.1090/S0002-9947-1979-0530062-6

Orthogonal polynomials defined by a recurrence relation
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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

K. M. Case, Orthogonal polynomials revisited, Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) Math. Res. Center, Univ. Wisconsin, Publ. No. 35, Academic Press, New York, 1975, pp. 289–304. MR 0390322

Sur les polynomes de Tchebicheff

200

Paul G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, v+185. MR 519926, DOI 10.1090/memo/0213

On orthogonal polynomials

Paul G. Nevai, Distribution of zeros of orthogonal polynomials, Trans. Amer. Math. Soc. 249 (1979), no. 2, 341–361. MR 525677, DOI 10.1090/S0002-9947-1979-0525677-5

Orthogonal polynomials