Abstract. The purpose of the paper is to investigate distribution of zeros of orthogonal polynomials given by a three term recurrence relation.Let a be a nondecreasing and bounded function on the real line such that the range of a is infinite and x" E L2(da) for every n £ N. Then there exists a unique system of polynomials {pn(da)}™=0 such that p"(da, x) = yn{da)xn + • • • , y "(da) > 0, and /oo pn(da, t)pm(da, t) da(t) = 8nm.-00

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R. Askey has conjectured that if a system of orthogonal polynomials is defined by the three term recurrence relation \[ x p n − 1 ( x ) = γ n − 1 γ n p n ( x ) + α n − 1 p n − 1 ( x ) + γ n − 2 γ n − 1 p n − 2 ( x ) x{p_{n\, - \,1}}\left ( x \right )\, = \,\frac {{{\gamma _{n\, - \,1}}}} {{{\gamma _{n\,}}}}\,{p_n}\left ( x \right )\, + \,{\alpha _{n\, - \,1}}{p_{n\, - \,1}}\left ( x \right )\, + \,\frac {{{\gamma _{n\, - \,2}}}} {{{\gamma _{n\, - \,1}}}}\,{p_{n\, - \,2}}\left ( x \right ) \] and \[ α n = ( − 1 ) n n const  +  O ( 1 n 2 ) , {\alpha _n}\, = \,\frac {{{{( - 1)}^n}}} {n}\,{\text {const}}\,{\text { + }}O\left ( {\frac {1} {{{n^2}}}} \right ), \] \[ γ n γ n + 1 = 1 2 + ( − 1 ) n n const  +  O ( 1 n 2 ) , \frac {{{\gamma _n}}} {{{\gamma _{n + 1}}}}\, = \,\frac {1} {2}\, + \,\frac {{{{( - 1)}^n}}} {n}\,{\text {const}}\,{\text { + }}O\left ( {\frac {1} {{{n^2}}}} \right ), \] 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.

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Orthogonal polynomials on the unit circle are completely determined by their reflection coefficients through the Szegő recurrences. We assume that the reflection coefficients converge to some complex number a with 0 < |a| < 1. The polynomials then live essentiallyWe analyze the orthogonal polynomials by comparing them with the orthogonal polynomials with constant reflection coefficients, which were studied earlier by Ya. L. Geronimus and N. I. Akhiezer. In particular, we show that under certain assumptions on the rate of convergence of the reflection coefficients the orthogonality measure will be absolutely continuous on the arc. In addition, we also prove the unit circle analogue of M. G. Krein's characterization of compactly supported nonnegative Borel measures on the real line whose support contains one single limit point in terms of the corresponding system of orthogonal polynomials.1991 Mathematics Subject Classification. 42C05.