Asymptotics of Orthogonal Polynomials for a Weight with a Jump on [−1,1]

By A. F. Moreno, Andrei Martínez-Finkelshtein, Vítor Luís Sousa. 2009

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AbstractWe consider the orthogonal polynomials on [−1,1] with respect to the weight $$w_c(x)=h(x)(1-x)^{\alpha}(1+x)^{\beta} \varXi _{c}(x),\quad\alpha,\beta>-1,$$ where h is real analytic and strictly positive on [−1,1] and Ξc is a step-like function: Ξc(x)=1 for x∈[−1,0) and Ξc(x)=c2, c>0, for x∈[0,1]. We obtain strong uniform asymptotics of the monic orthogonal polynomials in ℂ, as well as first terms of the asymptotic expansion of the main parameters (leading coefficients of the orthonormal polynomials and the recurrence coefficients) as n→∞. In particular, we prove for wc a conjecture of A. Magnus regarding the asymptotics of the recurrence coefficients. The main focus is on the local analysis at the origin. We study the asymptotics of the Christoffel–Darboux kernel in a neighborhood of the jump and show that the zeros of the orthogonal polynomials no longer exhibit clock behavior.For the asymptotic analysis we use the steepest descent method of Deift and Zhou applied to the noncommutative Riemann–Hilbert problems characterizing the orthogonal polynomials. The local analysis at x=0 is carried out in terms of confluent hypergeometric functions. Incidentally, we establish some properties of these functions that may have an independent interest.

Subfields: Applied Mathematics, Atomic and Molecular Physics, and Optics, Mathematical Physics

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