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JOURNALS // Trudy Instituta Matematiki i Mekhaniki UrO RAN // Archive

Trudy Inst. Mat. i Mekh. UrO RAN, 2010 Volume 16, Number 4, Pages 65–73 (Mi timm641)

Some properties of Jacobi polynomials orthogonal on a circle

V. M. Badkov

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: Let $\{\psi^{(\alpha,\beta)}_n(z)\}_{n=0}^\infty$ be a system of Jacobi polynomials that is orthonormal on the circle $|z|=1$ with respect to the weight $(1-\cos\tau)^{\alpha+1/2}(1+\cos\tau)^{\beta+1/2}$ ($\alpha,\beta>-1$), and let $\psi_n^{(\alpha,\beta)*}(z):=z^n\overline{\psi_n^{(\alpha,\beta)}(1/\overline z)}$. We establish relations between the polynomial $\psi_n^{(\alpha,-1/2)}(z)$ and the $n$-th $(C,\alpha-1/2)$-mean of the Maclaurin series for the function $(1-z)^{-\alpha-3/2}$ and also between the polynomial $\psi_n^{(\alpha,-1/2)*}(z)$ and the $n$-th $(C,\alpha+1/2)$-mean of the Maclaurin series for the function $(1-z)^{-\alpha-1/2}$. We use these relations to derive an asymptotic formula for $\psi_n^{(\alpha, -1/2)}(z)$; the formula is uniform inside the disk $|z|<1$. It follows that $\psi_n^{(\alpha,-1/2)}(z)\neq0$ in the disk $|z|\le\rho$ for fixed $\rho\in(0,1)$ and $\alpha>-1$ if $n$ is sufficiently large.

Keywords: Jacobi polynomials, Cesáaro means, asymptotic formula, zeros.

UDC: 517.5

Received: 11.02.2010


 English version:
Proceedings of the Steklov Institute of Mathematics (Supplement Issues), 2011, 273, suppl. 1, S49–S58

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