# Asymptotics of Polynomials Orthogonal on a Cross with a Jacobi-Type Weight

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## Abstract

We investigate asymptotic behavior of polynomials $$Q_n(z)$$ satisfying non-Hermitian orthogonality relations

\begin{aligned} \int _\Delta s^kQ_n(s)\rho (s){\mathrm {d}}s =0, \quad k\in \{0,\ldots ,n-1\}, \end{aligned}

where $$\Delta := [-\,a,a]\cup [-\,{\mathrm {i}}b,{\mathrm {i}}b]$$, $$a,b>0$$, and $$\rho (s)$$ is a Jacobi-type weight.

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1. 1.

Here and in what follows we state jump relations understanding that they hold outside the points of self-intersection of the considered arcs.

2. 2.

$$g_\Delta (z;\infty )$$ is equal to zero on $$\Delta$$, is positive and harmonic in $${\mathbb {C}}{\setminus }\Delta$$, and satisfies $$g(z;\infty )=\log |z|+{\mathcal {O}}(1)$$ as $$z\rightarrow \infty$$.

3. 3.

In what follows we write $$|g_1(z)|\sim |g_2(z)|$$ as $$z\rightarrow z_0$$ if there exists a constant $$C>1$$ such that $$C^{-1}|g_1(z)|\le |g_2(z)| \le C|g_1(z)|$$ for all $$z$$ close to $$z_0$$.

4. 4.

Hereafter, we set $$\sigma _3 := \left( \begin{matrix} 1 &{} 0 \\ 0 &{} -1 \end{matrix}\right)$$ and $${\varvec{I}}$$ to be the identity matrix.

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## Author information

Correspondence to Maxim L. Yattselev.