Asymptotics of Polynomials Orthogonal With Respect to a Generalized Freud Weight With Application to Special Function Solutions of Painlevé-IV
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917701135368192 |
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| author | Barhoumi, Ahmad |
| author_facet | Barhoumi, Ahmad |
| contents | We obtain asymptotics of polynomials satisfying the orthogonality relations $$ \int_{\mathbb{R}} z^k P_n(z; t , N) \mathrm{e}^{-N \left(\frac{1}{4}z^4 + \frac{t}{2}z^2 \right)} \mathrm{d} z = 0 \quad \text{ for } \quad k = 0, 1, ..., n-1, $$ where the complex parameter $t$ is in the so-called two-cut region. As an application, we deduce asymptotic formulas for certain families of solutions of Painlevé-IV which are indexed by a non-negative integer and can be written in terms of parabolic cylinder functions. The proofs are based on the characterization of orthogonal polynomials in terms of a Riemann-Hilbert problem and the Deift-Zhou non-linear steepest descent method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_11294 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Asymptotics of Polynomials Orthogonal With Respect to a Generalized Freud Weight With Application to Special Function Solutions of Painlevé-IV Barhoumi, Ahmad Classical Analysis and ODEs Mathematical Physics Complex Variables Primary: 34M55, 33C47, Secondary: 15B52, 30E15, 34E05, 34M50 We obtain asymptotics of polynomials satisfying the orthogonality relations $$ \int_{\mathbb{R}} z^k P_n(z; t , N) \mathrm{e}^{-N \left(\frac{1}{4}z^4 + \frac{t}{2}z^2 \right)} \mathrm{d} z = 0 \quad \text{ for } \quad k = 0, 1, ..., n-1, $$ where the complex parameter $t$ is in the so-called two-cut region. As an application, we deduce asymptotic formulas for certain families of solutions of Painlevé-IV which are indexed by a non-negative integer and can be written in terms of parabolic cylinder functions. The proofs are based on the characterization of orthogonal polynomials in terms of a Riemann-Hilbert problem and the Deift-Zhou non-linear steepest descent method. |
| title | Asymptotics of Polynomials Orthogonal With Respect to a Generalized Freud Weight With Application to Special Function Solutions of Painlevé-IV |
| topic | Classical Analysis and ODEs Mathematical Physics Complex Variables Primary: 34M55, 33C47, Secondary: 15B52, 30E15, 34E05, 34M50 |
| url | https://arxiv.org/abs/2312.11294 |