The Hankel Determinants from a Singularly Perturbed Jacobi Weight
We study the Hankel determinant generated by a singularly perturbed Jacobi weight <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mrow><mo>(</mo><mi>x</mi&...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/22/2978 |
| _version_ | 1827676189005709312 |
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| author | Pengju Han Yang Chen |
| author_facet | Pengju Han Yang Chen |
| author_sort | Pengju Han |
| collection | DOAJ |
| description | We study the Hankel determinant generated by a singularly perturbed Jacobi weight <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>s</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>)</mo></mrow><mi>α</mi></msup><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo></mrow><mi>β</mi></msup><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><mfrac><mi>s</mi><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow></mfrac></mrow></msup><mo>,</mo><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mi>x</mi><mo>∈</mo><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mo>,</mo><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mi>α</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mi>β</mi><mo>></mo><mn>0</mn><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mi>s</mi><mo>≥</mo><mn>0</mn><mo>.</mo></mrow></semantics></math></inline-formula> If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, it is reduced to the classical Jacobi weight. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, the factor <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><mfrac><mi>s</mi><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow></mfrac></mrow></msup></semantics></math></inline-formula> induces an infinitely strong zero at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. For the finite <i>n</i> case, we obtain four auxiliary quantities <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>r</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>R</mi><mo>˜</mo></mover><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>r</mi><mo>˜</mo></mover><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> by using the ladder operator approach. We show that the recurrence coefficients are expressed in terms of the four auxiliary quantities with the aid of the compatibility conditions. Furthermore, we derive a shifted Jimbo–Miwa–Okamoto <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-function of a particular Painlevé V for the logarithmic derivative of the Hankel determinant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. By variable substitution and some complicated calculations, we show that the quantity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> satisfies the four Painlevé equations. For the large <i>n</i> case, we show that, under a double scaling, where <i>n</i> tends to <i>∞</i> and <i>s</i> tends to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>0</mn><mo>+</mo></msup></semantics></math></inline-formula>, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>:</mo><mo>=</mo><msup><mi>n</mi><mn>2</mn></msup><mi>s</mi></mrow></semantics></math></inline-formula> is finite, the scaled Hankel determinant can be expressed by a particular <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mrow><mi>I</mi><mi>I</mi><msup><mi>I</mi><mo>′</mo></msup></mrow></msub></semantics></math></inline-formula>. |
| first_indexed | 2024-03-10T05:18:15Z |
| format | Article |
| id | doaj.art-d6b7b74391b445a7b9ac56a998206092 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T05:18:15Z |
| publishDate | 2021-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-d6b7b74391b445a7b9ac56a9982060922023-11-23T00:16:03ZengMDPI AGMathematics2227-73902021-11-01922297810.3390/math9222978The Hankel Determinants from a Singularly Perturbed Jacobi WeightPengju Han0Yang Chen1College of Science, Huazhong Agricultural University, Wuhan 430070, ChinaDepartment of Mathematics, Faculty of Science and Technology, University of Macau, Macau 999078, ChinaWe study the Hankel determinant generated by a singularly perturbed Jacobi weight <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>s</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>)</mo></mrow><mi>α</mi></msup><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo></mrow><mi>β</mi></msup><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><mfrac><mi>s</mi><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow></mfrac></mrow></msup><mo>,</mo><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mi>x</mi><mo>∈</mo><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mo>,</mo><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mi>α</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mi>β</mi><mo>></mo><mn>0</mn><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mi>s</mi><mo>≥</mo><mn>0</mn><mo>.</mo></mrow></semantics></math></inline-formula> If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, it is reduced to the classical Jacobi weight. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, the factor <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><mfrac><mi>s</mi><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow></mfrac></mrow></msup></semantics></math></inline-formula> induces an infinitely strong zero at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. For the finite <i>n</i> case, we obtain four auxiliary quantities <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>r</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>R</mi><mo>˜</mo></mover><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>r</mi><mo>˜</mo></mover><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> by using the ladder operator approach. We show that the recurrence coefficients are expressed in terms of the four auxiliary quantities with the aid of the compatibility conditions. Furthermore, we derive a shifted Jimbo–Miwa–Okamoto <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-function of a particular Painlevé V for the logarithmic derivative of the Hankel determinant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. By variable substitution and some complicated calculations, we show that the quantity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> satisfies the four Painlevé equations. For the large <i>n</i> case, we show that, under a double scaling, where <i>n</i> tends to <i>∞</i> and <i>s</i> tends to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>0</mn><mo>+</mo></msup></semantics></math></inline-formula>, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>:</mo><mo>=</mo><msup><mi>n</mi><mn>2</mn></msup><mi>s</mi></mrow></semantics></math></inline-formula> is finite, the scaled Hankel determinant can be expressed by a particular <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mrow><mi>I</mi><mi>I</mi><msup><mi>I</mi><mo>′</mo></msup></mrow></msub></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/9/22/2978random matrix theoryHankel determinantsingularly perturbed Jacobi weightladder operatorsPainlevé V |
| spellingShingle | Pengju Han Yang Chen The Hankel Determinants from a Singularly Perturbed Jacobi Weight Mathematics random matrix theory Hankel determinant singularly perturbed Jacobi weight ladder operators Painlevé V |
| title | The Hankel Determinants from a Singularly Perturbed Jacobi Weight |
| title_full | The Hankel Determinants from a Singularly Perturbed Jacobi Weight |
| title_fullStr | The Hankel Determinants from a Singularly Perturbed Jacobi Weight |
| title_full_unstemmed | The Hankel Determinants from a Singularly Perturbed Jacobi Weight |
| title_short | The Hankel Determinants from a Singularly Perturbed Jacobi Weight |
| title_sort | hankel determinants from a singularly perturbed jacobi weight |
| topic | random matrix theory Hankel determinant singularly perturbed Jacobi weight ladder operators Painlevé V |
| url | https://www.mdpi.com/2227-7390/9/22/2978 |
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