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The Relationship Between Semiclassical Laguerre Polynomials and the Fourth Painlevé Equation

Abstract

We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semiclassical Laguerre weight and classical solutions of the fourth Painlevé equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions that arise in the description of special function solutions of the fourth Painlevé equation.

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Acknowledgements

We thank the London Mathematical Society for support through a “Research in Pairs” grant. PAC thanks Ana Loureiro, Paul Nevai, James Smith, and Walter van Assche for their helpful comments and illuminating discussions. We also thank the referees for helpful suggestions and additional references.

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Correspondence to Peter A. Clarkson.

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Communicated by Percy Deift and Alexander Its.

Appendices

Appendix 1: Recurrence Coefficients and Polynomials for the Semiclassical Laguerre Weight

For the semiclassical Laguerre weight, the first few recurrence coefficients are given by

$$\begin{aligned} \alpha_0(t) =&\frac{1}{2}t- \frac{D_{-\lambda} (-\frac{1}{2}\sqrt{2} t )}{D_{-\lambda-1} (-\frac{1}{2}\sqrt{2}t )}\equiv\varPsi _\nu (t), \\ \alpha_1(t) =& \frac{1}{2}t-\varPsi_\nu(t)- \frac{\varPsi_\nu(t)}{2\varPsi _\nu^2(t)-t\varPsi_\nu(t)-\lambda-1}, \\ \alpha_2(t) =&\frac{1}{2}t+\frac{2\lambda+4}{t}+\frac{\varPsi_\nu (t)}{2\varPsi _\nu^2(t)-t\varPsi_\nu(t)-\lambda-1} \\ &{}-\frac{2[(\lambda+1)t^2+4(\lambda+2)(2\lambda+3)]\varPsi _\nu^2(t)-(\lambda +1)t[t^2+2(4\lambda+9)]\varPsi_\nu(t)- (\lambda+1)^2[t^2+8(\lambda +2)]}{2t [ 2t\varPsi_\nu^{3}(t)-({t}^{2}-4\lambda-6) \varPsi_\nu^{2}(t) -3(\lambda+1)t \varPsi_\nu(t) -2(\lambda+1) ^{2} ] }, \\ \beta_1(t) =&-\varPsi_\nu^2(t)+\frac{1}{2}t \varPsi_\nu(t)+\frac{1}{2}(\lambda +1), \\ \beta_2(t) =&-\frac{2t\varPsi_\nu^3(t)-(t^2-4\lambda-6)\varPsi_\nu ^2(t)-3(\lambda+1)t\varPsi_\nu(t)-2(\lambda+1)^2}{2 [\varPsi_\nu ^2(t)-\frac{1}{2}t\varPsi_\nu(t)-\frac{1}{2}(\lambda+1) ]^2}, \end{aligned}$$

and the first few monic orthogonal polynomials are given by

$$\begin{aligned} &P_1(x;t)=x-\varPsi_\nu, \\ &P_2(x;t)=x^{2}-\frac{2t\varPsi_\nu^2-(t^2+2)\varPsi_\nu-(\lambda +1)t}{2 [\varPsi_\nu^2-\frac{1}{2}t\varPsi_\nu -\frac{1}{2}(\lambda +1) ]}x\\ &\phantom{P_1(x;t)=}{} - \frac{2(\lambda+2)\varPsi_\nu^2-(\lambda+1)\varPsi_\nu-(\lambda +1)^2}{2 [\varPsi_\nu^2-\frac{1}{2}t\varPsi_\nu-\frac{1}{2}(\lambda +1) ]}, \\ &P_3(x;t)= x^3\\ &\quad\!{}- \biggl\{ \frac{4(t^2+2\lambda+4)\varPsi_\nu ^3-2t(t^2-\lambda -1)\varPsi_\nu^2-(\lambda+1)(5t^2+4\lambda+6)\varPsi_\nu-3(\lambda+1)^2t}{ 2 [ 2t\varPsi_\nu^{3}-({t}^{2}-4\lambda-6) \varPsi_\nu^{2} -3(\lambda+1)t \varPsi_\nu-2(\lambda+1) ^{2} ]} \biggr\} x^2 \\ &\quad\!{}+ \biggl\{ \frac{2t(t^2+2\lambda+4)\varPsi_\nu^3- [t^4+4(2\lambda +5)(\lambda+2) ]\varPsi_\nu^2-2(\lambda+1)t(t^2-\lambda-5)\varPsi _\nu-(\lambda +1)^2(t^2-4\lambda-12)}{4 [ 2t\varPsi_\nu^{3}-({t}^{2}-4\lambda -6) \varPsi _\nu^{2} -3(\lambda+1)t \varPsi_\nu-2(\lambda+1) ^{2} ]} \biggr\} x \\ &\quad\!{}+\frac{2 [(\lambda+1)t^2+4(\lambda+2)^2 ]\varPsi_\nu ^3-(\lambda +1)t(t^2+2\lambda+8)\varPsi_\nu^2-2(\lambda+1)^2(t^2+2\lambda +5)\varPsi_\nu-(\lambda+1)^3t}{ 4 [ 2t\varPsi_\nu^{3}-({t}^{2}-4\lambda-6) \varPsi_\nu^{2} -3(\lambda+1)t \varPsi_\nu-2(\lambda+1) ^{2} ]}. \end{aligned} $$

Appendix 2: Recurrence Coefficients and Polynomials for the Semiclassical Hermite Weight

For the semiclassical Hermite weight x 2exp(−x 2+tx), the first few recurrence coefficients are given by

$$\begin{aligned} \alpha_0(t) =&\frac{1}{2}t+{ \frac{2t}{t^{2}+2}}, \\ \alpha_1(t) =&\frac{1}{2}t+{\frac{4t^{3}}{t^{4}+12}}-{\frac {2t}{t^{2}+2}}, \\ \alpha_2(t) =&\frac{1}{2}t+{\frac{6t( t^{4}-4t^{2}+12)}{t^{6}-6t^{4}+36t^{2}+72}}-{\frac {4t^{3}}{t^{4}+12}}, \\ \alpha_3(t) =&\frac{1}{2}t+{\frac {8t^{3}(t^{4}-12t^{2}+60)}{t^{8}-16t^{6}+120t^{4}+720}}-{\frac{6t( t^{4}-4t^{2}+12)}{t^{6}-6t^{4}+36t^{2}+72}}, \\ \alpha_4(t) =&\frac{1}{2}t+{\frac {10t(t^{8}-24t^{6}+216t^{4}-480t^{2}+720)}{t^{10}-30t^{8}+360t^{6}-1200t^{4}+3600t^{2}+7200}} \\ &{} -{\frac{8t^{3}(t^{4}-12t^{2}+60)}{t^{8}-16t^{6}+120t^{4}+720}}, \\ \alpha_5(t) =&\frac{1}{2}t+{\frac {12t^{3}(t^{8}-40t^{6}+600t^{4}-3360t^{2}+8400)}{t^{12}-48t^{10}+900t^{8}-6720t^{6}+25200t^{4}+100800}}\\ &{} -{ \frac {10t(t^{8}-24t^{6}+216t^{4}-480t^{2}+720)}{t^{10}-30t^{8}+360t^{6}-1200t^{4}+3600t^{2}+7200}}, \\ \beta_1(t) =&\frac{1}{2}-{\frac{2({t}^{2}-2)}{({t}^{2}+2)^{2}}}, \\ \beta_2(t) =&1-{\frac {4{t}^{2}({t}^{2}-6)({t}^{2}+6)}{({t}^{4}+12)^{2}}}, \\ \beta_3(t) =&\frac{3}{2}-{\frac {6({t}^{4}-12{t}^{2}+12)({t}^{6}+6{t}^{4}+36{t}^{2}-72)}{({t}^{6}-6{t}^{4}+36{t}^{2}+72)^{2}}} \\ \beta_4(t) =&2-{\frac {8{t}^{2}({t}^{4}-20{t}^{2}+60)({t}^{8}+72{t}^{4}-2160)}{({t}^{8}-16{t}^{6}+120{t}^{4}+720)^{2}}}, \\ \beta_5(t) =&\frac{5}{2}-{\frac {10({t}^{6}-30{t}^{4}+180{t}^{2}-120)({t}^{12}-12{t}^{10}+180{t}^{8}-480{t}^{6}-3600{t}^{4}- 43200{t}^{2}+43200)}{({t}^{10}-30{t}^{8}+360{t}^{6}-1200{t}^{4}+3600{t}^{2}+7200)^{2}}}, \end{aligned}$$

and the first few monic orthogonal polynomials are given by

$$\begin{aligned} P_1(x;t)&=x-{\frac{t ( {t}^{2}+6 ) }{2({t}^{2}+2)}}, \\ P_2(x;t)&={x}^{2}-{\frac{t ( {t}^{4}+4{t}^{2}+12 )}{{t}^{4}+12}}x+{ \frac {{t}^{6}+6{t}^{4}+36{t}^{2}-72}{4({t}^{4}+12)}}, \\ P_3(x;t)&={x}^{3}-{\frac{3t ( {t}^{6}-2{t}^{4}+20{t}^{2}+120)}{2({t}^{6}-6{t}^{4}+36{t}^{2}+72)}}x^2+{ \frac{ 3( {t}^{8}+40{t}^{4}-240 )}{4({t}^{6}-6{t}^{4}+36{t}^{2}+72)}}x \\ &\phantom{=}{}-{\frac{ t( {t}^{8}+72{t}^{4}-2160 )}{8({t}^{6}-6{t}^{4}+36{t}^{2}+72)}}, \\ P_4(x;t)&= {x}^{4}-{\frac{2t ({t}^{8}-12{t}^{6}+72{t}^{4}+240{t}^{2} +720)}{{t}^{8}-16{t}^{6}+120{t}^{4}+720}} {x}^{3}\\ &\phantom{=}{} +{\frac{ 3( {t}^{10}-10{t}^{8}+80{t}^{6}+1200{t}^{2}-2400)}{2({t}^{8}-16{t}^{6}+120{t}^{4}+720)}} {x}^{2} \\ &\phantom{=}-{\frac{t( {t}^{10}-10{t}^{8}+120{t}^{6}-240{t}^{4}-1200{t}^{2}-7200 )}{2({t}^{8}-16{t}^{6}+120{t}^{4}+720)}}x \\ &\phantom{=}+{\frac {{t}^{12}-12{t}^{10}+180{t}^{8}-480{t}^{6}-3600{t}^{4}-43200{t}^{2}+43200}{16({t}^{8}-16{t}^{6}+120{t}^{4}+720)}}, \\ P_5(x;t)&={x}^{5}-{\frac{ 5t(t^{10}-26t^{8}+264t^{6}-336t^{4}+1680t^{2}+10080 )}{2(t^{10}-30t^{8}+360 t^{6}-1200t^{4}+3600t^{2}+7200)}}x^4 \\ &\phantom{= }+{\frac{5 ( t^{12}-24t^{10}+252t^{8}-672t^{6}+5040t^{4}-20160)}{2(t^{10}-30t^{8}+360t^{6}-1200t^{4}+3600t^{2}+7200)}}x^3 \\ &\phantom{= }-{\frac{ 5t( t^{12}-24t^{10}+300t^{8}-1440t^{6}+5040t^{4}-100800 )}{ 4(t^{10}-30t^{8}+360t^{6}-1200t^{4}+3600t^{2}+7200)}}x^2 \\ &\phantom{= }+{\frac{ 5( t^{14}-26t^{12}+396t^{10}-2520t^{8}+5040t^{6}-50400t^{4}-100800t^{2}+201600 )}{ 16(t^{10}-30t^{8}+360t^{6}-1200t^{4}+3600t^{2}+7200)}}x \\ &\phantom{= }-{\frac{ t( t^{14}-30t^{12}+540t^{10}-4200t^{8}+10800t^{6}-151200t^{4}-504000t^{2}+3024000)}{ 32(t^{10}-30t^{8}+360t^{6}-1200t^{4}+3600t^{2}+7200)}}. \end{aligned}$$

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Clarkson, P.A., Jordaan, K. The Relationship Between Semiclassical Laguerre Polynomials and the Fourth Painlevé Equation. Constr Approx 39, 223–254 (2014). https://doi.org/10.1007/s00365-013-9220-4

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Keywords

  • Semiclassical orthogonal polynomials
  • Recurrence coefficients
  • Painlevé equations
  • Wronskians
  • Parabolic cylinder functions
  • Hamiltonians

Mathematics Subject Classification (2010)

  • 34M55
  • 33E17
  • 33C47
  • 42C05
  • 33C15