Orthogonal polynomials, bi-confluent Heun equations and semi-classical weights

Abstract

In this paper, we focus on four weights $\omega (z,s)=z^{\lambda }{\mathrm{e}}^{-N(z+s(z^2-z\left)\right)},$ where $z\in (0,\mathrm{\infty })$, $\lambda >-1$, $0\le s\le 1$, N>0; $\omega (z,t)=z^{\lambda }{\mathrm{e}}^{-z^2+tz},$ where $z\in (0,\mathrm{\infty })$, $\lambda >-1$, $t\in \mathbf{R}$; $\omega (z,t_1)={\mathrm{e}}^{-z^2}\left(A+B\theta (z-t_1)\right),$ with $z\in \mathbf{R}$, $A\ge 0$, $A+B\ge 0$, $B\ne 0$, where $\theta \left(z\right)$ is the Heaviside step function; and $\omega \left(z\right)=|z{|}^{\alpha }{\mathrm{e}}^{-N(z^2+s(z^4-z^2\left)\right)},$ with $z\in \mathbf{R}$, $\alpha >-1$, N>0, $0\le s\le 1$. The second-order differential equations satisfied by $P_n\left(z\right)$, the degree-n polynomials orthogonal with respect to each of these weights, are shown to be asymptotically equivalent to the bi-confluent Heun equations as $n\to \mathrm{\infty }$. In most cases, a parameter other than n must simultaneously be sent to a limiting value.

Keywords:

• Orthogonal polynomials
• Bi-confluent Heun equation
• asymptotic
• semi-classical

2010 Mathematics Subject Classifications:

• 33C47
• 34M55
• 35C20
• 65Q99

Acknowledgments

D. Wang, M. Zhu and Y. Chen would like to give thanks the Science and Technology Development Fund of the Macau SAR for providing FDCT 023/2017/A1. They would also like to thank the University of Macau for MYRG 2018-00125 FST.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

D. Wang, M. Zhu and Y. Chen would like to give thanks the Science and Technology Development Fund of the Macau SAR for providing FDCT 023/2017/A1. They would also like to thank the University of Macau for MYRG 2018-00125 FST.