A generalized sextic Freud weight

ABSTRACT

We discuss the recurrence coefficients of orthogonal polynomials with respect to a generalized sextic Freud weight $\omega (x;t,\lambda )=|x{|}^{2\lambda +1}\exp \left(-x^6+t{x}^2\right),x\in \mathbb{R},$ with parameters $\lambda >-1$ and $t\in \mathbb{R}$. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of generalized hypergeometric functions ${}_1{F}_2(a_1;b_1,b_2;z)$. We derive a nonlinear discrete as well as a system of differential equations satisfied by the recurrence coefficients and use these to investigate their asymptotic behaviour. We conclude by highlighting a fascinating connection between generalized quartic, sextic, octic and decic Freud weights when expressing their first moments in terms of generalized hypergeometric functions.

Keywords:

• Generalized sextic freud weight
• generalized hypergeometric functions
• semi-classical orthogonal polynomials
• recurrence coefficients
• discrete Painlevé equation
• Painlevé equation

2010 Mathematics Subject Classifications:

• 33C20
• 33C47
• 34M55
• 65Q99

Acknowledgments

We gratefully acknowledge the support of a Royal Society Newton Advanced Fellowship NAF∖R2∖180669. P.A.C. thanks Evelyne Hubert, Arieh Iserles, Ana Loureiro and Walter Van Assche for their helpful comments and illuminating discussions. P.A.C. would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme ‘Complex analysis: techniques, applications and computations’ when the work on this paper was undertaken. This work was supported by EPSRC grant number EP/R014604/1. We also thank the referees for helpful suggestions and additional references.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

We gratefully acknowledge the support of a Royal Society Newton Advanced Fellowship [grant number NAF∖R2∖180669]. This work was supported by EPSRC [grant number EP/R014604/1].