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.

摘要

我们讨论关于广义六性弗洛伊德权重的正交多项式的递推系数 $\omega (X;吨,\lambda )=|X{|}^{2\lambda +1}\mathrm{经验值}\left(-X^6+吨X^2\right),X\in \mathbb{电阻},$ 带参数 $\lambda >-1$ 和 $吨\in \mathbb{电阻}$. 我们表明这些递推关系中的系数可以用广义超几何函数的 Wronskians 表示${}_1{F}_2({\mathrm{一种}}_1;{乙}_1,{乙}_2;z)$. 我们推导出非线性离散以及由递推系数满足的微分方程组，并使用它们来研究它们的渐近行为。最后，我们强调了广义四次、六次、八次和十次弗洛伊德权重在用广义超几何函数表达第一矩时之间的迷人联系。