Polynomials that are orthogonal with respect to a perturbation of the Freud weight function by some parameter, known to be modified Freudian orthogonal polynomials, are considered. In this contribution, we investigate certain properties of semi-classical modified Freud-type polynomials in which their corresponding semi-classical weight function is a more general deformation of the classical scaled sextic Freud weight ${\left|x\right|}^{\alpha }exp(-c{x}^6),c>0,\alpha >-1$. Certain characterizing properties of these polynomials such as moments, recurrence coefficients, holonomic equations that they satisfy, and certain non-linear differential-recurrence equations satisfied by the recurrence coefficients, using compatibility conditions for ladder operators for these orthogonal polynomials, are investigated. Differential-difference equations were also obtained via Shohat’s quasi-orthogonality approach and also second-order linear ODEs (with rational coefficients) satisfied by these polynomials. Modified Freudian polynomials can also be obtained via Chihara’s symmetrization process from the generalized Airy-type polynomials. The obtained linear differential equation plays an essential role in the electrostatic interpretation for the distribution of zeros of the corresponding Freudian polynomials.

中文翻译：

---

与修正性弗洛伊德型权重相关的半经典正交多项式

考虑相对于通过某些参数对弗洛伊德权函数的微扰正交的多项式，已知为修正的弗洛伊德正交多项式。在此贡献中，我们研究了半经典修正的弗洛伊德型多项式的某些性质，其中它们对应的半经典权重函数是经典比例性弗洛伊德权重的更一般变形${\left|X\right|}^{\alpha }经验值（-C{X}^6），C>0，\alpha >-\mathrm{1个}$。使用这些正交多项式的梯形算子的相容性条件，研究了这些多项式的某些特征，例如矩，递归系数，它们满足的完整方程以及递归系数所满足的某些非线性微分递归方程。还通过Shohat的拟正交方法以及这些多项式满足的二阶线性ODE（具有有理系数）获得了微分方程。修改后的弗洛伊德多项式也可以通过Chihara的对称化过程从广义的Airy型多项式获得。所获得的线性微分方程在静电解释中对于相应的弗洛伊德多项式零点的分布起着至关重要的作用。