Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. It consists of a four-parameter polynomial with continuous spectrum on the whole real line and two of its discrete versions; one with a finite spectrum and another with countably infinite spectrum. A second class of these new orthogonal polynomials appeared recently while solving a Heun-type equation. Based on these results and on our recent study of the solution space of an ordinary differential equation of the second kind with four singular points, we introduce a modification of the Askey scheme of hyper-geometric orthogonal polynomials. Up to now, these polynomials are defined by their three-term recursion relations and initial values. However, their other properties like the weight functions, generating functions, orthogonality, Rodrigues-type formulas, etc. are yet to be derived analytically. Due to the prime significance of these polynomials in physics and mathematics, we call upon experts in the field of orthogonal polynomials to study them, derive their properties and write them in closed form (e.g., in terms of hypergeometric functions).

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正交多项式中的开放问题

使用代数方法求解量子力学中的波动方程，我们在实线上遇到了一类新的正交多项式。它由在整条实线上具有连续谱的四参数多项式及其两个离散形式组成；一种具有有限谱，另一种具有可数无限谱。最近在求解 Heun 型方程时出现了第二类这些新的正交多项式。基于这些结果以及我们最近对具有四个奇异点的第二类常微分方程的解空间的研究，我们引入了对超几何正交多项式的 Askey 格式的修改。到目前为止，这些多项式是由它们的三项递归关系和初始值定义的。然而，它们的其他属性如权重函数，生成函数、正交性、罗德里格斯式公式等还有待解析导出。由于这些多项式在物理和数学中的重要意义，我们呼吁正交多项式领域的专家研究它们，推导出它们的性质并将它们写成封闭形式（例如，根据超几何函数）。