Abstract "hypergeometric" orthogonal polynomials
Abstract "hypergeometric" orthogonal polynomials
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发表时间：
2014-01
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通讯作者：
A. Zhedanov
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作者：
A. Zhedanov
我们求出了抽象的“超几何”方程$LP_n（x）= \lambda_n P_n（x）$的所有多项式解$P_n（x）$，其中$L$是一个线性算子，它把任意n次多项式变换成一个具有在单项式基上L$是双对角的性质的同次多项式，即，$L x^n = \lambda_n x^n + \mu_n x^{n-1}$，其中系数$\lambda_n，\mu_n$为任意非零。在明显的非退化条件下，多项式本征解Lp_n（x）= \lambda_nPn（x）$是唯一的.本文的主要结果是这类多项式P_n（x）$的一个分类，即P_n（x）$被假定为关于一个非退化线性泛函$\sigma$正交.我们证明了唯一的解是：Jacobi，Laguerre（相应的小$q$-Jacobi和小$q$-Laguerre以及其他特殊和退化的情况），Bessel和小-1 Jacobi多项式.
We find all polynomials solutions $P_n(x)$ of the abstract "hypergeometric" equation $L P_n(x) = \lambda_n P_n(x)$, where $L$ is a linear operator sending any polynomial of degree $n$ to a polynomial of the same degree with the property that $L$ is two-diagonal in the monomial basis, i.e. $L x^n = \lambda_n x^n + \mu_n x^{n-1}$ with arbitrary nonzero coefficients $\lambda_n, \mu_n$ . Under obvious nondegenerate conditions, the polynomial eigensolutions $L P_n(x) = \lambda_n P_n(x)$ are unique. The main result of the paper is a classification of all {\it orthogonal} polynomials $P_n(x)$ of such type, i.e. $P_n(x)$ are assumed to be orthogonal with respect to a nondegenerate linear functional $\sigma$. We show that the only solutions are: Jacobi, Laguerre (correspondingly little $q$-Jacobi and little $q$-Laguerre and other special and degenerate cases), Bessel and little -1 Jacobi polynomials.