Generalized Bochner theorem: Characterization of the Askey-Wilson polynomials
Generalized Bochner theorem: Characterization of the Askey-Wilson polynomials
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DOI：
10.1016/j.cam.2006.11.004
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发表时间：
2007-12
影响因子：
2.4
通讯作者：
L. Vinet;A. Zhedanov
中科院分区：
文献类型：
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作者：
L. Vinet;A. Zhedanov
设存在一组满足二阶差分方程的一元多项式Pn（z），其中z（s），A（s），B（s），C（s）是离散自变量s的函数，N可以是有限的也可以是无限的.对于s的所有容许值，假设不可约条件A（s-1）C（s）<$0。在有限情况下，我们假设有N+1个不同的网格点z（s），s= 0，1，.，N，使得z（i）<$z（j），i <$j。如果N=∞，我们假设网格z（s）对于不同的s值有无穷多个不同的值。在有限和无限的情况下，我们还假设问题是非退化的，即，λn <$λm，n <$m。然后我们证明，必然：（i）网格z（s）至多是s的二次或q-二次;（ii）对应的多项式Pn（z）至多是对应于网格z（s）的Askey-Wilson多项式。这一结果可以看作是Bochner定理（刻画普通经典多项式）在任意网格上任意差分算子的一般情形的推广。
Assume that there is a set of monic polynomials Pn(z) satisfying the second-order difference equation where z(s),A(s),B(s),C(s) are some functions of the discrete argument s and N may be either finite or infinite. The irreducibility condition A(s-1)C(s)≠0 is assumed for all admissible values of s. In the finite case we assume that there are N+1 distinct grid points z(s),s=0,1,…,N such that z(i)≠z(j),i≠j. If N=∞ we assume that the grid z(s) has infinitely many different values for different values of s. In both finite and infinite cases we assume also that the problem is non-degenerate, i.e., λn≠λm,n≠m. Then we show that necessarily: (i) the grid z(s) is at most quadratic or q-quadratic in s; (ii) corresponding polynomials Pn(z) are at most the Askey–Wilson polynomials corresponding to the grid z(s). This result can be considered as generalizing of the Bochner theorem (characterizing the ordinary classical polynomials) to generic case of arbitrary difference operator on arbitrary grids.