Generalized Hahn's theorem
One of the famous methods of classification of orthogonal polynomial systems goes back to \textit{W. Hahn} [Math. Z. 39, 634-638 (1935; Zbl 0011.06202)]: If \(\{P_n(x)\}_{n=0}^{\infty}\) and \(\{P_n'(x)\}_{n=0}^{\infty}\) are positive-definite orthogonal polynomial systems, then \(\{P_n(x)\}_{n=0}^{\infty}\) must be one of the classical systems (Jacobi, Laguerre or Hermite). This classification was later extended by several authors (to include the Bessel case too) and the conditions were weakened to: if \(\{P_n(x)\}_{n=0}^{\infty}\) is an orthogonal polynomial system and \(\{P_n^{(r)}(x)\}_{n=0}^{\infty}\) is a weakly orthogonal polynomial system, then \(\{P_n(x)\}_{n=0}^{\infty}\) must be classical. The authors now solve the following problems: (1) given an orthogonal polynomial system \(\{P_n(x)\}_{n=0}^{\infty}\) with respect to a moment functional \(\sigma\), (2) given a linear differential operator \(L[\cdot]=\sum_{i=0}^k a_i(x)D^i\) with polynomial coefficients, when is the polynomial sequence \(\{Q_n(x)\}_{n=0}^{\infty}\) defined by \(Q_n(x)=L[P_{n+r}^{(r)}(x)]\) also (weakly) orthogonal (\(r\) any non-negative integer)? The main result is: A. The \(Q\)'s are weakly orthogonal if and only if there exists a moment functional \(\tau\) and \(k+r+1\) polynomials \(\{b_i(x)\}_{i=r}^{k+2r}\) with deg\((b_i)\leq i\) and \[ \sum_{i=j}^{k+r} (-1)^j{i\choose j} (a_{i-r}(x)\tau)^{(i-j)}=b_{j+r}(x)\sigma,\quad 0\leq j\leq k+r, \] where \(\text{deg}(b_r)=r\), \(\langle\sigma,b_2r\rangle\not= 0\), \(b_{2r}(x)\not\equiv 0\). B. The \(Q\)'s are orthogonal if the \(b\)'s also satisfy \[ \sum_{i=0}^{k+r} b_{i+r,i+r}n_{(i)}\not= 0,\quad n\geq 0, \] where \(b_i(x)=\sum_{j=0}^i b_{ij}x^j\) (now deg\((b_r)=r\) and \(b_{k+2r}(x)\not= 0\)) and \(n_{(i)}=n(n-1)\cdots (n-i+1)\) for \(i\geq 1,\;n_{(0)}=1\). Furthermore, the authors give the complete analysis for the cases \(k=0,1,2\); \(r=1\) and \(k=r=1\). A nicely written paper.
scientific article; zbMATH DE number 1455710
scientific article