Classical \(2\)-orthogonal polynomials and differential equations
The authors study the differential equations for so-called \textit{classical \(2\)-orthogonal polynomials} as a first step in order to find a complete Bochner classification for \(d\)-orthogonal polynomials (\(d\geq 2\)). This type of orthogonal polynomial is intimately connected with Hermite-Padé approximation of type II (`German' polynomials). A known classification describes these polynomials by the form and length of their recurrence relation (`Favard'-type theorem). A monic sequence \(\{P_n(x)\}_{n\geq 0}\) of classical \(2\)-orthogonal polynomials is studied, which is equivalent to the existence of a recurrence relation of order \(3\) (\(4\) terms) for both the \(P_n(x)\) and the \(P'_n(x)\): \[ \begin{cases} P_0(x)=1,\;P_1(x)=x-\beta^0_0,\;P_2(x)=(x-\beta^0_1)P_1(x) - \gamma^0_1,\cr P_{n+3}(x) = (x-\beta^{0}_{n+2})P_{n+2}(x) - \gamma^{0}_{n+2}P_{n+1}(x) - \delta^{0}_{n+1}P_n(x),\;n\geq 0,\end{cases} \eqno{(1)} \] with regularity condition \[ \delta^0_n\not= 0,\;n\geq 1, \] and \[ \begin{cases} P'_1(x)=1,\;P'_2(x)=2(x-\beta^1_1),\;P'_3(x)={3\over 2}\, [(x-\beta^1_2)P'_2(x) - \gamma^1_2,]\cr {n+3\over n+4}\,P'_{n+4}(x) = (x-\beta^1_{n+3})P'_{n+3}(x) - \gamma^1_{n+3}P'_{n+2}(x) - \delta^1_{n+2}P'_{n+1}(x),\;n\geq 0\end{cases} \eqno{(2)} \] with regularity condition \[ \delta^1_{n+1}\not= 0,\;n\geq 1. \] The sequence is called \textit{\(2\)-symmetric \(2\)-orthogonal} when the recurrence relation takes the form \[ \begin{cases} P_0(x)=1,\;P_1(x)=x,\;P_2(x)=x^2,\cr P_{n+3}(x)=xP_{n+2}(x)-\delta^0_{n+1}P_n(x),\;n\geq 0.\end{cases}\eqno{(3)} \] The main results are \textbf{A.} The classical \(2\)-orthogonal polynomials satisfy a differential equation of order \(3\) with polynomial coefficients of the form \[ \sum_{k=0}^3\,R_{k,n}(x)P^{(k)}_{n+3}(x)=0,\;n\geq 0,\eqno{(4)} \] with deg\(\,R_{k,n}\leq k+1\). Moreover, \(R_{3,n}(x)=F_{1,n}(x)S_3(x)\) with \(S_3(x)\) a polynomial of degree \(\leq 3\) with coefficients independent of \(n\). \textbf{B.} Each of the \(5\) `canonical' \(2\)-orthogonal polynomials [described in \textit{K. Douak, P. Maroni}, Analysis 12, No. 1--2, 71--107 (1992; Zbl 0767.33004)] satisfy a differential equation (4) with deg\(\,R_{k,n}\leq 1\) and \(R_{3,n},\,R_{2,n}\) being independent of \(n\). \textbf{C.} Each of the \(4\) `canonical' \(2\)-symmetric \(2\)-orthogonal polynomials described in K. Douak, P. Maroni [loc. cit.] satisfies a differential equation (4) with \[ \begin{aligned} \text{type A.1:}\;&R_{3,n}=1,\,R_{2,n}=0,\,R_{1,n}=-x,\,R_{0,n}=n+3,\cr \text{type B:\;\;\(\,\)}\;&\text{deg}\,R_{3,n}=0, \,\text{deg}\,R_{2,n}=2, \,\text{deg}\,R_{1,n}\leq 1, \,\text{deg}\,R_{0,n}=0,\cr \text{type C:\;\;\(\,\)}\;&\text{deg}\,R_{3,n}=0, \,\text{deg}\,R_{2,n}=2, \,\text{deg}\,R_{1,n}=0, \,\text{deg}\,R_{0,n}=0,\cr \text{type D:\;\;\(\,\)}\;&\text{deg}\,R_{3,n}=3, \,\text{deg}\,R_{2,n}=2, \,\text{deg}\,R_{1,n}\leq 1, \,\text{deg}\,R_{0,n}=0. \end{aligned} \]
scientific article; zbMATH DE number 5164026
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