Three lectures on classical orthogonal polynomials (Q2712194)

The aim of the author is to establish a unified theory for the polynomials satisfying certain functional equations which seemingly are of different character. To this end he defines first two operators \(D\) and \(\overline D\) which unifies the differential, difference and \(q\)-difference (or Hahn) operators, namely: NEWLINE\[NEWLINE\begin{aligned} Df(x) & ={d\over dx}f(x) =f'(x)\text{ or } \Delta f(x) =f(x+1)- f(x) \text{ or }D_q f(x)=\bigl[f(qx) -f(x)\bigr]/ \bigl[(q-1)x \bigr]\\ \overline Df(x) & =Df(x) \text{ or }\Delta f(x-1) \text{ or } D_{1/ q} f(x)\end{aligned}NEWLINE\]NEWLINE and then consider a linear second order operator \(L\) written as follows: NEWLINE\[NEWLINEL=\sigma(x) D\overline D+\tau (x)D+\lambda I.NEWLINE\]NEWLINE Here I stands for the unit operator while \(\lambda\) is a constant and \(\sigma(x)=\alpha x^2+ \delta x+\varepsilon\), \(\tau(x)= px+r\). Here \(\alpha,\delta, \varepsilon\), \(p\) and \(r\) are constants. In the first part of the paper he establishes the polynomial solutions of degree \(n\) to the equation \(Lp_n=0\) and discuss the cases where the coefficients of these polynomials satisfy 3-terms recurrence relations. In the second part he defines new families of orthogonal polynomials and finds the fourth-order linear equations which admit these polynomials as solutions. In the third (and last) part he introduces some scalar products in Sobolev sense end extends the results obtained in the first two parts to this case.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00041].