The complementary polynomials and the Rodrigues operator. A distributional study

Abstract

We can write the polynomial solution of the second order linear differential equation of hypergeometric-type $$ \phi(x)y''+\psi(x)y'+\lambda y=0, $$ where $\phi$ and $\psi$ are polynomials, $\deg \phi\le 2$, $\deg \psi=1$ and $\lambda$ is a constant, among others, by using the Rodrigues operator $R_k(\phi,{\bf u})$ (see \cite{coma2}) where $\bf u$ is certain linear operator which satisfies the distributional equation \begin{equation} \label{1} \frac{d}{dx}[\phi {\bf u}]=\psi {\bf u}, \end{equation} as $$ P_n(x)=B_n R_n(\phi,{\bf u})[1], \qquad B_n\ne 0,\quad n=0, 1, 2, ... $$ Taking this into account we construct the complementary polynomials. Among the key results is a generating functional function in closed form leading to derivations of recursion relations and addition theorem. The complementary polynomials satisfy a hypergeometric-type differential equation themselves, have a three-term recursion among others and Rodrigues formulas.