The complementary polynomials and the Rodrigues operator. A distributional study

TL;DR

This paper explores the use of the Rodrigues operator to construct complementary polynomials related to hypergeometric-type differential equations, providing new generating functions, recursion relations, and Rodrigues formulas.

Contribution

It introduces the concept of complementary polynomials, derives their differential equations, recursion relations, and Rodrigues formulas, expanding the understanding of polynomial solutions to hypergeometric-type equations.

Findings

Derived a closed-form generating functional for complementary polynomials.

Established that complementary polynomials satisfy hypergeometric-type differential equations.

Provided explicit Rodrigues formulas and recursion relations for the complementary polynomials.

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.