Darboux characterization conjecture for polynomial solutions of bilinear hypergeometric equations

Let $p$ and $q$ be polynomial solutions of the bilinear hypergeometric equations considered in propositions and. Let

$$and$$

denote the respective families of polynomial pairs constructed in the paper.

Darboux characterization conjecture. Any polynomial solutions to the bilinear hypergeometric equations of propositions and are given by

$$and$$

, respectively.

The conjecture is presented by analogy with the Burchnall–Chaundy result for Adler–Moser polynomials. The paper proves that the polynomial families obtained from ordinary hypergeometric eigenfunctions by finitely many Darboux transformations solve the bilinear hypergeometric equation, but whether these exhaust all polynomial solutions remains open.