Classification of 2-orthogonal polynomials with Brenke type generating functions

Abstract

The Brenke type generating functions are the polynomial generating functions of the form $\sum _{n=0}^{\infty }\frac{P_n\left(x\right)}{n!}{t}^n=A\left(t\right)B\left(xt\right),$ where $A$ and $B$ are two formal power series subject to the conditions $A\left(0\right)\cdot B^{\left(k\right)}\left(0\right)\ne 0,k=0,1,2,\dots$. In this work, we shall determine all Brenke-type polynomials when they are also 2-orthogonal polynomial sequences, that is to say, polynomials with Brenke type generating function and satisfying one standard four-term recurrence relation. That allows us, on one hand, to obtain new 2-orthogonal sequences generalizing known orthogonal families of polynomials, and on the other hand, to recover particular cases of polynomial sequences discovered in the context of $d$-orthogonality.

The classification is based on the discussion of a three-order difference equation induced by the four-term recurrence relation satisfied by the considered polynomials. This study is motivated by the work of Chihara (1968) who gave all pairs $(A\left(t\right),B\left(t\right))$ for which ${\left\{P_n\left(x\right)\right\}}_{n\ge 0}$ is an orthogonal polynomial sequence. In some cases, we give the expression of the moments associated to the two-dimensional functional of orthogonality.