Classification of 2-Orthogonal Polynomials with Brenke Type Generating Functions

The Brenke type generating functions are the polynomial generating functions of the form $$\sum_{n=0}^{\infty}{P_n(x )\over n!}t^n=A(t)B(xt),$$ where $A$ and $B$ are two formal power series subject to the conditions $A(0)\;B^{(k)}(0)\neq0,\, k=0,1,2\ldots$.\\ In this work, we determine all Brenke-type polynomials when they are also $2$-orthogonal polynomial sets, that is to say, polynomials 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 resolution 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 who gave all pairs $(A (t), B(t))$ for which $\{P_n(x)\}_{n\geq 0}$ is an orthogonal polynomial sequence.