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The Brenke type generating functions are the polynomial generating functionsof 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 allBrenke-type polynomials when they are also $2$-orthogonal polynomial sets, thatis to say, polynomials satisfying one standard four-term recurrence relation.That allows us, on one hand, to obtain new 2-orthogonal sequences generalizingknown orthogonal families of polynomials, and on the other hand, to recoverparticular cases of polynomial sequences discovered in the context of$d$-orthogonality.\\ The classification is based on the resolution of athree-order difference equation induced by the four-term recurrence relationsatisfied by the considered polynomials. This study is motivated by the work ofChihara who gave all pairs $(A (t), B(t))$ for which $\{P_n(x)\}_{n\geq 0}$ isan orthogonal polynomial sequence.

Classification of 2-Orthogonal Polynomials with Brenke Type Generating Functions