On the classical \(d\)-orthogonal polynomials defined by certain generating functions. I (Q1971671)

The paper is devoted to \(d\)-orthogonal polynomials \((p_n)_{n\in \mathbb{N}}\) defined by \[ \sum^\infty_{n=0} a_n\bigl[(d+1) xt-t^{d+1} \bigr]= \sum^\infty_{n=0} b_np_n (x)t^n , \] where \(d\) is a fixed positive integer. It is proved that the polynomials \((p_n)_{n \in\mathbb{N}}\) are \(d\)-symmetric and ``classical'' in the Hahn's sense. Finally, some properties of the zeros of the polynomials \((p_n)_{n\in \mathbb{N}}\) as well as a \((d+1)\)-order differential equation satisfied by each polynomial are given.