Characterization of Gegenbauer Polynomials via Raising Operator

We explore a connection between Legendre and Gegenbauer polynomials through differential operators. It is well known that the derivative of a Legendre polynomial yields a multiple of a Gegenbauer polynomial. Here we establish an inverse relation using a linear differential operator that acts as a raising operator. This operator, denoted by $$\mathcal{G}_\xi$$ , is defined as $$ \mathcal{G}_\xi = x(xD + 2\mathbb{I}) + \xi D, $$ where $$D$$ is the derivative and $$\mathbb{I}$$ the identity. We prove that the scaled Gegenbauer polynomials are the unique monic orthogonal sequence that is $$\mathcal{G}_\xi$$ -classical.