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 G_ξ , is defined as G_ξ = x(xD + 2I) + ξ D, where D is the derivative and I the identity. We prove that the scaled Gegenbauer polynomials are the unique monic orthogonal sequence that is G_ξ -classical.