Classical 2 -orthogonal polynomials and differential equations

We construct the linear differential equations of third order satisfied by the classical 2 -orthogonal polynomials. We show that these differential equations have the following form: R 4 , n ( x ) P n + 3 ( 3 ) ( x ) + R 3 , n ( x ) P ″ n + 3 ( x ) + R 2 , n ( x ) P ′ n + 3 ( x ) + R 1 , n ( x ) P n + 3 ( x ) = 0 , where the coefficients { R k , n ( x ) } k = 1 , 4 are polynomials whose degrees are, respectively, less than or equal to 4 , 3 , 2 , and 1 . We also show that the coefficient R 4 , n ( x ) can be written as R 4 , n ( x ) = F 1 , n ( x ) S 3 ( x ) , where S 3 ( x ) is a polynomial of degree less than or equal to 3 with coefficients independent of n and deg ⠡ ( F 1 , n ( x ) ) ≤ 1 . We derive these equations in some cases and we also quote some classical 2 -orthogonal polynomials, which were the subject of a deep study.