Several characterizations of third-degree semiclassical linear forms of class two appearing via cubic decomposition

Abstract

ABSTRACT In this work, we consider orthogonal polynomials via cubic decomposition in the framework of the third-degree semiclassical class. Based on their third-degree character, we give a complete description, by using the formal Stieltjes function and the moments, of semiclassical linear forms of class two arising from the cubic decomposition W 3 n ( x ) = P n ( x 3 ) , n ≥ 0 . We focus our attention on the link between these forms and the strict third-degree classical forms V q k , l = J ( k + q / 3 , l − q / 3 ) , q ∈ { 1 , 2 } , k , l ∈ Z with k + l ≥ − 1 . All of them are rational transformations of the Jacobi form V = J ( − 2 / 3 , − 1 / 3 ) .