SO(3) quasi-monomial polynomial families - Samaruk, N.M. (Carpathian Mathematical Publications, 2024)
Let H be a subgroup of the affine space group Aff 3), considered with its natural action on the vector space of three-variable polynomials. The polynomial family { B_m,n,k(x,y,z) } is called quasi-monomial with respect to H if the group operators in two different bases { x^m y^n z^k } and { B_m,n,k(x,y,z) } have identical atrices. We derive a criterion for quasi-monomiality when the group H is the special orthogonal group SO(3). This criterion is expressed through the exponential generating function of the polynomial family {B_m,n,k(x,y,z)}. It has been proven that Appel's biorthogonal polynomials are quasi-monomials with respect to SO(3) and recurrence relations have been found for them.