@article{Samaruk_2024, title={$SO(3)$ quasi-monomial polynomial families}, volume={16}, url={https://journals.pnu.edu.ua/index.php/cmp/article/view/6763}, DOI={10.15330/cmp.16.1.40-52}, abstractNote={<p>Let $H$ be a subgroup of the affine space group ${\rm Aff} 3)$, 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.</p>}, number={1}, journal={Carpathian Mathematical Publications}, author={Samaruk, N.M.}, year={2024}, month={Apr.}, pages={40–52} }