On Polyconvolutions Generated by the Hankel Transform

In this paper, we construct two polyconvolutions (generalized convolutions) with weight $\gamma=x^{-\nu}$ generated by the Hankel transform possessing the factorization relations
$$ \text{H}_\nu[h_1](x)=x^{-\nu}\text{H}_\mu[f](x)\text{H}_\mu[g](x), \qquad \text{H}_\mu[h_2](x)=x^{-\nu}\text{H}_\nu[f](x)\text{H}_\mu[g](x). $$
Here $\text{H}_\mu$ is the Hankel transform operator of order $\mu$. Conditions for the existence of the constructed polyconvolutions are found. On their basis, using the differential properties of the Hankel transform, we obtain two more polyconvolutions. The derived constructions allow us to solve new classes of integral and integro-differential equations and systems of equations.