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The aim of this paper is to clear up the problem of the connection between the 3D geometric moments invariants and the invariant theory, considering a problem of describing of the 3D geometric moments invariants as a problem of the classical invariant theory. Using the remarkable fact that the groups $SO(3)$ and $SL(2)$ are locally isomorphic, we reduced the problem of deriving 3D geometric moments invariants to the well-known problem of the classical invariant theory. We give a precise statement of the 3D geometric invariant moments computation, introducing the notions of the algebras of simultaneous 3D geometric moment invariants, and prove that they are isomorphic to the algebras of joint $SL(2)$-invariants of several binary forms. To simplify the calculating of the invariants we proceed from an action of Lie group $SO(3)$ to an action of its Lie algebra $\mathfrak{sl}_2$. The author hopes that the results will be useful to the researchers in the fields of image analysis and pattern recognition., Comment: 19 pages

UDC 519.114; 512.622We introduce the notion of Chebyshev derivations of the first and second kinds based on the polynomial algebra andcorresponding specific differential operators, derive the elements of their kernels, and prove that any element of the kernelof the derivations defines a polynomial identity for Chebyshev polynomials of both kinds. We obtain several polynomialidentities involving the Chebyshev polynomials of both kinds, a partial case of the Jacobi polynomials, and the generalizedhypergeometric function.

We deduce formulas for finding the Poincare multiseries $ \mathcal{P}\left( {{\mathcal{C}_d},{z_1},{z_2}, \ldots, {z_n},t} \right) $ and $ \mathcal{P}\left( {{\mathcal{I}_d},{z_1},{z_2}, \ldots, {z_n}} \right) $ , where $ {\mathcal{C}_d} $ and $ {\mathcal{I}_d} $ , d = (d 1, d 2, . . . , d n ), are multigraded algebras of joint covariants and joint invariants for n binary forms of degrees d 1, d 2, . . . , d n .

For the Lie algebras of Cartan type , , a simple condition is obtained that must be satisfied by every nontrivial invariant polynomial function on the coadjoint module for the algebra.Bibliography: 8 titles.