Luno N. Locally nilpotent derivations and combinatorial identities for classical polynomials

The dissertation is devoted to the constructing of the polynomial identities for certain types of the classical polynomials with the usage of the kernel of the locally nilpotent derivations of complex value polynomial algebra in several indeterminates. The description of the Chebyshev first kind derivation, the Chebyshev second kind derivation and the Laguerre derivation are given. The kernels of the Chebyshev first kind derivation the Laguerre derivation are specified from boh algebraical and combinatorical point of view. As far as each such derivation generates some identities for the corresponding polynomials, the set of the new combinatorial identities for Laguerre polynomials is derived. The problem od description of the locally nilpotent derivation kernel is equivalent to the problem of obtaining GL_2 invariants, which implies the existence of the locally nilpotent operators regarding the corresponding Lie algebra sl_2 and its subalgebras. In the case of UT_2 the corresponding Lie algebra acts with the Weitzenbock basic derivation. The latter is a well-known locally nilpotent derivation. Appell polynomials arise in various fields of mathematics and in the other branches of science, in particular, quantum physics and theoretical chemistry, and have a wide range of properties. Since both the Appell derivation and the Weitzenbock basis derivation are expressed by the same formulas, there exists an isomorphism which is intertwining with the action of these derivations. The closed form of such isomorphism is obtained for Weitzenbock derivation and Chebyshev second kind derivation. On the other hand, Appell polynomials are in big interest regarding special functions. There exist some polynomial families which could be represented in the terms of the generalized hypergeometric function. Moreover, we were motivated to find a new Appell polynomial family having the generalized hypergeometric function’s representation. Such a family was constructed and we call it the generalized hypergeometric Appell polynomials. The partial cases of the generalized hypergeometric Appel polynomials are the classical Gould-Hopper polynomials and the Hermite polynomials. The properties of the generalized hypergeometric Appell polynomials are established, in particular, the for of the generating function and the differential operator expansion. As a new result, the reccurence equation for the generalized hypergeometric Appell polynomials is obtained, as well as the solutions of the inverse problem of the generalized hypergeometric Appel polynomials and the solutions of connection problems between the generalized hypergeometric Appell polynomialsn and other Appell families. Most of the formulas occurred to be reccurance, but some explicit forms are derived as well. The examples of the explicit form of generalized hypergeometric Appel polynomials connection problem solutions are given in the case of exponenta and Kummer function.