A generalized Binomial theorem and a summation formulae

Abstract

The paper is based on the Binomial theorem and its generalizations to the polynomials of
binomial type. Thus, we give some applications to the generalized Waring problemm (Loo-Keng
Hua) and Hilbert-Kamke problem (G.I.Arkhipov). We also prove Taylor-Maclaurin formula
for the polynomials and smooth functions and give its applications to the numerical analysis
(Newton’s root-finding algorithm, Hensel lemma in full non-archimedian fields, approximate
evaluaion of the function at given point). Next, we prove an analogue of Binomial theorem
for Bernoulli polynomials, Euler-Maclaurin summation formula over integers and Poisson
summation formula for the lattice and consider some examples of binomial-type polynomials
(monomials, rising and falling factorials, Abel and Laguerre polynomials). We prove some
binomial properties op Appel and Euler polynomials and establish the multidimensional Taylor
formula and the analogues of Euler-Maclaurin and Poisson summation formulas over the lattices.
Finally, we consider the multidimensional analogues of these formulas for the multidimensional
complex space and prove some properties of binomial-type polynomials of several variables.