Mathematics & Physics 2016, 9(3), 384–392 УДК 517.55+517.962.26 Multidimensional Analog of the Bernoulli Polynomials and its Properties Olga A. Shishkina∗ Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041 Russia Received 07.04.2016, received in revised form 18.05.2016, accepted 20.06.2016 We consider a generalization of the Bernoulli numbers and polynomials to several variables, namely, we define the Bernoulli numbers associated with a rational cone and the corresponding Bernoulli polynomials. <...> Keywords: Bernoulli numbers and polynomials, generating functions, Todd operator, rational cone. <...> Introduction The Bernoulli numbers bµ are the coefficients of the Taylor series expansion of the function T (ξ) = ξ eξ −1: T (ξ) =∑ µ0 bµ ξµ µ! . <...> The Bernoulli numbers were introduced by J. Bernoulli in connection with the problem of summation of powers of consecutive integers: 1µ +2µ +.+xµ. <...> The Bernoulli polynomials Bµ (x) = ∑ k=0 µ Ck µbµ−kxk, Ck µ = µ! k! (µ−k)! , where bµ = Bµ (0) are the Bernoulli numbers were considered by J. Bernoulli [1] for natural x, and for any x these polynomials were first studied by Euler [2] who used the generating function in 1738: eξ −1exξ =∑ ξ µ0 Bµ (x) ξµ µ! . <...> The Bernoulli and Euler polynomials were later systematically studied by N.N¨ orlund [4]. <...> The Bernoulli numbers have wide applications in computer technology [5], combinatorial analysis [6,7] and in numerical analysis [8]. <...> Gould [9] remarks that many sums involving binomial coefficients greatly benefit from the use of Bernoulli numbers. <...> There is a number of papers about ∗olga_a_sh@mail.ru ⃝ Siberian Federal University. <...> All rights reserved c – 384 – Olga A. Shishkina Multidimensional Analog of the Bernoulli Polynomials and its Properties different generalizations of Bernoulli numbers and polynomials [10–15]. <...> For the Bernoulli polynomials this formula can be called the differentiation formula. where D is a differential operator, now this operator is called the Todd operator and denoted It was Euler who first defined a differential operator of infinite order D eD −1 = ∑ Td <...>