Bernoulli's discrete periodic functions | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2019. № 43. DOI: 10.17223/20710410/43/2

This paper is a survey of known and some new properties of the discrete periodic Bernoulli functions bn(j ) of order n introduced by V. N. Malozemov and viewed as elements x = x(0)x(1) . . . x(N - 1) ∈ C₀^N ⊂ C^N with the normalization condition _N-1⁰ x(k) = 0. It is proved that the operator ∆ : C₀^N → C₀^N where ∆[x] = y = k=0 = y(0)y(1) . . . y(N - 1), y(k) = x(k + 1) - x(k), is a bijection and ∆[bn] = bn-1. Moreover, according to Malozemov's result, the set of the discrete periodic Bernoulli functions is an infinite cyclic group relative to the cyclic convolution x * y(s) = N-1 = x(j)y(s - j) with a neutral element b0, and bn * bm = bn+m. It is proved j=0 that either the set of N - 1 cyclic shifts x^k→(j) = x(j - k) of any discrete periodic Bernoulli function or the set {bm, bm+1, . . .,bm+N-2} yields a basis of the space C₀^N. ∞ The generating function bntⁿ of a sequence of discrete periodic Bernoulli functions n=0 N-1 is calculated. Formulas sin^2m(πk∕N), m ∈ Z, for calculating the sums of even de- k=1 grees of sinuses at equidistant nodes of a circle are found by means of these functions and the discrete Fourier transform. It has been established that a cyclic shift by 1 and the multiplication by -N transform these functions of positive order into special polynomials Pn(k), which were introduced by Bespalov and Korobov and have become popular as the Korobov polynomials of the first kind in the form Kn(x) = n!Pn(x). We have calculated the Korobov numbers K_n = -n! ∙ N ∙ b_n(1) up to K₁₃ and the Korobov polynomials up to K7(x) for any array size (parameter) N.