On two combinatorial identities

Abstract

Two combinatorial identities are proved. The first one allows easy to establish connections between Bernoulli and Stirling numbers and values of zeta function of integer arguments and to find numerous identities, including the new ones, for these quantities. It also leads to a simple derivation of Abel’s analog of the Newton binomial formula, to the shortest proof of Staudt-Clausen theorem on Bernoulli numbers, and to a convenient summation formula for powers of natural numbers and series including such powers. The second identity presents the sum of two prime powers $x^p+y^p$ ( $p=2n+1=6m\pm 1$ ) of real or natural numbers as Chebyshev polynomial of $4xy/{(x+y)}^2$ . The same sum is expressed as a sum of $m$ subsequent powers of ${[xy(x+y)]}^2{(xy+x^2+y^2)}^3$ with coefficients $(n-j-1|2j)/(2j+1)$ .