WEIGHTED SYLVESTER SUMS ON THE FROBENIUS SET IN MORE VARIABLES

Let $a_1,a_2,\dots,a_k$ be positive integers with $\gcd(a_1,a_2,\dots,a_k)=1$. ${\rm NR}={\rm NR}(a_1,a_2,\dots,a_k)$ denote the set of nonrepresentable in terms $a_1,a_2,\dots,a_k$. The largest integer $\max{\rm NR}$, number $\sum_{n\in{\rm NR}}1$ and sum NR}}n$ have been widely studied for a long time as related to famous Frobenius problem. In this paper by using Eulerian numbers, we give formulas weighted NR}}\lambda^{n}n^\mu$, where $\mu$ is nonnegative $\lambda$ complex number. We also examine power sums numbers some formulae three variables. Several examples illustrate support our results.