On the k-polygonal numbers and the mean value of Dedekind sums

For any positive integer k a parts per thousand yen 3, it is easy to prove that the k-polygonal numbers are a (n) (k) = (2n+n(n-1)(k-2))/2. The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet L-functions and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums S(a (n) (k)A (m) (k), p) for k-polygonal numbers with 1 a parts per thousand currency sign m, n a parts per thousand currency sign p - 1, and give an interesting computational formula for it.