Abstract
It is known that certain convolution sums can be expressed as a combination of divisor functions and Bernoulli formula. One of the main goals in this paper is to establish combinatoric convolution sums for the divisor sums . Finally, we find a formula of certain combinatoric convolution sums and Bernoulli polynomials. MSC:11A05, 33E99.
Highlights
1 Introduction The symbols N and Z denote the set of natural numbers and the ring of integers, respectively
The Bernoulli polynomials Bk(x), which are usually defined by the exponential generating function text tk et – = Bk(x) k!, k=
The study of convolution sums and their applications is classical, and they play an important role in number theory
Summary
The symbols N and Z denote the set of natural numbers and the ring of integers, respectively. We find a formula of certain combinatoric convolution sums and Bernoulli polynomials. The Bernoulli polynomials Bk(x), which are usually defined by the exponential generating function text tk et – = Bk(x) k! The Bernoulli polynomials satisfy the following well-known identities: N jk = Bk+ (N + ) – Bk+ ( ) (k ≥ ) k+ For n ∈ N, k ∈ Z, we define some divisor functions σk(n) := dk, d|n σk∗(n) :=
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