Certain combinatoric Bernoulli polynomials and convolution sums of divisor functions
© Kim and Yildiz Ikikardes; licensee Springer. 2013
Received: 26 July 2013
Accepted: 23 September 2013
Published: 8 November 2013
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.
can be evaluated explicitly in terms of divisor functions and a combinatorial convolution sum. We prove the following.
Equations (1.3) and (1.4) are in (1.2) and [, Corollary 2.4]. Using these combinatoric convolution sums, we obtain the following.
where and .
Thus, we can pose a general question regarding Bernoulli polynomials.
The problem of convolution sums of the divisor function and the theory of Eisenstein series has recently attracted considerable interest with the emergence of quasimodular tools. In connection with the classical Jacobi theta and Euler functions, other aspects of the function are explored by Simsek in . Finally, we prove the following.
2 Properties of convolution sums derived from divisor functions
Proposition 5 ()
This proves the theorem. □
(ii) and (iii) are applied in a similar way. □
3 Bernoulli polynomials and convolution sums
Proposition 9 ()
It is well known that . Using Proposition 9, we get this lemma.
- (ii)If n is an odd integer, then(3.1)
- (iii)In (3.1), put , we get
Others cases follow in a similar way. This completes the proof. □
and (iii) are applied in a similar way. □
by (3.1) and (3.2).
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