Certain combinatoric Bernoulli polynomials and convolution sums of divisor functions
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RESEARCH
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Certain combinatoric Bernoulli polynomials and convolution sums of divisor functions Daeyeoul Kim1 and Nazli Yildiz Ikikardes2* * Correspondence: [email protected] 2 Department of Elementary Mathematics Education, Necatibey Faculty of Education, Balikesir University, Balikesir, 10100, Turkey Full list of author information is available at the end of the article
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 ton establish combinatoric convolution sums for the divisor sums σˆ s (n) = d|n (–1) d –1 ds . Finally, we find a formula of certain combinatoric convolution sums and Bernoulli polynomials. MSC: 11A05; 33E99 Keywords: Bernoulli numbers; convolution sums
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 ∞
tk text = , B (x) k et – k! k=
play an important and quite mysterious role in mathematics and various fields like analysis, number theory and differential topology. The Bernoulli polynomials satisfy the following well-known identities: N
jk =
j=
=
Bk+ (N + ) – Bk+ () k+
(k ≥ )
k k+ Bj N k+–j . (–)j j k + j=
(.)
The Bernoulli numbers Bk are defined to be Bk := Bk (). For n ∈ N, k ∈ Z, we define some divisor functions σk (n) :=
σk∗ (n) :=
dk ,
d|n
σˆ k (n) :=
d|n
dk ,
d|n n odd d n
(–) d – dk ,
σk,l (n; ) :=
σk (n) :=
(–)d– dk ,
d|n
dk .
d|n d≡l(mod )
©2013 Kim and Yildiz Ikikardes; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Kim and Yildiz Ikikardes Advances in Difference Equations 2013, 2013:310 http://www.advancesindifferenceequations.com/content/2013/1/310
Page 2 of 11
It is well known that σk∗ (n) = σk (n) – σk ( n ) and σˆ k (n) = σk (n) – σk ( n ) [, (.)]. The identity n–
σ (k)σ (n – k) =
k=
σ (n) + – n σ (n)
for the basic convolution sum first appeared in a letter from Besge to Liouville in []. Hahn [, (.)] considered
σˆ (m)σˆ (n – m) =
m . We check that k k + j
j=
=
k+–j Bj · a
k k + j
j=
k+–j
k k + (–)j Bj · a B · a +
a
= (k + )
j=
jk + (k + )
– a k
= Bk+ a
(.)
by (.). (ii) and (iii) are applied in a similar way.
Remark If p is a prime integer, then
v–
u+v+w=k+
by (.) and (.).
p– k– k + k m w Bv · p = (k + ) ; σs+ (p – m) σk–s–, u, v, w s + m= s=
Kim and Yildiz Ikikardes Advances in Difference Equations 2013, 2013:310 http://www.advancesindifferenceequations.com/content/
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