Certain combinatoric bernoulli polynomials and convolution sums of divisor functions

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R E S E A R C H

Open Access

Certain combinatoric Bernoulli polynomials

and convolution sums of divisor functions

Daeyeoul Kim

1

_{and Nazli Yildiz Ikikardes}

2*

*_{Correspondence:} nyildizikikardes@gmail.com 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 to establish combinatoric convolution sums for the divisor sums

σ

ˆs(n) =

d|n(–1)

n d–1ds.

Finally, we ﬁnd a formula of certain combinatoric convolution sums and Bernoulli polynomials.

MSC: 11A05; 33E99

Keywords: Bernoulli numbers; convolution sums

1 Introduction

The symbolsN and Z denote the set of natural numbers and the ring of integers, respec-tively. The Bernoulli polynomials Bk(x), which are usually deﬁned by the exponential

gen-erating function text et_{– }= ∞ k= Bk(x) tk 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 j= jk=Bk+(N + ) – Bk+() k+  (k≥ ) =  k+  k j= (–)j k+  j BjNk+–j. (.)

The Bernoulli numbers Bk are deﬁned to be Bk:= Bk(). For n∈ N, k ∈ Z, we deﬁne

some divisor functions

σk(n) := d|n dk, σ_k∗(n) := d|n n dodd dk, σk(n) := d|n (–)d–dk, ˆσk(n) := d|n (–)nd–_dk_, _σ_k_,l_{(n; ) :=} d|n d≡l(mod ) dk.

©2013 Kim and Yildiz Ikikardes; licensee Springer. This is an Open Access article distributed under the terms of the Creative Com-mons Attribution License (http://creativecomCom-mons.org/licenses/by/2.0), which permits unrestricted use, distribution, and repro-duction in any medium, provided the original work is properly cited.

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It is well known that σ_k∗(n) = σk(n) – σk(n_) and ˆσk(n) = σk(n) – σk(n_) [, (.)]. The identity n– k= σ(k)σ (n – k) =  σ(n) +  –  n σ(n)

for the basic convolution sum ﬁrst appeared in a letter from Besge to Liouville in  []. Hahn [, (.)] considered  m<n ˆσ (m) ˆσ(n – m) = ⎧ ⎨ ⎩ –ˆσ(n) + σ(n) if n is odd, –ˆσ(n) – σ(n) + σ(n_) if n is even. (.)

For some of the history of the subject, and for a selection of these articles, we mention [, ] and [], and especially [, ] and []. The study of convolution sums and their ap-plications is classical, and they play an important role in number theory. In this paper, we investigate the combinatorial Bernoulli numbers and convolution sums. For k and n being positive integers, we show that the sum

k j= k +  j Bjˆσk+–j(n)

can be evaluated explicitly in terms of divisor functions and a combinatorial convolution sum. We prove the following.

Theorem  Let k, n be positive integers. Then

k j= k +  j Bjˆσk+–j(n) = (k + )σ_k+∗ (n) – k +   ˆσk+(n) – k(k + )  ˆσk–(n) – (k + ) k– s= k s +  n– m= ˆσk–s–(m)ˆσs+(n – m).

Remark  Let n be positive integers. In Theorem , replace k by , we ﬁnd easily that

n– m= ˆσ(m) ˆσ(n – m) = σ ∗ (n) –  ˆσ(n) –  ˆσ(n), (.) and in particular, if q∈ N, p = q + , an odd prime integer, then

q m= ˆσ(m) ˆσ(p – m) =  q(q + )(q + ) = q m= k=  B(q + ). (.) Equations (.) and (.) are in (.) and [, Corollary .]. Using these combinatoric convolution sums, we obtain the following.

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Theorem  If k is a positive integer, then u+v+w=k+ v–k– k +  u, v, w Bv· (l + )w= Bk+(l + ), where_uk+_,v,w=(k+)!_u_!v!w! and l= , , , , , , , , , , , .

Thus, we can pose a general question regarding Bernoulli polynomials.

Question For all k, l∈ N, does the identity u+v+w=k+ v–k– k +  u, v, w Bv· (l + )w= Bk+(l + ) hold?

The problem of convolution sums of the divisor function σ(n) and the theory of

Eisenstein series has recently attracted considerable interest with the emergence of quasi-modular tools. In connection with the classical Jacobi theta and Euler functions, other aspects of the function σ(n) are explored by Simsek in []. Finally, we prove the

follow-ing.

Theorem  If a(≥ ) and k are positive integers, then (i) k– s= k s +  a– m= σk–s–, m ;  σs+ a– m =  (k + ) Bk+ a+  – k (k + ) a– i= Bk+ i+  (k+)a_{– } k+_{– } –  a+_{+ } , (ii) k– s= k s +  a– m= σk–s–, m ;  σs+ a– m =  + k (k + ) a– i= Bk+ i+  (k+)a_{– } k+_{– } + 

k(a+)– k(a+)–+ ka––  k_{– } + a–  – (k–)(a+)_{– }(k–)a+ k–_{– } + a–– , (iii) k– s= k s +  a– m= σk–s– m  σs+ a– m =  (k + ) Bk+ a+  (k + ) a– i= Bk+ i

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+  (k+)a–  k+_{– } + 

k(a+)– k(a+)–+ ka––  k_{– } + a–  – (k–)(a+)_{– }(k–)a+ k–_{– } .

2 Properties of convolution sums derived from divisor functions Proposition ([]) Let k, n be positive integers. Then

k j= k +  j Bjσk+–j(n) = – k +   σk+(n) – (k + ) k – n σk–(n) + (k + ) k– s= k s +  n– m= σk–s–(m)σs+(n – m).

Proposition ([, ]) Let k, n be positive integers. Then

(i) k– s= k s +  n– m= σ_{k–s–}∗ (m)σ_s+∗ (n – m) =   σ_k+∗ (n) – nσ_k–∗ (n), (ii) k– s= k s +  n m= σk–s–(m – )σs+(n – m + ) =  σ ∗ k+(n).

Proof of Theorem Let k, n∈ N. By Proposition  and Proposition , we obtain

T := k– s= k s +  n– m= ˆσk–s–(m)ˆσs+(n – m) = k– s= k s +  n– m= σk–s–(m) – σk–s– m  × σs+(n – m) – σs+ n– m  . It is easily checked that

σk–s– m  σs+(n – m) + σk–s–(m)σs+ n– m  = σk–s– m  σs+ n– m  + σk–s–(m)σs+(n – m) + σk–s–(m) – σk–s– m  σs+(n – m) – σs+ n– m  . Thus, T = k– s= k s +  n– m= σk–s–(m)σs+(n – m) + σk–s– m  σs+ n– m  –  k– s= k s +  n– m= σk–s– m  σs+ n– m  + σk–s+(m)σs+(n – m)

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–  k– s= k s +  n– m= σk–s–(m) – σk–s– m  × σs+(n – m) – σs+ n– m  =  k– s= k s +  n– m= σk–s–(m) – σk–s– m  σs+(n – m) – σs+ n– m  – k– s= k s +  n– m= σk–s–(m)σs+(n – m) – σk–s– m  σs+ n– m  =  k– s= k s +  n– m= σ_{k–s–}∗ (m)σ_s+∗ (n – m) – k +  k +  σk+(n) + k – n σk–(n) +  k +  k j= k +  j Bjσk+–j(n) +  k +  k +  σk+ n  + k – n  σk– n  +  k +  k j= k +  j Bjσk+–j n  =σ_k+∗ (n) – nσ_k–∗ (n)– k +  k +  σk+(n) – σk+ n  –k  σk–(n) – σk– n  + n σk–(n) – σk– n  –  k +  k j= k +  j Bj σk+–j(n) – σk+–j n  = σ_k+∗ (n) – k +  k +  ˆσk+(n) – k ˆσk–(n) –  k +  k j= k +  j Bjˆσk+–j(n).

This proves the theorem.

Example  Let n be a positive integer. In Theorem , put k = , we get

n– m= ˆσ(m) ˆσ(n – m) =  σ ∗ (n) –  ˆσ(n) –  ˆσ(n) +  ˆσ(n).

Corollary  Let k, n be positive integers. Then, we obtain (i) k j= k +  j Bjˆσk+–j(n) = k +   σ_k+∗ (n) – k +   ˆσk+(n) – k(k + )  ˆσk–(n) – (k + ) k– s= k s +  n– m= ˆσk–s–(m)ˆσs+(n – m),

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(ii) k j= k +  j Bj σk+–j(n) + σk+–j n  = –σk+(n) – (k + )σk+ n  –(k + )(k – n)  σk–(n) –(k + )(k – n)  σk– n  + (k + ) k– s= k s +  n– m= σk–s– m  σs+(n – m), (iii) k j= k +  j Bjσk+–j∗ (n) = kσ_k+∗ (n) –k(k + )  σ ∗ k–(n) – n(k + )  ˆσk–(n) – (k + ) k– s= k s +  n– m= σ_{k–s–}∗ (m)ˆσs+(n – m) = –(k + )σ_k+∗ (n) –(k – n)(k + )  σ ∗ k–(n) + n(k + )  σk– n  + (k + ) k– s= k s +  n– m= σ_{k–s–}∗ (m)σs+(n – m).

Proof (i) We note that

k– s= k s +  n– m= σ_{k–s–}∗ (m)σ_s+∗ (n – m) = k– s= k s +  n– m= σ_{k–s–}∗ (m)σ_s+∗ (n – m) – k– s= k s +  n– m= σ_{k–s–}∗ (m – )σ_s+∗ n– (m – ).

(ii) and (iii) are applied in a similar way.

3 Bernoulli polynomials and convolution sums Proposition ([]) Let k, n be positive integers. Then

k– s= k s +  n– m= k–s–σk–s–(m/)σs+(n – m) = σk+(n/) –   σk(n) – k+σk(n/) – kσk(n/)

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–n  σk–(n) + kσk–(n/) + σk,(n; ) + k– k +  k j= k +  j Bjσk+–j(n/) +  (k + ) k j= k +  j Bjk+–jσk+–j(n/) +  (k + ) k j= k +  j Bjσk+–j,(n; ) –  (k + ) u+v+w=k+ v– k +  u, v, w Bvσw,(n; ).

It is well known that σk–s–,(m_; ) = k–s–σk–s–(m_). Using Proposition , we get

this lemma.

Lemma  Let k, n be positive integers. Then (i) k– s= k s +  n– m= σk–s–, m ;  σs+(n – m) = σk+ n  +  (k + )σk+,(n; ) +  (k + )σk+, n ;  +  (k + )σk,(n; ) –  σk(n) +  σk,(n; ) –  σk,(n; ) +  σk,(n; ) –n σk–(n) + k σk–,(n; ) + k – n  σk–, n ;  + k σk–,(n; ) + k (k + ) k j= k +  j Bjσk+–j n  +  (k + ) k j= k +  j Bjσk+–j, n ;  +  (k + ) k j= k +  j Bjσk+–j,(n; ) –  (k + ) u+v+w=k+ v– k +  u, v, w Bvσw,(n; ), (ii) k– s= k s +  n– m= σk–s–, m ;  σs+(n – m) =  (k + ) σk+(n) + k+  k +  σk+ n  + k– n  σk–(n) + k– n  σk– n  –  σk+ n 

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–  (k + ) σk+,(n; ) –  (k + ) σk+, n ;  –  (k + ) σk,(n; ) +  σk(n) –  σk,(n; ) +  σk,(n; ) –  σk,(n; ) +n σk–(n) – k σk–,(n; ) – k – n  σk–, n ;  – k σk–,(n; ) –  k (k + ) k j= k +  j Bjσk+–j n  –  (k + ) k j= k +  j Bjσk+–j, n ;  –  (k + ) k j= k +  j Bjσk+–j,(n; ) +  (k + ) u+v+w=k+ v– k +  u, v, w Bvσw,(n; ).

Remark  (i) Using Lemma , we obtain u+v+w=k+ v– k +  u, v, w Bvσw,(n; ) = k +   σk+ n  – σk(n) + σk,(n; ) + σk, n ;  – n(k + )  σk–(n) + σk–, n ;  + (k + )σk,(n; ) + k k j= k +  j Bjσk+–j n  + k j= k +  j Bjσk+–j, n ;  + k j= k +  j Bjσk+–j,(n; ) – (k + ) k– s= k s +  n– m= σk–s–, m ;  σs+(n – m) = – k +   σk+ n  – nσk–, n ;  + σk, n ;  – k j= k +  j Bjσk+–j(n) + k–  k j= k +  j Bjσk+–j n  + k j= k +  j Bjσk+–j, n ;  + k j= k +  j Bjσk+–j,(n; ) + (k + ) k– s= k s +  n– m= σk–s–, m ;  σs+(n – m).

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(ii) If n is an odd integer, then u+v+w=k+ v– k +  u, v, w Bvσw,(n; ) = (k + ) k– s= k s +  n– m= σk–s–, m ;  σs+(n – m). (.)

(iii) In (.), put k = , we get u+v+w= v–  u, v, w Bvσw,(n; ) =  n– m= σ, m ;  σ(n – m), and thus, n– m= σ, m ;  σ(n – m) =   σ(n) – σ(n) . In (.), replace k by , we ﬁnd that u+v+w= v–  u, v, w Bvσw,(n; ) =  _n_– m= σ, m ;  σ(n – m) + n– m= σ, m ;  σ(n – m) , and thus, n– m= σ, m ;  σ(n – m) + n– m= σ, m ;  σ(n – m) =   σ(n) – σ(n) + σ(n) .

Proof of Theorem If n = , compare both sides of (.), we obtain u+v+w=k+ v– k +  u, v, w Bv= . (.) If we put n =  in (.), we obtain u+v+w=k+ v– k +  u, v, w Bv·  + w = u+v+w=k+ v– k +  u, v, w Bv· w = (k + ) k– s= k s +  . (.)

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From (.) and (.), we get u+v+w=k+ v– k +  u, v, w Bv· w= (k + )k. (.)

By combining (.) and (.), we obtain u+v+w=k+ v– k +  u, v, w Bv· w= kBk+().

Others cases follow in a similar way. This completes the proof.

Proof of Theorem (i) If a is a positive integer, then u+v+w=k+ v– k +  u, v, w Bv· σw, a; = u+v+w=k+ v– k +  u, v, w Bv= 

by (.). According to Remark (i), we deduce that

 = (k + ) k– s= k s +  a– m= σk–s–, m ;  σs+ a– m – k j= k +  j Bjσk+–j a+k–  k j= k +  j Bjσk+–j a  + k j= k +  j Bjσk+–j, a ;  – k +   σk+ a  – a+  .

If a = , it is clearly evident. We suppose that a > . We check that

k j= k +  j Bj· ak+–j = k j= k +  j (–)jBj· ak+–j+  k +   B· ak = (k + ) a j= jk+ (k + ) –  ak = Bk+ a (.) by (.).

(ii) and (iii) are applied in a similar way.

Remark  If p is a prime integer, then u+v+w=k+ v– k +  u, v, w Bv· pw= (k + ) k– s= k s +  p– m= σk–s–, m ;  σs+(p – m) by (.) and (.).

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Competing interests

The authors declare that they have no competing interests. Authors’ contributions

All authors contributed equally and signiﬁcantly in writing this article. All authors read and approved the ﬁnal manuscript. Author details

1_{National Institute for Mathematical Sciences, Yuseong-daero 1689-gil, Yuseong-gu, Daejeon, 305-811, South Korea.} 2_{Department of Elementary Mathematics Education, Necatibey Faculty of Education, Balikesir University, Balikesir, 10100,} Turkey.

Received: 26 July 2013 Accepted: 23 September 2013 Published:08 Nov 2013 References

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Cite this article as: Kim and Yildiz Ikikardes: Certain combinatoric Bernoulli polynomials and convolution sums of