Vol. 37, No. 2, 2007, 123-128
A NOTE ABOUT THE SUMS OF PRODUCTS OF BERNOULLI NUMBERS
Aleksandar Petojevi´c1
Abstract. This paper presents the formula to calculate the sums of products of the Bernoulli numbers in the form
X
1≤k1≤[n2]
1≤k2≤
·n−k1 2
¸
...
1≤km≤
·n−k1−···−km−1 2
¸
Ak1,k2,...,kmB2k1· · ·B2kmB2(n−k1−···−km),
whereAk1,k2,...,km is a certain sequence of rational numbers.
AMS Mathematics Subject Classification (2000): 11B68 Key words and phrases:Bernoulli numbers, sums of products
1. Introduction
The Bernoulli numbersBn are defined by the generating function:
t et−1 =
X∞
n=0
Bn
tn
n! (|t|<2π).
Euler’s identity involves the sum of products of two Bernoulli numbers as follows
n−1X
k=1
µ2n 2k
¶
B2kB2n−2k= −(2n+ 1)B2n (n≥2).
Eie [4] and Sitaramachandrarao and Davis [10] considered the sum of products of 3 and 4 Bernoulli numbers. Dilcher [3] proved form≥2
X
j1+···+jm=n j1,...,jm≥0
µ 2n 2j1, ...,2jm
¶
B2j1· · ·B2jm =
=
(2n)!
(2n−m)!
[m−1P2 ]
k=0
b(m)k B2n−2k2n−2k 2n > m ,
(2n)!
4n +n−1P
k=0 (2n)!
2n−2kb(2n)k B2n−2k 2n=m , (−1)m−1(m−2n−1)!(2n)!b(m)n 0≤n≤£m−1
2
¤.
1University of Novi Sad, Faculty of Education, Podgoriˇcka 4, 25000 Sombor, Serbia, e-mail:
where
µ 2n 2j1, ...,2jm
¶
:= (2n)!
(2j1)!· · ·(2jm)!
is the multinomial coefficient andb(m)k is the sequence of rational numbers de- fined by
b(1)0 := 1, b(m+1)k :=−1
mb(m)k +1
4b(m−1)k−1 ,
with b(m)k = 0 for k <0 and fork >[(m−1)/2]. Here [x] denotes the integer part ofx.
Finally, Petojevi´c [8] established the following sums of product of m ∈ N Bernoulli numbers
X
k1+···+km+1=n k1,...,km≥0
km+1≥1
µ n
k1, k2, ..., km+1
¶
Bk1Bk2· · ·Bkm
B2km+1+k1
2km+1+k1 =
cn m= 1
Pn k=0
¡n
k
¢ak,m−1cn−k m >1, (1.1)
and form, n∈N0
X
k1+···+k2m+1=n k1,...,k2m≥0
k2m+1≥1
µ n
k1, k2, ..., k2m+1
¶
Bk1Bk2· · ·Bkm+1
B2k2m+1+k1
2k2m+1+k1 =
1
3nBnB3n n > m= 0
Pn k=0
¡n
k
¢cn−k(−1)k¡m−1
k
¢−1
s(m−1, m−k−1) 0≤n < m
m−1P
k=0
¡n
k
¢cn−k(−1)k¡m−1
k
¢−1
s(m−1, m−k−1) +(−1)(m−1)!m+1 Pn
k=m
¡n
k
¢cn−kk!m−1P
r=0
s(m−1,r)
(k−m+r+1) (k−m)!Bk−m+r+1 n≥m >0 wheres(n, k) are Stirling numbers of the first kind andαn, cnandan,mare the
sequences defined by αn := (−1)n
µ2n−1 n
¶−1
, n∈N.
cn :=
0 n= 0,
1
12 n= 1,
αn
4nB2n−2n1B2n n≥2.
an,m :=
1 n= 0,
m¡n
m
¢m−1P
k=0
(−1)m−k−1s(m, m−k)Bn−kn−k n≥m∈N,
n!
(m−1)!
m−n−1P
k=0
(m−n−1)!
k!mm−n−k
×Pm
r=0
(−1)r−k+1s(m, r)(r)k m > n >0.
2. New sum of products
The relation (1.1) produces X
k1+k2=n k1≥0, k2≥1
µ n k1, k2
¶ Bk1
B2k2+k1
2k2+k1 = ( 1
12, n= 1,
αn
4nB2n−2n1B2n, n >1, or, it can be rewritten as
(2.2)
n−1X
k=0
µn k
¶
BkB2n−k
2n−k =
1
12, n= 1,
α(n)
4n B2n−2n1Bn2, n >1.
Let us specify new generalization of the relation (2.2). Before that, we will specify the notation: form∈Nandn >1, Qm(n) is defined by
Qm(n) = X
1≤k1≤[n2]
1≤k2≤[n−k21] ...
1≤km≤
·n−k1−···−km−1 2
¸
Ak1,k2,...,kmB2k1· · ·B2kmB2(n−k1−···−km)
where Ak1,k2,...,km is the sequence of rational numbers, defined as:
Ak1,k2,...,km=
· n 2k1, . . . ,2km
¸2−αn−k1−···−km
n−k1− · · · −kmf(n, k1, . . . , km).
Here· n 2k1, . . . ,2km
¸
=
¡n
2k1
¢
2−αn−k1
·
¡n−k
1
2k2
¢
2−αn−k1−k2
· · ·
¡n−k
1−···−km−1
2km
¢
2−αn−k1−···−km
,
f(n, k1, . . . , km) =
½ 1, 2km6=n−k1− · · · −km−1 or m= 1
1
2, 2km=n−k1− · · · −km−1andm6= 1 Next, form∈Nandn >1 the sumsPm(n) and ˜Pm(n) are defined by
Pm(n) = X
1≤k1≤[n2]
1≤k2≤[n−k21] ...
1≤km≤
·n−k1−···−km−1 2
¸
· n 2k1, . . . ,2km
¸2−αn−k1−···−km
n−k1− · · · −km
×
× B2k1· · ·B2kmB2(n−k1−···−km), and
P˜m(n) = X
1≤k1≤[n2]
1≤k2≤[n−k21] ...
1≤km≤
·n−k1−···−km−1 2
¸
· n 2k1, . . . ,2km
¸
B2k1· · ·B2km
Bn−k2 1−...−km n−k1− · · · −km.
Finally, we will define the sequencedk(n) (n≥2 ):
d1(n) = hn
2 i
, dk(n) =
n−k−1P
j=1
dj(n) 2
,
so that, clearly, ifdk(n) = 1 anddk+1(n) = 0 then maxm=k.Hence 1≤m <1 + log2n .
After the appropriate substitutionB2(n−k1), . . . , B2(n−k1−k2−···−km−1)in for- mula (2.2) we get
Pm(n) = αn
n(−2)mB2n+ 1
n(−2)m−1Bn2+
m−1X
s=1
(−2)s+1−mP˜s(n), or recursively
P1(n) = −2−αn
2n B2n+ 1 nBn2, Pm(n) = −1
2Pm−1(n) + ˜Pm−1(n).
For 2km6=n−k1− · · · −km−1>1,the equationBn−k1−···−km−1 = 0 produces Pm(n)−P˜m−1(n) =
= X
1≤k1≤[n2]
1≤k2≤[n−k21] ...
1≤km≤
·n−k1−···−km−1 2
¸ 2km6=n−k1−···−km−1
· n 2k1, . . . ,2km
¸2−αn−k1−···−km
n−k1− · · · −km
×
× B2k1· · ·B2kmB2(n−k1−···−km)
+ X
1≤k1≤[n2]
1≤k2≤[n−k21] ...
1≤km≤
·n−k1−···−km−1 2
¸ 2km=n−k1−···−km−1
· n 2k1, . . . ,2km
¸2−αkm
2km ×
× B2k1· · ·B2km−1B2k2m
= Qm(n).
That is how the following statement is proved.
Theorem 2.1. Forn≥2 and1≤m <1 + log2n we have:
Q1(n) = 1
nBn2+αn−2 2n B2n, Qm(n) = −1
2Pm−1(n)
= 2−αn
n(−2)mB2n+ 1
n(−2)m−1B2n+
m−2X
s=1
(−2)s+1−mP˜s(n).
Acknowledgement
This work was supported in part by the Ministry of Science of the Republic of Serbia under Grant No. 149011D.
References
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Received by the editors May 3, 2007
