We observe that the classical Faulhaber's theorem on sums of odd powers also
holds for an arbitrary arithmetic progression, namely, the odd power sums of
any arithmetic progression $a+b, a+2b, ..., a+nb$ is a polynomial in
$na+n(n+1)b/2$. While this assertion can be deduced from the original
Fauhalber's theorem, we give an alternative formula in terms of the Bernoulli
polynomials. Moreover, by utilizing the central factorial numbers as in the
approach of Knuth, we derive formulas for $r$-fold sums of powers without
resorting to the notion of $r$-reflexive functions. We also provide formulas
for the $r$-fold alternating sums of powers in terms of Euler polynomials.Comment: 12 pages, revised version, to appear in Discrete Mathematic

Chen, William Y. C.

Fu, Amy M.

Zhang, Iris F.

English

arXiv

AbstractWe observe that the classical Faulhaber’s theorem on sums of odd powers also holds for an arbitrary arithmetic progression, namely, the odd power sums of any arithmetic progression a+b,a+2b,…,a+nb is a polynomial in na+n(n+1)b/2. While this assertion can be deduced from the original Fauhalber’s theorem, we give an alternative formula in terms of the Bernoulli polynomials. Moreover, by utilizing the central factorial numbers as in the approach of Knuth, we derive formulas for r-fold sums of powers without resorting to the notion of r-reflective functions. We also provide formulas for the r-fold alternating sums of powers in terms of Euler polynomials

Chen, William Y.C.

Elsevier - Publisher Connector 

Discrete Mathematics 309 (2009) 2974–2981
Contents lists available at ScienceDirect
Discrete Mathematics
journal homepage: www.elsevier.com/locate/disc
Faulhaber’s theorem on power sums
William Y.C. Chen ∗, Amy M. Fu, Iris F. Zhang
Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, PR China
a r t i c l e i n f o
Article history:
Received 12 October 2006
Received in revised form 23 July 2008
Accepted 23 July 2008
Available online 28 August 2008
Keywords:
Faulhaber’s theorem
Power sum
Alternating sum
r-fold power sum
r-fold alternating power sum
Bernoulli polynomial
Euler polynomial
a b s t r a c t
We observe that the classical Faulhaber’s theorem on sums of odd powers also holds
for an arbitrary arithmetic progression, namely, the odd power sums of any arithmetic
progression a+b, a+2b, . . . , a+nb is a polynomial in na+n(n+1)b/2.While this assertion
can be deduced from the original Fauhalber’s theorem, we give an alternative formula in
terms of the Bernoulli polynomials. Moreover, by utilizing the central factorial numbers as
in the approach of Knuth, we derive formulas for r-fold sums of powers without resorting
to the notion of r-reflective functions. We also provide formulas for the r-fold alternating
sums of powers in terms of Euler polynomials.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
The classical theorem of Faulhaber states that the sums of odd powers
12m−1 + 22m−1 + · · · + n2m−1
can be expressed as a polynomial of the triangular number Tn = n(n+1)/2; See Beardon [2], Knuth [12].Moreover, Faulhaber
observed that the r-fold summation of nm is a polynomial in n(n+ r)whenm is positive andm− r is even [12]. The classical
Faulhaber theorem for odd power sums was proved by Jacobi [11]; See also Edwards [4]. Let us recall the notation on the
r-fold power sums:
∑0 nm = nm, and
r∑
nm =
r−1∑
1m +
r−1∑
2m + · · · +
r−1∑
nm. (1.1)
For example,
∑1 nm = 1m + 2m + · · · + nm, and
2∑
nm =
1∑
1m +
1∑
2m + · · · +
1∑
nm =
n∑
i=1
(n+ 1− i)im.
For even powers, it has been shown that the sum12m+22m+· · ·+n2m is a polynomial in the triangular number Tn, multiplied
by a linear factor in n. Gessel and Viennot [6] had a remarkable discovery that the alternating sum
∑n
i=1(−1)n−ii2m is also a
polynomial in the triangular number Tn.
Faulhaber’s theorem has drawn much attention from various points of view. Grosset and Veselov [7] investigated a
generalization of the Faulhaber polynomials related to elliptic curves. Warnaar [15], Schlosser [14] and Zhao and Feng [16]
∗ Corresponding author.
E-mail addresses: chen@nankai.edu.cn (W.Y.C. Chen), fu@nankai.edu.cn (A.M. Fu), zhangfan03@mail.nankai.edu.cn (I.F. Zhang).
0012-365X/$ – see front matter© 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.disc.2008.07.027
W.Y.C. Chen et al. / Discrete Mathematics 309 (2009) 2974–2981 2975
studied the q-analogues of the formulas for the first few power sums. Garrett [5] found a combinatorial proof of the formula
for sums of q-cubes. Guo and Zeng [8] obtained the q-analogue formula in the general case. Furthermore, Guo, Rubey and
Zeng [9] have shown that the q-Faulhaber and q-Salié coefficients are nonnegative and symmetric in a combinatorial setting
of nonintersecting lattice paths.
In this paper, we first formulate Faulhaber’s theorem in a more general framework, that is, in terms of power sums of an
arithmetic progression [10]. Given an arithmetic progression:
a+ b, a+ 2b, . . . , a+ nb,
Faulhaber’s theorem implies that odd power sums of the above series are polynomials in na+ n(n+ 1)b/2. In particular, an
odd power sum of the first n odd numbers
12m−1 + 32m−1 + · · · + (2n− 1)2m−1
is a polynomial in n2, and the sum
12m−1 + 42m−1 + 72m−1 + · · · + (3n− 2)2m−1
is a polynomial in the pentagonal number n(3n− 1)/2.
Because of the relation (a+ bi)m = bm(a/b+ i)m, there is no loss of generality to consider the series
x+ 1, x+ 2, . . . , x+ n. (1.2)
Let
λ = n(n+ 2x+ 1) (1.3)
be the sum of the sequence x+ 1, x+ 2, . . . , x+ n. Then the power sums
S2m−1 = (x+ 1)2m−1 + (x+ 2)2m−1 + · · · + (x+ n)2m−1
is a polynomial in λ. For exmaple,
S3 = λ
2
4
+ (x
2 + x)
2
λ;
S5 = λ
3
6
+ 1
12
(6x2 + 6x− 1)λ2 + 1
6
(3x4 + 6x3 + 2x2 − x)λ;
S7 = λ
4
8
+ 1
6
(3x2 + 3x− 1)λ3 + 1
12
(9x4 + 18x3 + 3x2 − 6x+ 1)λ2 + 1
6
(3x6 + 9x5 + 6x4 − 3x3 − 2x2 + x)λ.
It should be noticed that the abovemore general setting of Faulhaber’s theorem can be deduced from the original version
of Faulhaber’s theorem. When x is a positive integer, we have the relation
n∑
i=1
(x+ i)m =
n+x∑
i=1
im −
x∑
i=1
im.
By Faulhaber’s theorem,
∑n+x
i=1 im and
∑x
i=1 im are polynomials in (n + x)(n + x + 1) and x(x + 1), respectively. Using the
following relation
(n+ x)(n+ x+ 1) = n(n+ 2x+ 1)+ x(x+ 1), (1.4)
we see that
[(n+ x)(n+ x+ 1)]i − [x(x+ 1)]i =
i∑
k=1
(
i
k
)
[n(n+ 2x+ 1)]k[x(x+ 1)]i−k, (1.5)
which is a polynomial in n(n+ 2x+ 1). Clearly, one sees that the above assertion holds for all real numbers x.
Although in principle, Faulhaber’s theorem is valid for any arithmetic progression, from a computational point of view it
still seems worthwhile to find a formula for the coefficients in terms of Bernoulli polynomials. The main result of this paper
is an approach to the study of the r-fold sums of powers without resorting to the properties of r-reflective functions, as in
the approach of Knuth [12]. In the last section, we obtain formulas for the r-fold alternating sums of powers in terms of the
Euler polynomials.
2976 W.Y.C. Chen et al. / Discrete Mathematics 309 (2009) 2974–2981
2. An alternative formula
In this section, we give an explicit formula for the coefficients regarding Faulhaber’s theorem for the series x + 1, x +
2, . . . , x + n, which reduces to an alternative formula to the Gessel–Viennot formula when setting x = 0. We first recall
some basic facts about Bernoulli polynomials Bn(x)which are defined by the following generating function:
∞∑
n=0
Bn(x)tn
n! =
text
et − 1 . (2.1)
The power sums of the first n positive integers can be expressed in terms of Bi(x):
n∑
i=1
im = 1
m+ 1 (Bm+1(n+ 1)− Bm+1(1)).
Moreover, we have, see [3],
n∑
i=1
(x+ i)2m−1 = 1
2m
(B2m(x+ n+ 1)− B2m(x+ 1)) . (2.2)
The Bernoulli numbers Bn are given by Bn = Bn(0). Note that the Bernoulli polynomials satisfy the following relations [1].
Bn(x+ 1)− Bn(x) = nxn−1, (2.3)
Bn(1− x) = (−1)nBn(x), (2.4)
d
dx
Bn(x) = nBn−1(x), (2.5)
Bn(x+ y) =
n∑
i=0
(n
i
)
Bi(x)yn−i. (2.6)
The evaluation of Bernoulli polynomials at 1/2 is of special interest. For n ≥ 0, we have
B2n+1(1/2) = 0, B2n(1/2) = (21−2n − 1)B2n. (2.7)
From (2.6) and (2.7), we can deduce the following form of Faulhaber’s theorem.
Theorem 2.1. Let λ = n(n+ 2x+ 1). Then we have
n∑
i=1
(x+ i)2m−1 =
m∑
k=1
F (m)k (x)λ
k, (2.8)
where
F (m)k (x) =
1
2m
m∑
i=k
(
2m
2i
)(
i
k
)(
x+ 1
2
)2i−2k
B2m−2i
(
1
2
)
. (2.9)
Proof. From the binomial expansion (2.6), we get
B2m(x+ n+ 1) =
2m∑
i=0
(
2m
i
)
B2m−i
(
1
2
)
·
(
x+ n+ 1
2
)i
. (2.10)
It follows from (2.2) and (2.7) that∑
(x+ n)2m−1 = 1
2m
m∑
i=0
(
2m
2i
)
B2m−2i
(
1
2
)
·
(
x+ n+ 1
2
)2i
− 1
2m
B2m(x+ 1).
Since (
x+ n+ 1
2
)2i
=
(
λ+
(
x+ 1
2
)2)i
=
i∑
k=0
(
i
k
)(
x+ 1
2
)2i−2k
λk, (2.11)
we immediately get (2.9) for k > 1. For k = 0, we have
F (m)0 (x) =
1
2m
m∑
i=0
(
2m
2i
)
B2m−2i
(
1
2
)(
x+ 1
2
)2i
= 1
2m
B2m(x+ 1).
This completes the proof. 
W.Y.C. Chen et al. / Discrete Mathematics 309 (2009) 2974–2981 2977
The above formula can be viewed as an alternative form of the formula of Gessel and Viennot [6] for the coefficients
A(m)k = F (m)k (0):
A(m)k = (−1)m−k
∑
j
(
2m
m− k− j
)(
m− k+ j
j
)
m− k− j
m− k+ j Bm+k+j, 0 ≤ k < m.
Note that B0 = 1 and B1 = −1/2 are used in the above formula whereas the formula (2.9) does not involve B1. The
equivalence between the formulas for F (m)k (0) and A
(m)
k can be established via the following generating function for the
coefficients F (m+1)k (x). The proof is analogous to that given by Gessel and Viennot [6] for the case x = 0.
Theorem 2.2. We have
∞∑
m=0
∞∑
k=1
F (m+1)k (x)t
k y
2m+1
(2m+ 1)! =
cosh y
√
(x+ 12 )2 + t − cosh y(x+ 12 )
2 sinh( y2 )
. (2.12)
Similarly, the theorem of Gessel and Viennot on the alternating sums of even powers of the first n natural numbers can
be extended to an arithmetic progression x+ 1, x+ 2, . . . , x+ n. It turns out that the Euler polynomials play the same role
as the Bernoulli polynomials for sums of odd powers.
The Euler polynomials En(x) are defined by
∞∑
n=0
En(x)
tn
n! =
2ext
et + 1 . (2.13)
The following expansion formula holds:
En(x+ y) =
n∑
k=0
(n
k
)
Ek(x)yn−k. (2.14)
For positive even number n, we have En(1) = 0. The Euler numbers En and the Euler polynomials are related by
E2n+1 = 0, En = 2nEn(1/2), n ≥ 0. (2.15)
Theorem 2.3. Let λ = n(n+ 2x+ 1). Then we have
n∑
i=1
(−1)n−i(x+ i)2m =
m∑
k=0
G(m)k (x)λ
k,
where
G(m)k (x) =
1
2
m∑
i=k
(
2m
2i
)(
i
k
)
E2m−2i
(
1
2
)(
x+ 1
2
)2i−2k
, 1 ≤ k ≤ m;
G(m)0 (x) =
1
2
(1− (−1)n)E2m(x+ 1).
The generating function for G(m)k (x) is given below, which is a straightforward extension of the formula of Gessel and
Viennot [6].
Theorem 2.4. We have
∞∑
m=0
∞∑
k=1
G(m)k (x)t
k y
2m
(2m)! =
cosh y
√
(x+ 12 )2 + t − cosh y(x+ 12 )
2 cosh( y2 )
. (2.16)
3. r-fold sums of powers
In this section, we derive a formula for the r-fold sums of powers of the series x + 1, x + 2, . . . , x + n in terms of the
central factorial numbers as used in the approach of Knuth [12]. A key step in our approach is the r-fold summation formula
for the lower factorials. It can be seen from our formula, that if r and m have the same parity, then the r-fold power sum∑r
(n+ x)m is a polynomial in n(n+ 2x+ r) plus a term that vanishes when x = 0. The cases when r andm have different
parities can be dealt with some care, and the details are omitted.
2978 W.Y.C. Chen et al. / Discrete Mathematics 309 (2009) 2974–2981
Recall the notation for the lower factorials (x)k = x(x− 1) · · · (x− k+ 1). The definition of the central factorials x[k] [13,
p. 213], is given by
x[k] = x(x+ k/2− 1)k−1.
The central factorial numbers T (m, k) are determined by the following relation:
xm =
m∑
k=1
T (m, k)x[k], m ≥ 1. (3.1)
Note that T (m, k) = 0 whenm− k is odd. In particular, we need the following relation
x2m−1 =
m∑
k=1
T (2m, 2k)(x+ k− 1)2k−1. (3.2)
We first give a formula for the r-fold sums of lower factorials. From the recursive definition (1.1) of r-fold summations,
we have
r∑
(x+ n)l =
∑
16i16i26···6ir6n
(x+ i1)l. (3.3)
The above multiple summation can be simplified to a single sum.
Theorem 3.1. We have
r∑
(x+ n)l = (x+ n+ r)l+r
(l+ r)r −
r∑
i=1
(
n+ r − i− 1
r − i
)
(x+ i)l+i
(l+ i)i .
Proof. Setting
F(n) = 1
l+ 1 (x+ n+ 1)l+1
gives F(i)− F(i− 1) = (x+ i)l. Hence we get∑
16i16i2
(x+ i1)l =
i2∑
i1=1
(F(i1)− F(i1 − 1))
= 1
l+ 1 (x+ i2 + 1)l+1 −
1
l+ 1 (x+ 1)l+1. (3.4)
Iterating (3.4) r − 1 times and using the following identity
n∑
i=1
(
l+ i− 1
l
)
=
(
l+ n
l+ 1
)
,
we obtain the desired formula. 
From Theorem 3.1 and the relation (3.2), we derive two formulas for the r-fold sums of the m-th powers when r and m
have the same parity.
Theorem 3.2. For m ≥ 1, we have
2r+1∑
(x+ n)2m−1 =
m∑
k=1
T (2m, 2k)
{
(x+ n+ k+ 2r)2k+2r
(2k+ 2r)2r+1 −
2r+1∑
i=1
(
n+ 2r − i
2r − i+ 1
)
(x+ k+ i− 1)2k+i−1
(2k+ i− 1)i
}
.
Weremark that the second summation in the above formula vanisheswhen x = 0, and the lower factorial (n+k+2r)2k+2r
can be rewritten as
k+r∏
i=1
(n+ k+ 2r + 1− i)(n− k+ i) =
k+r∏
i=1
[n(n+ 2r + 1)− (k+ 2r − i+ 1)(k− i)].
Hence we obtain Faulhaber’s theorem for the (2r + 1)-fold sums of odd powers of the first n positive integers.
Applying Theorem 3.1 together with the following relation
(x+ n)(x+ n+ k− 1)2k−1 = 12 (x+ n+ k)2k +
1
2
(x+ n+ k− 1)2k, (3.5)
we arrive at the following formula for the (2r)-fold summation of even powers.
W.Y.C. Chen et al. / Discrete Mathematics 309 (2009) 2974–2981 2979
Theorem 3.3. For m ≥ 1, we have,
2r∑
(x+ n)2m =
m∑
k=1
T (2m, 2k)
{
(x+ n+ r)(x+ n+ k+ 2r − 1)2k+2r−1
(2k+ 2r)2r
−
2r∑
i=1
(
n+ 2r − i− 1
2r − i
)
(2x+ i)(x+ k+ i− 1)2k+i−1
2(2k+ i)i
}
.
Setting x = 0 in the above formula, the second summation vanishes. One sees that the (2r)-fold summation becomes a
polynomial in n(n+ 2r) because (n+ r)(n+ k+ 2r − 1)2k+2r−1 can be rewritten as
k+r∏
i=1
(n+ k+ 2r − i)(n− k+ i) =
k+r∏
i=1
[n(n+ 2r)+ (k+ 2r − i)(i− k)].
4. r-fold alternating sums of powers
In this section, we investigate the r-fold alternating sums of powers. Following the notation of Faulhaber, we define
r∑
(−1)n(n+ x)m :=
∑
1≤i1≤···≤ir≤n
(−1)i1(i1 + x)m. (4.1)
We will show that the 2r-fold alternating sum of even powers
∑2r
(−1)nn2m is a polynomial in n(n + r). For other cases
concerning the parities of r,m, we will outline the results without proofs.
Define
E(r)m (x1, . . . , xr) =
∑
i1+···+ir=m
(
m
i1, . . . , ir
)
Ei1(x1) · · · Eir (xr). (4.2)
The following lemma holds. The proof is based on induction and is omitted.
Lemma 4.1. Let r,m be positive integers. Then
r∑
(−1)n(n+ x)m = (−1)n2−rE(r)m (1/2, . . . , 1/2, (x+ n+ r/2+ 1/2))
−
r∑
k=1
(
n+ r − k− 1
r − k
)
2−kE(k)m (1/2, . . . , 1/2, x+ (k+ 1)/2) .
We now give recursive formulas for E(k)2m in order to compute the multiple sums in the above lemma.
Lemma 4.2. Let k,m be positive integers. Then E(k)2m (1/2, . . . , 1/2, (k+ 1)/2) equals
k/2∑
i=0
(
k
2i
) i∑
j=0
(
i
j
)
(−1)jE(2j)2m (1/2, . . . , 1/2) , (4.3)
and E(k)2m+1 (1/2, . . . , 1/2, (k+ 1)/2) equals
(k−1)/2∑
i=0
(
k
2i+ 1
) i∑
j=0
(
i
j
)
(−1)j+1E(2j+1)2m+1 (1/2, . . . , 1/2, 1) . (4.4)
Proof. From the generating function of Em(x), one sees that the exponential generating function of E
(k)
m (1/2, . . . , 1/2,
(k+ 1)/2) equals
∞∑
m=0
E(k)m (1/2, . . . , 1/2, (k+ 1)/2)
tm
m! =
(
2et
1+ et
)k
.
Observe that(
2et
1+ et
)2
= 2 · 2e
t
1+ et −
(
2e
t
2
1+ et
)2
.
2980 W.Y.C. Chen et al. / Discrete Mathematics 309 (2009) 2974–2981
Denote the exponential generating functions of En(1) and En(1/2) by A(t) and B(t), respectively. Then the above relation
implies that A(t)2 = 2A(t)− B(t)2, which yields
A(t) = 1−
√
1− B(t)2.
Taking the k-th power, we obtain
A(t)k =
k/2∑
i=0
(
k
2i
) i∑
j=0
(
i
j
)
(−1)jB(t)2j +
(k−1)/2∑
i=0
(
k
2i+ 1
) i∑
j=0
(
i
j
)
(−1)jB(t)2j(1− A(t)).
Since E2m−1(1/2) = 0 and E2m(1) = 0 for m ≥ 1, equating the coefficients of both sides of above identity yields the
desired recurrence relations. 
The following theorem is concerned with 2r-fold alternating sums of even powers.
Theorem 4.3. Let r,m be positive integers. Then the 2r-fold alternating sum
∑2r
(−1)nn2m is of the form (−1)nF(n(n+ 2r))+
G(n(n+ 2r)), where F and G are polynomials of degree m and r − 1 respectively.
Proof. Applying Lemma 4.1 with r replaced by 2r ,m replaced by 2m, and setting x = 0, we get
2r∑
(−1)nn2m = (−1)n2−2rE(2r)2m (1/2, . . . , 1/2, (n+ r + 1/2)) (4.5)
+
2r∑
k=1
(
n+ 2r − k− 1
2r − k
)
2−kE(k)2m (1/2, . . . , 1/2, (k+ 1)/2) . (4.6)
In the expansion of (4.5) indexed by 2i1 + 2i2 + · · · + 2i2r = 2m, each term contains a factor of the form E2ir (n+ r + 1/2).
According to (2.14), we find that
E2i2r
(
n+ r + 1
2
)
=
i2r∑
k=0
(
2i2r
2k
)
E2i2r−2k
(
1
2
)
(n+ r)2k.
Since (n + r)2k = (n(n + 2r) + r2)k, it is easily seen that (4.5) is a polynomial in n(n + 2r) of degree m. We need to show
that (4.6) is also a polynomial in n(n+ 2r). Applying (4.3), we find that (4.6) equals
2r∑
k=1
(
n+ 2r − k− 1
2r − k
)
2−k
k/2∑
i=0
(
k
2i
) i∑
j=0
(
i
j
)
(−1)jE(2j)2m (1/2, . . . , 1/2)
=
r∑
j=0
(−1)jE(2j)2m (1/2, . . . , 1/2)
r∑
k=j
(
n+ 2r − k− 1
2r − k
)
2−k
k/2∑
i=j
(
k
2i
)(
i
j
)
. (4.7)
Using the identity
n/2∑
i=j
( n
2i
)( i
j
)
= 2n−2j−1
(
n− j
j
)
n
n− j , (4.8)
we deduce that the second sum in (4.7) equals
r∑
k=j
(
n+ 2r − k− 1
2r − k
)(
k− j− 1
j− 1
)
k
j
2−1−2j. (4.9)
In view of the following relation,∑
k
(
n− k
i
)(
m+ k
j
)
=
(
m+ n+ 1
i+ j+ 1
)
.
the above sum (4.9) simplifies to(
n+ 2r − j− 1
2r − 2j− 1
)
n+ r
r − j 2
−1−2j. (4.10)
Now, (n+ r)(n+ 2r − j− 1)2r−2j−1 can be rewritten as
r−j∏
i=1
(n+ i+ j)(n+ 2r − i− j) =
r−j∏
i=1
[n(n+ 2r)+ (2r − j− i)(i+ j)].
W.Y.C. Chen et al. / Discrete Mathematics 309 (2009) 2974–2981 2981
which is a polynomial in n(n+ 2r) of degree r − j. Since E(0)2m(·) = 0, (4.6) is a polynomial in n(n+ 2r) of degree r − 1. This
completes the proof. 
For the remaining three caseswith respect to the parities of r andm, we have the following theorem. The proof is omitted.
Theorem 4.4. For r,m ≥ 0, we have
2r+1∑
(−1)nn2m = (−1)nF (1)m (n(n+ 2r + 1))+ (2n+ 2r + 1)G(1)r−1(n(n+ 2r + 1)), (4.11)
2r∑
(−1)nn2m+1 = (−1)n(n+ r)F (2)m (n(n+ 2r))+ (n+ r)G(2)r−1(n(n+ 2r)), (4.12)
2r+1∑
(−1)nn2m+1 = (−1)n(n+ r)F (3)m (n(n+ 2r + 1))+ G(3)r (n(n+ 2r + 1)), (4.13)
where F (i)m (x) and G
(i)
r (x) (i = 1, 2, 3) stand for polynomials of degrees m and r respectively, and G(i)−1 = 0 for i = 1, 2.
Acknowledgments
We are grateful to the referees for helpful comments leading to an improvement of an earlier version. This work was
supported by the 973 Project, the PCSIRT Project of the Ministry of Education, the National Science Foundation, and the
Ministry of Science and Technology of China.
References
[1] M. Abramowitz, I.A. Stegun, Bernoulli and Euler polynomials and the Euler–Maclaurin formula, Section 23.1, in: Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972, pp. 804–806.
[2] A.F. Beardon, Sums of powers of integers, Amer. Math. Monthly 103 (1996) 201–213.
[3] G. Dattoli, C. Cesarano, S. Lorenzutta, Bernoulli numbers and polynomials from a more general point of view, Rend. Math. (7) 22 (2002) 193–202.
[4] A.W.F. Edwards, A quick route to sums of powers, Amer. Math. Monthly 93 (1986) 451–455.
[5] K.C. Garrett, K. Hummel, A combinatorial proof of the sum of q-cubes, Electron. J. Combin. 11 (2004) #R9.
[6] I.M. Gessel, G. Viennot, Determinants, paths, and plane partitions, 1989, preprint.
[7] M.-P. Grosset, A.P. Veselov, Elliptic Faulhaber polynomials and Lemé densities of states, Int. Math. Res. Not. (2006) 31. Art. ID 62120.
[8] V.J.W. Guo, J. Zeng, A q-analogue of Faulhaber’s formula for sums of powers, Electron. J. Combin. 11 (2) (2004–2006) #R19.
[9] V.J.W.Guo,M. Rubey, J. Zeng, Combinatorial interpretations of the q-Faulhaber and q-Salié coefficients, J. Combin. Theory, Ser. A 113 (2006) 1501–1515.
[10] M.D. Hirschhorn, Evaluating
∑N
n=1(a+ nd)p , Math. Gaz. (in press).
[11] C.G.J. Jacobi, De usu legitimo formulae summatoriae Maclaurinianae, J. Reine Angew. Math. 12 (1834) 263–272.
[12] D.E. Knuth, Johann Faulhaber and sums of powers, Math. Comput. 61 (1993) 277–294.
[13] J. Riordan, Combinatorial Identities, John Wiley & Sons, New York, 1968.
[14] M. Schlosser, q-Analogues of the sums of consecutive integers, squares, cubes, quarts and quints, Electron. J. Combin. 11 (2004) #R71.
[15] S.O. Warnaar, On the q-analogue of the sum of cubes, Electron. J. Combin. 11 (2004) #N13.
[16] G.J. Zhao, H. Feng, A new q-analogue of the sum of cubes, Discrete Math. 307 (22) (2007) 2861–2865.


Faulhaber’s theorem on power sums 

file:///data/core-remote/dit/data/elsevier/pdf/488/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDAxMjM2NXgwODAwNDc5Mg==.pdf

Faulhaber's Theorem on Power Sums

Abstract

Similar works

Full text

Available Versions

Elsevier - Publisher Connector

Elsevier - Publisher Connector