We determine the explicit formulas for the sum of products of homogeneous
multiple harmonic sums $\sum_{k=1}^n \prod_{j=1}^r H_k(\{1\}^{\lambda_j})$ when
$\sum_{j=1}^r \lambda_j\leq 5$. We apply these identities to the study of two
congruences modulo a power of a prime

Tauraso, Roberto

English

arXiv

Tauraso, R

Archivio della Ricerca - Università di Roma Tor vergata

Se´minaire Lotharingien de Combinatoire 63 (2010), Article B63g
NEW HARMONIC NUMBER IDENTITIES WITH APPLICATIONS
Roberto Tauraso
Dipartimento di Matematica
Universita` di Roma “Tor Vergata”, Italy
e-mail: tauraso@mat.uniroma2.it
WWW: http://www.mat.uniroma2.it/∼tauraso
Abstract. We determine the explicit formulas for the sum of products of homoge-
neous multiple harmonic sums
∑n
k=1
∏r
j=1Hk({1}λj ) when
∑r
j=1 λj ≤ 5. We apply
these identities to the study of two congruences modulo a power of a prime.
1. Introduction
Let s = (s1, s2, . . . , sd) be a vector whose entries are positive integers, and define the
multiple harmonic sum (MHS for short) for n ≥ 0 by
Hn(s) =
∑
1≤k1<k2<···<kd≤n
1
ks11 k
s2
2 · · · ksdd
.
We call d and |s| = ∑di=1 si its depth and its weight, respectively. This kind of sums
have a long history, see for example [4], [12], and the references therein.
In this note we present an algorithmic procedure to determine a closed formula for
n∑
k=1
m∏
j=1
Hk(sj)
involving products of MHS evaluated at n and of total weight less or equal to
∑n
j=1 |sj|.
(i) The first step is to expand recursively any product Hn(s)Hn(t) as a non-negative
integral combination of MHS of weight |s|+ |t|:
Hn(s)Hn(t) =
∑
r∈s t
Hn(r),
where s t is the so-called stuﬄe product (or quasi-shuﬄe product). If we
consider the vectors s = (s1, s
′) and t = (t1, t′) as words formed by concatenating
their components then the stuﬄe product is recursively defined by
s ∅ = ∅ s = s, s t = s1(s′ t) + t1(s t′) + (s1 + t1)(s′ t′).
So, for example,
Hn(1)Hn(2, 3) = Hn(1, 2, 3) +Hn(2, 1, 3) +Hn(2, 3, 1) +Hn(2, 1 + 3) +Hn(1 + 2, 3).
2000 Mathematics Subject Classification. Primary 11A07 11B65; Secondary 05A10 05A19.
2 ROBERTO TAURASO
(ii) The second step is to sum over k any MHS by using recursively the following
rule:
n∑
k=1
Hk(s) =
n∑
k=1
k∑
j=1
Hj−1(s′)
jsd
=
n∑
j=1
Hj−1(s′)
jsd
n∑
k=j
1 =
n∑
j=1
Hj−1(s′)
jsd
(n+ 1− j)
= (n+ 1)Hn(s)−

Hn(s
′, sd − 1) if sd > 1
n∑
k=1
Hk(s
′)−Hn(s′) if sd = 1
where s = (s′, sd). It is easy to see that the MHS involved in the final formula
have weight less than or equal to |s|.
(iii) The last step is to simplify the formula by collecting terms by using the stuﬄe
product relations. This can be done in various ways and the choices depend on
the application of the identity.
This procedure will be employed in the next section, and it will generate a number
of identities. We give two applications of these identities (we hope that many more will
come out in the future). The first is about the sum
n∑
k=0
(−1)ak
(
n
k
)a
,
which seems to have a closed form only for very few values of the parameter a ∈ Z,
namely for −1 ≤ a ≤ 3. In [2], Cai studied a related congruence by showing that for
any a ∈ Z and any prime p ≥ 5
p−1∑
k=0
(−1)ak
(
p− 1
k
)a
≡
(
ap− 2
p− 1
)
(mod p4).
Here we present the following extension (see also [5] and [3] for similar results):
Theorem 1.1. Let p > 5 be a prime. For any a ∈ Z, we have
p−1∑
k=0
(−1)ak
(
p− 1
k
)a
≡ (a− 1)p
ap− 1
(
1 +
a(a+ 1)(3a− 2)
6
p3Xp
)
(mod p6),
where Xp =
Bp−3
p−3 − B2p−44p−8 and Bn denotes the n-th Bernoulli number.
Moreover, by using the previous theorem for a = −2 we get the following corollary.
Corollary 1.2. For any prime p > 5, we have
p−1∑
k=1
1
k
(
2k
k
)
≡ −16
3
p2Xp (mod p
4).
This improves the congruence modulo p3 contained in [9].
NEW HARMONIC NUMBER IDENTITIES WITH APPLICATIONS 3
2. MHS: identities
By following the procedure introduced in the previous section, we find an explicit
formula for any sum of products of homogeneous MHS like Hk({1}d) up to total weight
5 ({1}d means that the number 1 is repeated d times):
n∑
k=1
Hk(1) = (n+ 1)Hn(1)− n , (1)
n∑
k=1
Hk({1}2) = (n+ 1)Hn({1}2)− nHn(1) + n , (2)
n∑
k=1
H2k(1) = (n+ 1)H
2
n(1)− (2n+ 1)Hn(1) + 2n , (3)
n∑
k=1
Hk({1}3) = (n+ 1)Hn({1}3) + n
(
Hn(1)− 1
2
H2n(1)
)
+
n
2
Hn(2)− n , (4)
n∑
k=1
Hk(1)Hk({1}2) = (n+ 1)Hn(1)Hn({1}2) + (3n+ 1)
(
Hn(1)− 1
2
H2n(1)
)
+
n+ 1
2
Hn(2)− 3n , (5)
n∑
k=1
H3k(1) = (n+ 1)H
3
n(1) + (6n+ 3)
(
Hn(1)− 1
2
H2n(1)
)
+
1
2
Hn(2)− 6n. (6)
It is interesting to note that the formulas for
∑n
k=1H
r
k(1) when r = 1, 2, 3 appear as
Entry 8 on page 94 in [1]. To illustrate the procedure, we show how to obtain (5).
By (i), we have
Hk(1)Hk({1}2) = 3Hk({1}3) +Hk(2, 1) +Hk(1, 2).
Moreover, by (ii), we obtain
n∑
k=1
Hk({1}3) = (n+ 1)Hn({1}3)− nHn({1}2) + nHn(1)− n ,
n∑
k=1
Hk(2, 1) = (n+ 1)Hn(2, 1)− nHn(2) +Hn(1) ,
n∑
k=1
Hk(1, 2) = (n+ 1)Hn(1, 2)−Hn({1}2).
Finally, since by (i), we have
Hn(1)Hn({1}2) = 3Hn({1}3) +Hn(2, 1) +Hn(1, 2) and 2Hn({1}2) = H2n(1)−Hn(2),
by applying (iii), we easily get (5).
The formulas for the total weight being 4 and 5 are contained in Tables 1 and 2,
respectively: the sum
∑n
k=1 fk− (n+1)fn, where fn is an entry in the first row, is equal
to the sum of the entries of the first column, each multiplied by the linear polynomial
an+ b contained in the intersection of the chosen row and column.
4 ROBERTO TAURASO
Table 1∑n
k=1 fk − (n+ 1)fn Hn({1}4) H2n({1}2) Hn(1)Hn({1}3) H2n(1)Hn({1}2) H4n(1)∑3
k=1
(−1)k−1
k!
Hkn(1) −n −6n− 2 −4n− 1 −12n− 5 −24n− 12
1
2
Hn(2) −n −2n− 2 −2n− 1 −2n− 3 −4
1
3
Hn(3) −n 1 −n− 1 1 3
1
2
Hn(1)Hn(2) n 2n 2n+ 1 2n+ 1 0
Hn(1, 2) 0 1 0 1 2
n 1 6 4 12 24
Table 2∑n
k=1 fk − (n+ 1)fn Hn({1}5) Hn(1)Hn({1}4) Hn(1, 1)Hn({1}3) H2n(1)Hn({1}3)∑4
k=1
(−1)k−1
k!
Hkn(1) n 5n+ 1 10n+ 3 20n+ 7
1
2
Hn(2) n 3n+ 1 4n+ 3 6n+ 5
1
3
Hn(3) n 2n+ 1 n 2n+ 1
1
2
Hn(1)Hn(2) −n −3n− 1 −4n− 1 −6n− 3
Hn(1, 2) 0 0 −1 −1
1
4
Hn(4) n n+ 1 −1 −1
1
8
H2n(2) −n −(n+ 1) −(2n− 1) 1
1
4
H2n(1)Hn(2) n 3n+ 1 4n+ 1 6n+ 3
1
3
Hn(1)Hn(3) −n −2n− 1 −n −2n− 1
Hn(1, 3) 0 0 0 0
Hn(1, 1, 2) 0 0 1 1
n −1 −5 −10 −20
NEW HARMONIC NUMBER IDENTITIES WITH APPLICATIONS 5∑n
k=1 fk − (n+ 1)fn Hn(1)H2n({1}2) H3n(1)Hn({1}2) H5n(1)∑4
k=1
(−1)k−1
k!
Hkn(1) 30n+ 12 60n+ 27 120n+ 60
1
2
Hn(2) 6n+ 8 6n+ 13 20
1
3
Hn(3) −3 −6 −15
1
2
Hn(1)Hn(2) −6n− 2 −6n− 3 0
Hn(1, 2) −3 −5 −10
1
4
Hn(4) −2 −3 −4
1
8
H2n(2) −2n+ 4 9 20
1
4
H2n(1)Hn(2) 6n+ 2 6n+ 3 0
1
3
Hn(1)Hn(3) 0 0 0
Hn(1, 3) 1 2 5
Hn({1}2, 2) 3 5 10
n −30 −60 −120
3. MHS: congruences
Among the various known results about MHS modulo powers of a prime, the following
ones will be crucial for us: for any prime p > 5, we have
Hp−1(1) ≡ 2p2Xp (mod p4) ,
Hp−1(2) ≡ −4pXp (mod p3) ,
Hp−1(3) ≡ 0 (mod p2) ,
Hp−1(1, 2) ≡ −6Xp (mod p2) ,
Hp−1(4) ≡ Hp−1({1}2, 2) ≡ Hp−1(1, 3) ≡ 0 (mod p),
where Xp =
Bp−3
p−3 − B2p−44p−8 (see [7] for the MHS of depth 1, see [12] and [4] for all the MHS
of depth > 1 with the exception of Hp−1(1, 2) modulo p2 which has been established in
[10]).
Note that every homogeneous MHS can be expressed in terms of MHS of depth
1. More precisely (see Theorem 2.3 in [4]): given a positive integer d, then for any
unordered partition λ = {λ1, λ2, . . . , λr} of d there is some integer cλ such that
d!Hn({1}d) =
∑
λ∈P (d)
cλ
r∏
i=1
Hn(λi).
For example:
2Hn({1}2) = H2n(1)−Hn(2) ,
6Hn({1}3) = H3n(1)− 3Hn(1)Hn(2) + 2Hn(3) ,
24Hn({1}4) = H4n(1)− 6H2n(1)Hn(2) + 8Hn(1)Hn(3) + 3H2n(2)− 6Hn(4).
6 ROBERTO TAURASO
Hence, for any prime p > 5, we have
Hp−1({1}2) ≡ 2pXp (mod p3) ,
Hp−1({1}3) ≡ 0 (mod p2) ,
Hp−1({1}4) ≡ 0 (mod p).
Therefore, by Equations (1)–(6), Tables 1 and 2, it follows that, for any prime p > 5,
we have:
Table 3∑p−1
k=1Hk(1) ≡ −(p− 1) + 2p3Xp (mod p5)∑p−1
k=1Hk({1}2) ≡ (p− 1) + (4− 2p)p2Xp (mod p4)∑p−1
k=1H
2
k(1) ≡ 2(p− 1) + (2− 4p)p2Xp (mod p4)∑p−1
k=1Hk({1}3) ≡ −(p− 1) + (2− 4p)pXp (mod p3)∑p−1
k=1Hk(1)Hk({1}2) ≡ −3(p− 1) + (−6p)pXp (mod p3)∑p−1
k=1H
3
k(1) ≡ −6(p− 1) + (−2− 6p)pXp (mod p3)∑p−1
k=1Hk({1}4) ≡ (p− 1) + (−2p)Xp (mod p2)∑p−1
k=1H
2
k({1}2) ≡ 6(p− 1) + (−6)Xp (mod p2)∑p−1
k=1Hk(1)Hk({1}3) ≡ 4(p− 1) + (−2p)Xp (mod p2)∑p−1
k=1H
2
k(1)Hk({1}2) ≡ 12(p− 1) + (−6 + 2p)Xp (mod p2)∑p−1
k=1H
4
k(1) ≡ 24(p− 1) + (−12 + 8p)Xp (mod p2)∑p−1
k=1Hk({1}5) ≡ 1 (mod p)∑p−1
k=1Hk(1)Hk({1}4) ≡ 5 (mod p)∑p−1
k=1Hk({1}2)Hk({1}3) ≡ 10 + 6Xp (mod p)∑p−1
k=1H
2
k(1)Hk({1}3) ≡ 20 + 6Xp (mod p)∑p−1
k=1Hk(1)H
2
k({1}2) ≡ 30 + 18Xp (mod p)∑p−1
k=1H
3
k(1)Hk({1}2) ≡ 60 + 30Xp (mod p)∑p−1
k=1H
5
k(1) ≡ 120 + 60Xp (mod p)
Note that
∑p−1
k=1H
r
k(1) (mod p
4−r) for r = 1, 2, 3 have been established by Z. W. Sun in
[8].
4. Proof of Theorem 1.1 and Corollary 1.2
Proof of Theorem 1.1. Assume that p > 5 is a prime, then for k = 1, . . . , p− 1 we have
that
(−1)k
(
p− 1
k
)
=
k∏
j=1
(
1− p
j
)
≡ 1 +
5∑
j=1
(−p)jHk({1}j) (mod p6).
NEW HARMONIC NUMBER IDENTITIES WITH APPLICATIONS 7
Hence we have
(−1)ak
(
p− 1
k
)a
≡ 1+
5∑
j=1
(−p)j
j∑
r=1
(
a
r
) ∑
λ∈P(j,r)
(
j
λ1, . . . , λr
) r∏
i=1
Hk({1}λi) (mod p6),
where P(j, r) is the set of the integer partitions λ of j into r parts.
By summing over k, we find
p−1∑
k=0
(−1)ak
(
p− 1
k
)a
= p+
5∑
j=1
(−p)j
j∑
r=1
(
a
r
) ∑
λ∈P(j,r)
(
j
λ1, . . . , λr
) p−1∑
k=1
r∏
i=1
Hk({1}λi) (mod p6).
Finally, by Table 3, we can compute
(−p)j
p−1∑
k=1
r∏
i=1
Hk({1}λi) (mod p6)
for any partition λ ∈ P(j, r), and we get easily the result. 
Proof of Corollary 1.2. In [6] Staver proved that for any integer n ≥ 1
n∑
k=1
1
k
(
2k
k
)
=
(
2n
n
)
2n+ 1
3n2
n−1∑
k=0
(
n− 1
k
)−2
.
Letting n = p, by Theorem 1.1 for a = −2 we have
p−1∑
k=1
1
k
(
2k
k
)
=
1
p
(
2p
p
)(
2p+ 1
3p
p−1∑
k=0
(
p− 1
k
)−2
− 1
)
≡ 2
p
(
2p− 1
p− 1
)((
1− 8
3
p3Xp
)
− 1
)
≡ −16
3
p2Xp (mod p
4),
where in the last step we used the fact that
(
2p−1
p−1
) ≡ 1 (mod p3) by Wolstenholme’s
theorem. 
References
[1] B. C. Berndt, Ramanujan’s Notebooks Part I, Springer-Verlag, New York, 1998.
[2] T. X. Cai, On the residues of binomial coefficients and their products modulo prime powers, Acta
Math. Sin. (Engl. Ser.) 18 (2002), 277–288.
[3] M. Chamberland and K. Dilcher, Divisibility properties of a class of binomial sums, J. Number
Theory 120 (2006), 349–371.
[4] M. Hoffman, Quasi-symmetric functions and mod p multiple harmonic sums, preprint
arχiv:0401319 [math.NT] (2007).
[5] H. Pan, On a generalization of Carlitz’s congruence, Int. J. Mod. Math. 4 (2009), 87–93.
[6] T. B. Staver, Om summasjon av potenser av binomialkoeffisienten, Norsk Mat. Tidsskrift 29
(1947), 97–103.
[7] Z. H. Sun, Congruences concerning Bernoulli numbers and Bernoulli polynomials, Discrete Appl.
Math. 105 (2000), 193–223.
[8] Z. W. Sun, Arithmetic theory of harmonic numbers, preprint arχiv:0911.4433 [math.NT] (2009).
[9] Z. W. Sun and R. Tauraso, New congruences for central binomial coefficients, Adv. in Appl. Math.
45 (2010), 125–148.
[10] R. Tauraso, More congruences for central binomial coefficients, to appear in J. Number Theory.
[11] R. Tauraso and J. Zhao, Congruences of alternating multiple harmonic sums, to appear in J.
Combin. Number Theory.
8 ROBERTO TAURASO
[12] J. Zhao, Wolstenholme type theorem for multiple harmonic sums, Int. J. Number Theory 4 (2008),
73–106.


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