Generalized
Harmonic
Series (NOT
MZVs)
ME Hoffman
Outline
Introduction
H-series and
Stirling
numbers of
the first kind
Sums of Generalized Harmonic Series
(NOT Multiple Zeta Values)
Michael E. Hoffman
Proof of the
sum formulas
USNA Basic Notions
6 February 2014
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
Outline
Generalized
Harmonic
Series (NOT
MZVs)
ME Hoffman
Outline
1 Introduction
Introduction
H-series and
Stirling
numbers of
the first kind
2 H-series and Stirling numbers of the first kind
Proof of the
sum formulas
3 Proof of the sum formulas
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
Generalized harmonic series
Generalized
Harmonic
Series (NOT
MZVs)
ME Hoffman
Outline
Introduction
H-series and
Stirling
numbers of
the first kind
Proof of the
sum formulas
The generalized harmonic series we will be concerned with are
H(a1 , a2 , . . . , ak ) =
∞
X
n=1
na1 (n
+
1)a2
1
,
· · · (n + k − 1)ak
where a1 , . . . , ak are nonnegative integers whose sum is at least
2. We call this quantity an H-series of length k and weight
a1 + · · · + ak . Of course H-series of length 1 are just values of
the Riemann zeta function. We will prove that
X
H(a1 , . . . , ak ) = kζ(m),
(1)
a1 +···+ak =m
where m ≥ 2 and the sum is over k-tuples of nonnegative
integers. This was conjectured by C. Moen last spring, and
proved by both of us over the summer.
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
First cases
Generalized
Harmonic
Series (NOT
MZVs)
Equation (1) is only interesting if k > 1. The first case is
H(2, 0) + H(1, 1) + H(0, 2) = 2ζ(2),
ME Hoffman
Outline
Introduction
H-series and
Stirling
numbers of
the first kind
Proof of the
sum formulas
which is pretty trivial: evidently H(2, 0) = ζ(2) and
H(0, 2) = ζ(2) − 1, and the usual telescoping-sum argument
shows
∞
X
1
= 1.
H(1, 1) =
n(n + 1)
n=1
More generally, equation (1) for k = 2 is
H(m, 0) + H(m − 1, 1) + · · · + H(1, m − 1) + H(0, m) = 2ζ(m),
which reduces to
H(m − 1, 1) + · · · + H(1, m − 1) = 1.
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
(2)
Multiple zeta values
Generalized
Harmonic
Series (NOT
MZVs)
ME Hoffman
Outline
Introduction
H-series and
Stirling
numbers of
the first kind
Proof of the
sum formulas
Equation (1) is superficially similar to the “sum theorem” for
multiple zeta values (conjectured by C. Moen 1988, proved by
A. Granville 1997):
X
ζ(a1 , a2 , . . . , ak ) = ζ(m)
(3)
a1 +···+ak =m, ai ≥1, a1 ≥2
for m ≥ 2, where
ζ(a1 , a2 . . . , ak ) =
X
1
.
· · · nkak
n a1 n a2
n1 >n2 >···>nk ≥1 1 2
But note that the multiple zeta value ζ(a1 , . . . , ak ) is a k-fold
sum, while H(a1 , . . . , ak ) is a simple sum. Also, the left-hand
side of (3) is a sum over k-tuples of positive integers, while
that of (1) is a sum over k-tuples of nonnegative integers.
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
Another sum of H-series
Generalized
Harmonic
Series (NOT
MZVs)
ME Hoffman
Outline
We could ask what happens if we sum H-series over all k-tuples
of positive integers with sum m ≥ 2. There is a nice formula in
this case, though by no means as simple as equation (1):
Introduction
H-series and
Stirling
numbers of
the first kind
Proof of the
sum formulas
X
H(a1 , . . . , ak ) =
a1 +···+ak =m, ai ≥1
k−2 X
1
k −2
(−1)i
. (4)
(k − 1)!
i
(i + 1)m−k+1
i=0
Note that the right-hand side is a rational number. The case
k = 2 is equation (2).
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
Telescoping
Generalized
Harmonic
Series (NOT
MZVs)
We shall build up a chain of lemmas on H-series. Using
1
1
1
n(n+1) = n − n+1 , note that for a, b ≥ 1,
ME Hoffman
Outline
H(a, b) =
n=1
Introduction
H-series and
Stirling
numbers of
the first kind
Proof of the
sum formulas
∞
X
∞ X
n=1
na (n
1
=
+ 1)b
1
1
−
= H(a, b−1)−H(a−1, b).
na (n + 1)b−1 na−1 (n + 1)b
Hence
H(m − 1, 1) + H(m − 2, 2) + · · · + H(1, m − 1) =
H(m − 1, 0) − H(m − 2, 1) + H(m − 2, 1) − H(m − 3, 2) + · · ·
+ H(1, m − 2) − H(0, m − 1) = H(m − 1, 0) − H(0, m − 1) = 1,
proving equation (2).
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
Sum over indices of fixed sum, fixed positions
Generalized
Harmonic
Series (NOT
MZVs)
ME Hoffman
Outline
Introduction
H-series and
Stirling
numbers of
the first kind
Proof of the
sum formulas
This generalizes as follows.
Lemma (1)
Let 1 ≤ i < j ≤ p, and let
a1 , . . . , ai−1 , ai+1 , . . . , aj−1 , aj+1 , . . . , ap be fixed nonnegative
integers. Then
m−1
X
H(a1 , . . . , ai−1 , k, ai+1 , . . . , aj−1 , m − k, aj+1 , . . . , ap ) =
k=1
1
[H(a1 , . . . , ai−1 , m − 1, ai+1 , . . . , aj−1 , 0, aj+1 , . . . , ap )
j −i
− H(a1 , . . . , ai−1 , 0, ai+1 , . . . , aj−1 , m − 1, aj+1 , . . . , ap )].
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
Sum over indices with fixed nonzero positions
ME Hoffman
We can use Lemma (1) to obtain a formula for the sum of all
H-series of fixed length and weight and nonzero entries at
P
(r )
specified positions. Let Hp,q = qj=p j −r for p ≤ q.
Outline
Lemma (2)
Generalized
Harmonic
Series (NOT
MZVs)
Introduction
H-series and
Stirling
numbers of
the first kind
Proof of the
sum formulas
For k ≥ 1 and 1 ≤ i0 < i1 < · · · < ik ≤ n, let P(i0 , . . . , ik ) be
the set of strings (a1 , . . . , an ) of nonnegative integers with
a1 + · · · + an = m and aj 6= 0 iff j = iq for some q. Then
X
H(a1 , . . . , an ) =
(a1 ,...,an )∈P(i0 ,...,ik )
(m−k)
(−1)j−1 Hi0 ,ij −1
k
X
j=1
(ij − i0 )(ij − i1 ) · · · (ij − ij−1 )(ij+1 − ij ) · · · (ik − ij )
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
.
Sum over indices with fixed number of nonzero
positions
Generalized
Harmonic
Series (NOT
MZVs)
ME Hoffman
Outline
Introduction
H-series and
Stirling
numbers of
the first kind
For 0 ≤ k ≤ n − 1, let C (k, n; m) be the sum of all H-series
H(a1 . . . , an ) of length n, weight m, and exactly k + 1 of the ai
nonzero. If k ≥ 1, C (k, n; m) is the sum over all sequences
1 ≤ i0 < · · · < ik ≤ n of the sums given by the preceding
(m−k)
1
with
result; since each Hi0 ,ij −1 in that result is a sum of pm−k
1 ≤ i0 ≤ p ≤ ij − 1 ≤ n − 1, we have
Proof of the
sum formulas
C (k, n; m) =
n−1 (n)
X
ck,j
j=1
(n)
(5)
j m−k
(n)
with ck,j rational. We can deduce results about the ck,j from
Lemma (2).
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
An (anti)symmetry property
Generalized
Harmonic
Series (NOT
MZVs)
Lemma (3)
(n)
(n)
For 1 ≤ k, j ≤ n − 1, ck,n−j = (−1)k−1 ck,j .
ME Hoffman
Outline
Introduction
H-series and
Stirling
numbers of
the first kind
Proof of the
sum formulas
Proof.
For 1 ≤ i0 < · · · < ik ≤ n, let Snm (i0 , . . . , ik ) be the left-hand
side of the equation in Lemma (2). Then if
Snm (i0 , . . . , ik ) = p1 +
p2
pn−1
+ ··· +
,
2m−k
(n − 1)m−k
it follows from Lemma (2) that
(−1)k−1 Snm (n + 1 − ik , . . . , n + 1 − i0 ) =
pn−2
p1
pn−1 + m−k + · · · +
.
2
(n − 1)m−k
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
Some explicit coefficients
Generalized
Harmonic
Series (NOT
MZVs)
(n)
Here are some of the coefficients ck,j .
(3)
ME Hoffman
c1,1 =
Outline
c2,1 =
(3)
3
2
1
2
(3)
c1,2 =
(3)
3
2
c2,2 = − 21
Introduction
(4)
c1,1 =
H-series and
Stirling
numbers of
the first kind
(4)
c2,1
(4)
c3,1
Proof of the
sum formulas
(5)
c1,1 =
(5)
c2,1 =
(5)
c3,1 =
(5)
c4,1 =
25
12
35
24
5
12
1
24
11
6
=1
=
1
6
(5)
(4)
c1,2 =
(4)
c2,2 = 0
(4)
c3,2 = − 62
35
12
(5)
c2,2 = 85
(5)
5
c3,2 = − 12
(5)
3
c4,2 = − 24
c1,2 =
ME Hoffman
7
3
(5)
(4)
11
6
(4)
c2,3 = −1
(4)
c3,3 = 61
c1,3 =
35
12
(5)
c2,3 = − 85
(5)
5
c3,3 = − 12
(5)
3
c4,3 = 24
c1,3 =
(5)
25
12
(5)
c2,4 = − 35
24
(5)
5
c3,4 = 12
(5)
1
c4,4 = − 24
c1,4 =
Generalized Harmonic Series (NOT MZVs)
More explicit coefficients
Generalized
Harmonic
Series (NOT
MZVs)
ME Hoffman
Outline
Introduction
H-series and
Stirling
numbers of
the first kind
Proof of the
sum formulas
(n)
For brevity, we give the matrices [(n − 1)!ci,j ] for n = 6, 7.


274 404 444 404
274
225 150
0
−150 −225


 85 −40 −90 −40
85 


 15 −30
0
30
−15 
1
−4
6
−4
1


1764 2688 3108 3108 2688
1764
1624 1330 490 −490 −1330 −1624


 735 −105 −630 −630 −105
735 


 175 −245 −140 140

245
−175


 21
−63
42
42
−63
21 
1
−5
10
−10
5
−1
Note that all row sums except the first are 0.
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
Stirling numbers of the first kind
Generalized
Harmonic
Series (NOT
MZVs)
ME Hoffman
Outline
Introduction
H-series and
Stirling
numbers of
the first kind
Proof of the
sum formulas
The first columns the
of the matrices above turn out to be
n
well-known. Let k be the number of permutations of
{1, 2, . . . , n} with exactly k disjoint cycles;
0this is the unsigned
Stirling number of the
first
kind.
We
set
0 = 1, and for n ≥ 1
n the Stirling number k is only nonzero if 1 ≤ k ≤ n. Some
useful facts are
n
n
n
n
= (n − 1)!,
= 1,
=
.
1
n
n−1
2
One has the generating function
x(x + 1) · · · (x + n − 1) =
n X
n
k=1
ME Hoffman
k
x k , n ≥ 1.
Generalized Harmonic Series (NOT MZVs)
(6)
Stirling numbers of the first kind cont’d
Generalized
Harmonic
Series (NOT
MZVs)
ME Hoffman
(n)
The following result relates the Stirling numbers to the ck,1 .
Lemma (4)
For 1 ≤ k ≤ n − 1,
Outline
Introduction
(n)
ck,1 = (−1)k−1 ck,n−1 =
H-series and
Stirling
numbers of
the first kind
Proof of the
sum formulas
n
1
.
(n − 1)! k + 1
Proof.
By Lemma (3), it suffices to prove the last equality. Lemma (2)
says
X
(−1)k−1
(n)
ck,n−1 =
(n − i0 ) · · · (n − ik−1 )
1≤i0 <···<ik−1 <n
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
Stirling numbers of the first kind cont’d
Generalized
Harmonic
Series (NOT
MZVs)
ME Hoffman
Outline
Proof cont’d.
so it’s enough to show
Introduction
X
H-series and
Stirling
numbers of
the first kind
Proof of the
sum formulas
1≤i0 <···<ik−1 <n
n
(n − 1)!
=
.
(n − i0 ) · · · (n − ik−1 )
k +1
Now the left-hand side is evidently the sum of all products of
n − k − 1 distinct
n factors from the set {1, 2 . . . , n − 1}, and
that this is k+1
follows from consideration of the generating
function (6).
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
A general formula
Generalized
Harmonic
Series (NOT
MZVs)
ME Hoffman
The preceding result generalizes as follows.
Lemma (5)
For 1 ≤ k, j ≤ n − 1,
Outline
Introduction
H-series and
Stirling
numbers of
the first kind
Proof of the
sum formulas
(n)
ck,j
=
j X
q
X
(−1)p−1
q=1 p=1
qn+1−q p
k+2−p
(q − 1)!(n − q)!
.
We only sketch the proof. The lemma amounts to
n+1−j j
X
(−1)p−1 pj k+2−p
(n)
(n)
.
ck,j − ck,j−1 =
(q − 1)!(n − q)!
p=1
(n)
(n)
By Lemma (3), we can make this about ck,n−j − ck,n−j+1 .
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
(7)
A general formula cont’d
Generalized
Harmonic
Series (NOT
MZVs)
(n)
ME Hoffman
Now think of the ck,i as sums of terms via Lemma (2). If
Outline
j ≤ k, then all terms contributing to ck,n−j+1 also contribute to
(n)
(n)
Introduction
ck,n−j , and in addition there are j classes of terms that
H-series and
Stirling
numbers of
the first kind
contribute to ck,n−j and not to ck,n−j+1 : these j classes of
terms can be matched up with the j terms on the right-hand
side of equation (7). If j > k, then there are also terms that
(n)
(n)
contribute to ck,n−j+1 and not to ck,n−j : one can use Lemmas
(2) and (4) to match up their sum with a term on the
right-hand side of equation (7).
Proof of the
sum formulas
(n)
ME Hoffman
(n)
Generalized Harmonic Series (NOT MZVs)
Proof of the second sum formula
Generalized
Harmonic
Series (NOT
MZVs)
ME Hoffman
Outline
Now we’ll prove the sum formulas (1) and (4). As a warm-up,
we’ll do the second of these first. Replacing k by n, equation
(4) is
Introduction
H-series and
Stirling
numbers of
the first kind
Proof of the
sum formulas
X
a1 +···+an =m, ai ≥1
n−1 X
n − 2 (−1)j−1
1
H(a1 , . . . , an ) =
.
(n − 1)!
j − 1 j m−n+1
j=1
The left-hand side is C (n − 1, n; m), so it suffices to show that
(−1)j−1 n − 2
(n)
cn−1,j =
.
(n − 1)! j − 1
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
Proof of the second sum formula cont’d
Generalized
Harmonic
Series (NOT
MZVs)
Since kn = 0 when k > n and nn = 1, this turns out to be an
easy consequence of Lemma (5):
ME Hoffman
Outline
(n)
cn−1,j =
Introduction
H-series and
Stirling
numbers of
the first kind
Proof of the
sum formulas
=
j X
q
X
(−1)p−1
qn+1−q
p
n+1−p
(q − 1)!(n − q)!
q=1 p=1
n+1−q
j
X
(−1)q−1 qq n+1−q
q=1
(q − 1)!(n − q)!
j
X
1
q−1 n − 1
=
(−1)
q−1
(n − 1)!
q=1
(−1)j−1 n − 2
=
.
(n − 1)! j − 1
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
Proof of the first sum formula
Generalized
Harmonic
Series (NOT
MZVs)
Next we attack equation (1). We have
X
ME Hoffman
Outline
Introduction
H-series and
Stirling
numbers of
the first kind
Proof of the
sum formulas
H(a1 , . . . , an ) = C (0, n; m) +
a1 +···+an =m
n−1
X
C (k, n; m).
k=1
Now C (0, n; m) is
H(m, 0, . . . , 0) + H(0, m, 0, . . . , 0) + · · · + H(0, . . . , 0, m)
1
1
= ζ(m) + ζ(m) − 1 + · · · + ζ(m) − 1 − m − · · · −
2
(n − 1)m
n−2
1
= nζ(m) − (n − 1) − m − · · · −
2
(n − 1)m
n−1
X
n−j
= nζ(m) −
jm
j=1
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
Proof of the first sum formula cont’d
Generalized
Harmonic
Series (NOT
MZVs)
ME Hoffman
Outline
Introduction
H-series and
Stirling
numbers of
the first kind
so (using equation (5))
X
H(a1 , . . . , an ) = nζ(m) −
a1 +···+an =m
n−1
X
n−j
j=1
jm
+
n−1 X
n−1 (n)
X
ck,j
k=1 j=1
j m−k
.
Hence, to prove the result requires
n−1
X
Proof of the
sum formulas
(n)
j k ck,j = n − j
k=1
for 1 ≤ j ≤ n − 1. In view of Lemma (5), this is
n+1−q j X
q
n−1 X
X
(−1)p−1 qp k+2−p
k
j
= n − j.
(q − 1)!(n − q)!
k=1 q=1 p=1
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
(8)
Proof of the first sum formula cont’d
Generalized
Harmonic
Series (NOT
MZVs)
ME Hoffman
The left-hand side of equation (8) is
j X
q
X
Outline
q=1 p=1
Proof of the
sum formulas
p
k=1
Introduction
H-series and
Stirling
numbers of
the first kind
q
n−1 X
n+1−q k
j .
(q − 1)!(n − q)!
k +2−p
(−1)p−1
If p = 1, the inner sum is
n−1 n X
n + 1 − q k X n + 1 − q k−1
n+1−q
j =
j
−
=
k +1
k
1
k=1
k=1
(n + j − q)!
n+1−q
n+j −q
−
= (n − q)!
−1 ,
j(j − 1)!
1
j
where we have used the generating function (6).
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
Proof of the first sum formula cont’d
Generalized
Harmonic
Series (NOT
MZVs)
ME Hoffman
If p ≥ 2, the inner sum is
X n + 1 − q (n + j − q)!
j k+p−2 = j p−2
,
k
(j − 1)!
k≥1
Outline
Introduction
again using equation (6), and so
H-series and
Stirling
numbers of
the first kind
n−1 X
n+1−q
Proof of the
sum formulas
k=1
k +2−p
k
j = (n − q)!j
p−1
n+j −q
.
j
Thus the left-hand side of equation (8) is
j j X
q
X
X
(−1)p−1 qp n + j − q p−1
n+j −q
−j +
j
.
j
(q − 1)!
j
q=1
q=2 p=2
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
Proof of the first sum formula cont’d
Generalized
Harmonic
Series (NOT
MZVs)
ME Hoffman
The double sum can be rewritten as
j q X
X
n+j −q
1
q
(−j)p−1 =
j
(q − 1)!
p
q=2
p=2
Outline
Introduction
j X
H-series and
Stirling
numbers of
the first kind
q=2
Proof of the
sum formulas
n+j −q
q
1
(1 − j)(2 − j) · · · (q − 1 − j) −
j
(q − 1)!
1
X
j j X
n+j −q
n+j −q
j −1
=−
+
(−1)q−1
j
j
q−1
q=2
q=2
using equation (6) again, so the left-hand side of equation (8) is
j X
n+j −q
j −1
−j +
(−1)q−1 .
j
q−1
q=1
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
One more lemma
Generalized
Harmonic
Series (NOT
MZVs)
This means we need one last lemma.
Lemma
ME Hoffman
For positive integers n and j,
Outline
j X
n+j −q
j −1
(−1)q−1 = n.
j
q−1
Introduction
H-series and
Stirling
numbers of
the first kind
Proof of the
sum formulas
q=1
Proof.
Use Wilf’s “snake oil method”: let
j ∞ X
X
n+j −q
j −1
F (x) =
(−1)q−1 x n .
j
q−1
n=1 q=1
ME Hoffman
Generalized Harmonic Series (NOT MZVs)
One more lemma cont’d
Generalized
Harmonic
Series (NOT
MZVs)
Proof cont’d.
ME Hoffman
Then
Outline
Introduction
H-series and
Stirling
numbers of
the first kind
Proof of the
sum formulas
j ∞ X
X
j −1
n+j −q n
q−1
F (x) =
(−1)
x
q−1
j
q=1
=
j X
q=1
n=1
j−1 X
j − 1 (−1)q−1 x q
x
j −1
=
(−x)p
q − 1 (1 − x)j+1
p
(1 − x)j+1
p=0
=
x)j−1
x(1 −
x
=
= x + 2x 2 + 3x 3 + · · ·
j+1
(1 − x)
(1 − x)2
and the result follows.
ME Hoffman
Generalized Harmonic Series (NOT MZVs)