A NOTE ON STIRLING SERIES
MARKUS KUBA AND HELMUT PRODINGER
Abstract. We study sums S = S(d, n, k) =
P
[dj ]
j≥1 j k (n+j )j!
j
with d ∈ N = {1, 2, . . . }
and n, k ∈ N0 = {0, 1, 2, . . . } and relate them with (finite) multiple zeta functions. As a
(d+1)
byproduct of our results we obtain asymptotic expansions of ζ(d + 1) − Hn
as n tends
to infinity. Furthermore, we relate sums S to Nielsen’s polylogarithm.
1. Introduction
The unsigned Stirling numbers of the first kind, also called Stirling cycle numbers, are
defined by the recurrence relation
n
n−1
n−1
n
= (n − 1)
+
, n ≥ 1, with
= δn,0 , n ≥ 0,
k
k
k−1
0
where δi,j denotes
the Kronecker delta function. Throughout this work we use Knuth’s
n
notation k . It is well known that Stirling numbers of the first kind are closely related
(2)
2
− Hn )/2, where for
to harmonic numbers, i.e. n2 = (n − 1)!Hn−1 , n3 = (n − 1)!(Hn−1
P
(1)
(s)
s, n ∈ N the values Hn = n`=1 1/`s denote n-th harmonic numbers of order s, Hn = Hn .
Furthermore, it is known (i.e. see Adamchik [1]) that Stirling numbers of the first kind are
expressible in terms of (finite) multiple zeta functions defined by
X
1
,
ζN (a1 , . . . , al ) =
a1 a2
n n . . . na` `
N ≥n1 >n2 >···>nl ≥1 1 2
X
1
ζ(a1 , . . . , a` ) =
a` ,
a1 a2
n
n
.
.
.
n
1
2
`
n >n >···>n ≥1
1
2
`
by the following formula
n
= (n − 1)!ζn−1 (1, . . . , 1) = (n − 1)! · ζn−1 ({1}k−1 ).
| {z }
k
k−1
ζ(∪ri=1 {ai }) =
(s)
ζn (s) = Hn .
We use the shorthand notations
ζ(a1 , . . . , ar ), and ζ(∪ri=1 {a}) = ζ({a}r ).
Note that for n, s ∈ N0 we have
We are interested in evaluations of sums
P
[dj ]
S = j≥1 j k n+j j! with d ∈ N = {1, 2, . . . } and n, k ∈ N0 = {0, 1, 2, . . . }. We assume that
( j )
2000 Mathematics Subject Classification. 11M06, 40B05.
Key words and phrases. Stirling numbers, multiple zeta functions, harmonic numbers.
The author M.K. was supported by the Austrian Science Foundation FWF, grant S9608-N13.
1
2
M. KUBA AND H. PRODINGER
n and k are choosen in way such that n + k ≥ 1 in order to ensure that the sum converges.
Special instances of this family of sums have been studied by Adamchik [1], and also by
Choi and Srivastava [6].
2. Evaluation of sum S
We obtain the following result.
Theorem 1. The sum S = S(d, n, k) with d ∈ N = {1, 2, . . . } and n, k ∈ N0 = {0, 1, 2, . . . }
can be evaluated in terms of harmonic numbers and (finite) multiple zeta functions,
S=
k+1
X
k+1−m
(−1)
ζ(m, {1}d−1 )
m=2
Pk+1−m
i=1
k
+ (−1)
k
X
X
k+1−m
Y
X
i·mi =k+1−m
r=1
(r)
(Hn )mr
rmr mr !
ζn (`1 , `2 − `1 , . . . , `h−1 − `h−2 , d + k − lh−1 ),
h=1 1≤`1 <`2 <···<`h−1 <k
subject to `0 := 0. We have the short equivalent expression
S=
(−1)k ζn∗ ({1}k−1 , d
+ 1) +
k+1
X
(−1)k+1−m ζ(m, {1}d−1 )ζn∗ ({1}k+1−m ).
m=2
Remark 1. The second expression for the sum S is given according to a variant of finite
multiple zeta functions, ζN∗ (a1 , . . . , ak ), which recently attracted some interest,[2, 12, 9, 7]
where the summation indices satisfy N ≥ n1 ≥ n2 ≥ · · · ≥ nk ≥ 1 in contrast to N ≥ n1 >
n2 > · · · > nk > 1, as in the usual definition (1),
X
1
ζN∗ (a1 , . . . , ak ) =
ak .
a1 a2
n
n
.
.
.
n
1
2
k
N ≥n ≥n ≥···≥n ≥1
1
2
k
The form stated above is due to the conversion formula below applied to ζn∗ ({1}k−1 , d + 1),
X
`1
`2
k
k
X
X
X
X
∗
ζN (a1 , . . . , ak ) =
ζN
ai 1 ,
ai 2 , . . . ,
ai h .
h=1 1≤`1 <`2 <···<`h−1 <k
`0 =0
i1 =1
i2 =`1 +1
ih =`h−1 +1
P
Note that the first term h = 1 should be interpreted as ζN ( ki1 =`0 +1 ai1 ), subject to `0 = 0.
The notation ζN∗ (a1 , . . . , ak ) is chosen in analogy with Aoki and Ohno [2] where infinite
counterparts of ζN∗ (a1 , . . . , ak ) have been treated; see also Ohno [12].
Remark 2. The sum ζ(m, {1}d−1 ) can be completely transformed into single zeta values.
By results of Borwein, Bradley and Broadhoarst [3]
ζ(2, {1}d ) = ζ(d + 2)
d
d+2
1X
ζ(d + 3) −
ζ(` + 1)ζ(d + 2 − `).
ζ(3, {1}d ) =
2
2 `=1
A NOTE ON STIRLING SERIES
3
Furthermore, in the general case of ζ(m + 2, {1}d ) = ζ(d + 2, {1}m ) one obtains products
of up to min{m + 1, d + 1} zeta values, according to the generating function, see [3],
X k
X
x + y k − (x + y)k
m+1 n+1
ζ(m + 2, {1}n )x
y
= 1 − exp
ζ(k) .
(1)
k
m,n≥0
k≥2
Below we state three specific evaluations of the sum S for special choices of d, n, k.
Corollary 1. For k = 0 and arbitrary n, d ∈ N we get
j X
1
d
= d.
S(d, n, 0) =
n+j
n
j!
j
j≥1
For k = 1 and arbitrary n, d ∈ N we get
j X
d
= ζ(2, {1}d−1 ) − ζn (d + 1) = ζ(d + 1) − Hn(d+1) ,
S(d, n, 1) =
n+j
j
j!
j
j≥1
(2)
For n = 0 and arbitrary d, k ∈ N we get
S(d, 0, k) = ζ(k + 1, {1}d−1 ).
In order to prove the results above we proceed as follows: Since
n
X
n − 1 (−1)`−1
1
n!
n
,
=
n+j =
n
`
−
1
(n
+
j)
j
+
`
j
`=1
we obtain
S=
X
j≥1
j n
X
X
n
−
1
`−1
d
d
=
n
(−1)
.
k (j + `)
`
−
1
j!j
j k n+j
j!
j
j≥1
`=1
j We use partial fraction decomposition and obtain
k
X
(−1)k−m
(−1)k+1 1
1 1
=
+
−
.
j k (j + `) m=2 j m `k+1−m
`k
j j+`
Consequently, we get by using the partial fraction decomposition above and the representation of Stirling numbers by finite multiple zeta functions
k+1
n
X
X
(−1)k+1−m X ζj−1 ({1}d−1 )
n−1
`−1
S=
n
(−1)
`k+2−m j≥1
jm
`−1
m=2
`=1
n
1
X
(−1)k X
n−1
1 +
n
(−1)`−1 k+1
ζj−1 ({1}d−1 )
−
= S1 + S2 .
`
−
1
`
j
j
+
`
j≥1
`=1
4
M. KUBA AND H. PRODINGER
By definition of the multiple zeta function we get
n
k+1
X
X
(−1)k+1−m
n−1
`−1
S1 =
n
(−1)
ζ(m, {1}d−1 )
k+2−m
`
`
−
1
m=2
`=1
k+1
n
X
X
n − 1 (−1)`−1
k+1−m
=
(−1)
ζ(m, {1}d−1 )
n
.
` − 1 `k+2−m
m=2
`=1
We rewrite the inner sum as
n n
X
n − 1 (−1)`−1 X n (−1)`−1
=
.
n
` `k+1−m
` − 1 `k+2−m
`=1
`=1
This sum can be evaluated by using the following result of Flajolet and Sedgewick [8],
n m
(r) mr
X
X
Y
n (−1)`−1
(Hn )
;
=
m
mr m !
`
r
`
Pm
r
r=1
`=1
i=1
(s)
i·mi =m
Pn
we recall that Hn = `=1 1/`s denotes the n-th harmonic number of order s; in other
(s)
words we have Hn = ζn (s), according to our previous definition of finite multiple zeta
Pn n (−1)`−1
functions (1). Furthermore, it is well known that
= ζn∗ ({1}m ), which
`=1 `
`m Pn
`−1
can immediately be deduced by repeated usage of the formula nk =
`=k k−1 . The
multiple zeta function ζ(m, {1}d ) is evaluated using a result of Borwein, Bradley and
Broadhoarst [3], see Remark 2. Consequently, we can write sum S1 as a finite sum involving
higher order harmonic numbers and products of zeta functions and obtain the first part of
our result. For the simplification of the inner sum
n
X
n−1
(−1)k X
1
1 S2 =
n
(−1)`−1 k+1
ζj−1 ({1}d−1 ) −
,
`
−
1
`
j
j
+
`
j≥1
`=1
P
1
1
we use the notation Tm,` =
ζ
({1}
)
−
. Subsequently, we interchange
j−1
m
j≥1
j
j+`
summation, compare with Panholzer and Prodinger [11]. First we start with the simple
case m = 1 and calculate T1,` , since it is most instructive.
1
X
1 X 1
1
T1,` =
Hj−1
−
=
Hj
−
.
j j+`
j+1 j+1+`
j≥1
j≥1
P
Since by definition Hj = jh=1 1/h we obtain after summation change (partial summation)
T1,` =
`
X1X
X 1 X 1
1
1
−
=
.
h
j
+
1
j
+
1
+
`
h
j
+
h
j=1
h≥1
j≥h
h≥1
By partial fraction decomposition we get
`
`
(2)
X
1 X1
1 X Hj
H 2 + H`
T1,` =
−
=
= `
.
j
h
j
+
h
j
2
j=1
j=1
h≥1
A NOTE ON STIRLING SERIES
5
Now we turn to the general case Tm,` . Shifting the index as before, and changing the order
of summation leads to
X ζh−1 ({1}m−1 ) X 1
1
Tm,` =
−
h
j+1 j+1+`
h≥1
j≥h
Consequently,
Tm,` =
`
1
1 X1
−
=
Tm−1,j .
ζh−1 ({1}m−1 )
j h≥1
h h+j
j
j=1
`
X
1X
j=1
Hence, the value Tm,` is a variant of the finite multiple zeta function ζ` ({1}m+1 ), where the
summation indices satisfy N ≥ n1 ≥ n2 ≥ · · · ≥ nm ≥ nm+1 ≥ 1 instead of N ≥ n1 > n2 >
· · · > nm > nm+1 > 1, see Remark 1, such that Tm,` = ζ`∗ ({1}m+1 ). We further obtain
Tm,` =
ζ`∗ ({1}m+1 )
=
` X
` (−1)h−1
h=1
h
hm+1
,
P
`−1
according to the well known formula nk = n`=k k−1
. Consequently, the sum S2 simplifies
to
` n n
n X
X
(−1)h−1 X n
` (−1)`−1
n (−1)`−1 X ` (−1)h−1
k
k
=
(−1)
,
S2 = (−1)
k
d
d
k
h
`
h
`
h
h
`
`
h=1
`=h
h=1
`=1
or equivalently
S2 = (−1)
k
n X
n (−1)`−1
`=1
`
`k
ζ`∗ ({1}d ).
In order to obtain the final form of S2 for k ∈ N we combine our previous considerations
as follows:
hk
h1
n
X
X
1 X
1
hk
k
S2 = (−1)
···
(−1)hk+1 −1 ζh∗k+1 ({1}d ).
h
h
h
1
2
k+1
h =1
h =1
h
=1
1
We use the fact that
Pn
`=h
2
k+1
n
`
`
(−1)`−1 = δh,n (−1)n−1 and the sum S2 simplifies to
h
S2 = (−1)k ζn∗ ({1}k−1 , d + 1).
In the case k = 0 we use
n n
X
1
`
(−1)h−1 X n
(−1)`−1 = d .
S2 =
d
h
`
h
n
`=h
h=1
6
M. KUBA AND H. PRODINGER
2.1. An application: asymptotic expansions. Following Romik [13] we note that the
P
[dj ]
limit limn→∞ j≥1 j k n+j
= 0 provides information about the convergence of the two
( j )j!
sums appearing in the results for S = S1 + S2 , stated in Theorem 1. This is of particular
interest in the special case k = 1 and arbitrary n, d ∈ N, where we have obtained ζ(d +
P
[dj ]
(d+1)
1) − Hn
= j≥1 j n+j
, see Corollary 1.
( j )j!
Proposition 1. We obtain the following asymptotic expansions for n → ∞
(−1)k ζn∗ ({1}k−1 , d
k+1
X
+ 1) +
(−1)k+1−m ζ(m, {1}d−1 )ζn∗ ({1}k+1−m )
m=2
=
n
X
j d
k n+j j!
j
j
j=1
1
+O √ n ,
n2
In the case of S = S(d, n, 1) with k = 1 and arbitrary n, d ∈ N we obtain in particular:
j n
X
1
(d+1)
d
+ O √ n , for n → ∞.
ζ(d + 1) − Hn
=
n+j
n2
j
j!
j
j=1
Proof. We can split the summation range j ≥ 1 into 1 ≤ j ≤ N and j ≥ N + 1 > k. We
get
j X
X
X
2
2
(n + 1 + N )
d
<
<
n+j =
.
n+j
n+j
j
n+1+N N +1
k
k
j
j
k!
j!
k!(k
−
1
+
N
)
j
j
k
j
n
k
j≥N +1
j≥N +1
j≥N +1
Consequently, we readily obtain, setting i.e. N = n and using Stirling’s formula,
1 nn √
,
n! = n 2πn 1 + O
e
n
the stated asymptotic expansions.
3. Relation to Nielsen’s polylogarithm
Nielsen’s polylogarithm Lk,d (z) is defined by
Z
(−1)k−1+d 1 logk−1 (t) logd (1 − zt)
Lk,d (z) =
dt
(k − 1)!d! 0
t
By definition of the generating function of the Stirling cycle numbers
X n z n
(−1)k logk (1 − z)
=
,
k n!
k!
n≥k
it is evident that Lk,d (z) =
P
[dj ]zj
j≥1 j k j!
. Hence, we obtain the following result.
A NOTE ON STIRLING SERIES
Proposition 2. The series S(z) = Sd,n,k (z) =
P
j≥1
7
[dj ]zj
can be expressed by Nielsen’s
j!
j k (n+j
j )
polylogarithm Lk,d (z) in the following way.
j j
Z
X
z
n z
u n−1
d
=
1−
Lk,d (u)du.
z 0
z
j k n+j
j!
j
j≥1
Note that
n
X
Z
Z
n (−1)k−1 1 z `−1 1 logk−1 (t) logd (1 − ut)
u
dtdu
Sd,n,k (z) =
`(−1)
l
(k
−
1)!d!
z
t
`
0
0
`=1
Z z
n
X
`−1 n 1
=
`(−1)
u`−1 Lk,d (u)du.
l
`
z
0
`=1
`−1
Interchanging summation and integration gives the desired result.
3.1. Generalized r-Stirling numbers of the first kind. In a recent work Mező [10]
considered series involving so-called r-Stirling
n numbers of the first kind, see Broder [5]. For
any positive integer r ∈ N the quantity m r denotes the number of permutations of the
set {1, . . . , n} having m cycles such that the first r element are in distinct cycles. These
numbers obey the recurrence relation
n
n
n−1
n−1
n
= 0, n < r.
= δk,r , n = r,
, n > r,
+
= (n − 1)
k r
k r
k−1 r
k r
k r
For r = 0 and r = 1 these numbers coincide with the ordinary Stirling numbers of the first
kind. We will consider the series
j
X j+`+r
z
(r)
d+r
r
S (r) (z) = Sd,n,k,` (z) =
,
j k n+j
j!
j
j≥1
which generalizes the series considered by Mező [10] (case n = 0) and our previously
(r)
considered series S (case ` = r = 0). Subsequently, we obtain representations of Sd,n,k,0 (z)
(r)
(r)
and also of Sd,n,k,` (z). We introduce the quantity Ln,k (z), which generalizes Nielsen’s
polylogarithm
Z
(−1)k−1+d 1 logk−1 (t) logd (1 − zt)
(r)
dt.
Lk,d (z) =
(k − 1)!d! 0
(1 − zt)r t
(r)
Proposition 3. The series Sd,n,k,0 (z) =
P
j≥1
j+r
[d+r
]r zj
(r)
can be expressed by Lk,d (z) in the
n+j
k
j ( j )j!
following way.
(r)
Sd,n,k,0 (z)
The series
0 ≤ h ≤ d.
(r)
Sd,n,k,` (z)
n
=
z
Z
0
z
u n−1 (r)
1−
Lk,d (u)du.
z
(r+`)
can be expression as a linear combination of the sums Sh,n,k,0 (z), with
8
M. KUBA AND H. PRODINGER
First we note that the r-Stirling numbers of the first kind have the generating function
X n + r z n
(−1)k logk (1 − z)
=
.
r
k
+
r
n!
k!(1
−
z)
r
n≥k
We observe that
j+r (r)
Lk,d (z) =
zj
d+r r
j k n+j
j!
j
j≥1
X
(r)
= Sd,0,k,0 (z).
Consequently, we get
j+r (r)
Sd,n,k,0 (z)
=
zj
d+r r
j k n+j
j!
j
j≥1
X
Z
=
0
z
n(1 − uz )n (r)
L (u)du.
(z − u) k,d
Next we turn to the general case ` ∈ N. Since
X n + r z n
(−1)d logd (1 − z)
=
,
r
n!
d!(1
−
z)
d
+
r
r
n≥k
d logd (1−z)
we obtain the exponential generating function of n+`+r
by differentiating (−1)d!(1−z)
r
d+r r
`-times with respect to z and a subsequent shift of the index,
X n + r z n−`
X
n + ` + r zn
∂ ` (−1)d logd (1 − z)
=
=
.
d + r r (n − `)!
d + r r n!
∂z `
d!(1 − z)r
n≥d+`
n≥max{d−`,0}
By Faà di Bruno’s formula we get
`
`
∂ ` (−1)d logd (1 − z) X dh (−1)h logd−h (1 − z) X `−i
r Bi,h (0!, 1!, 2!, . . . , (i − h)!),
=
r+`
∂z `
d!(1 − z)r
(1
−
z)
h=0
i=h
where Bi,h (x1 , x2 , . . . , xi−h+1 ) denote the Bell polynomials. Consequently, we can express
(r)
(r)
the sum Sd,n,k,` (z) as a linear combination of the sums Sh,n,k,0 (z), with 0 ≤ h ≤ d, which
proves the stated result.
P
[j+`+r] zj
(r)
Remark 3. Note that the sums Sd,n,k,` (1) = j≥1 j kd+rn+jr j! can in principle also be treated
( j )
using our previous approach; however, the expression become much more involved, therefore we refrain from going into this matter. Furthermore, one can evaluate sums of the
P
[dj ]
form j≥1 j k n+j
g , with g ∈ N; however, the expressions get more and more involved.
( j ) j!
4. A generalization of series S
In the following we will briefly consider the more general series V defined by
X ζj−1 (a1 , . . . , ar )
,
V = V (a1 , . . . , ar , n, k) =
k+1 n+j
j
j
j≥1
with ai ∈ N for 1 ≤ i ≤ r, and n, k ∈ N0 = {0, 1, 2, . . . } such that n + k ≥ 1. We reobtain
our previous Stirling cycle number series S choosing r = d − 1 and ai = 1, 1 ≤ i ≤ d − 1.
A NOTE ON STIRLING SERIES
9
Before we state our result for the series V we introduce one more series, namely a variant
of the finite multiple zeta star function
N X
X
N (−1)a1 −1 ∗
N
(−1)a1 −1
∗
=
AN (a1 , . . . , ar ) =
ζn1 (a2 , . . . , ar ),
a1 a2
n1
n1 n1 n2 . . . nar r
na11
n =1
N ≥n ≥n ≥···≥n ≥1
1
2
1
r
which can be expressed in terms of ζN∗ (a1 , . . . , ar ) by the relation
A∗N (a1 , {1}b1 −1 , ∪ri=2 {ai + 1, {1}bi −1 }) = ζN∗ (∪r−1
i=1 {{1}ai −1 , bj + 1}, {1}ar −1 , br ),
which is due to Bradley [4].
Theorem 2. The sum V = V (a1 , . . . , ar , n, k) with ai ∈ N for 1 ≤ i ≤ r, and n, k ∈
N0 = {0, 1, 2, . . . } such that n + k ≥ 1 can be evaluated in terms of (finite) multiple zeta
functions,
ag
r
X
X
Pg−1
(af +1)
f
=1
(−1)
(−1)ag −m ζ(m, ∪ri=g+1 {ai })An (k, ∪g−1
V = (−1)
i=1 {ai }, ag + 1 − m)
k
g=1
+ (−1)
k+r+
m=2
Pr
f =1
af
An (k, ∪ri=1 {ai }, 1)
+
k+1
X
(−1)k+1−m ζ(m, a1 , . . . , ar )ζn∗ ({1}k+1−m ).
m=2
Proof (Sketch). The proof is analogous to the proof of Theorem 1; therefore it is only
sketched. We
new difficulty—the evaluation of the sum
P only elaborate on the 1main
. Proceeding as before, i.e. interchanging sumTa1 ,...,ar ;` = j≥1 ζj−1 (a1 , . . . , ar ) 1j − j+`
mation and using partial fraction decomposition, we obtain the recurrence relation
Ta1 ,...,ar ;` =
a1
X
(−1)
a1 −m
ζ(m, a2 , . . . , ar )ζ`∗ (a1
a1 +1
+ 1 − m) + (−1)
m=2
`
X
1
Ta ,...,a ;i .
ia1 2 r
i=1
One can show that
Ta1 ,...,ar ;`
ag
r
X
X
Pg−1
(a
+1)
f
(−1) f =1
(−1)ag −m ζ(m, ∪ri=g+1 {ai })ζ`∗ (∪g−1
=
i=1 {ai }, ag + 1 − m)
g=1
m=2
r+
+ (−1)
Pr
f =1 af ζ ∗ (∪r
`
i=1 {ai }, 1),
which implies the stated result for the series V .
Historical remark
The author H.P. has found the formula (2) empirically in 2003. He contacted several
specialists about it and got feedback from Christian Krattenthaler who provided a hypergeometric proof for it. Eventually it turned out that it was known already [6]. We are
happy that in 2009 we could put new life into this project.
10
M. KUBA AND H. PRODINGER
References
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Publishers, Dordrecht, x+388 pp., 2001.
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of Number Theory, vol. 129, 908-921, 2009.
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Integrals. Theoretical Computer Science, vol. 144 (1-2), 01–124, 1995.
[9] M. E. Hoffmanm, Multiple harmonic series. Pacific J. Math. vol. 152 (2), 275–290, 1992.
[10] I. Mező, New properties of r-Stirling series, Acta Math. Hungar., vol. 119 (4), 341–358, 2008.
[11] A. Panholzer and H. Prodinger, Computer-free evaluation of a double infinite sum via Euler sums,
Séminaire Lotharingien de Combinatoire, B55a, 2005.
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Number Theory, vol. 74, 39–43, 1999.
[13] D. Romik, Some comments on Eulers series for π 2 /6. Math. Gazette, 281–284, 2002.
Markus Kuba, Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstr. 8-10/104, 1040 Wien, Austria
E-mail address: [email protected]
Helmut Prodinger, Department of Mathematics, University of Stellenbosch, 7602 Stellenbosch, South Africa
E-mail address: [email protected]