Table Of ContentPower sums of Coxeter exponents
John M. Burns
1
1
0 School of Mathematics, Statistics and Applied Mathematics
2
National University of Ireland, Galway
n
a University Road, Galway, Ireland
J
6 e-mail: [email protected]
2
] Ruedi Suter
O
C
. Departement Mathematik
h
t
a ETH Zu¨rich
m
R¨amistrasse 101, 8092 Zu¨rich, Switzerland
[
1 e-mail: [email protected]
v
2
8
0
5
. Abstract
1
0
ConsideranirreduciblefiniteCoxetersystem. Weshowthatforanynonnegative
1
1 integer n the sum of the nth powers of the Coxeter exponents can be written uni-
: formly as a polynomial in four parameters: h (the Coxeter number), r (the rank),
v
i α, β (two further parameters).
X
r
a
1 Introduction
Let (W,S) be an irreducible finite Coxeter system of rank r with S = {s ,...,s } its set
1 r
of simple reflections. The Coxeter transformation c := s ...s ∈ W has order |c| = h
1 r
known as the Coxeter number, and the eigenvalues of c in the reflection representation
of W are of the form e2πim1/h,...,e2πimr/h with 1 = m 6 m 6 ··· 6 m = h − 1 the
1 2 r
exponents of (W,S). Furthermore, for any permutation σ of {1,...,r} the elements c and
s ...s are conjugate in W. Hence the exponents do not depend on the enumeration
σ(1) σ(r)
of the simple reflections. Recall that the symmetry m + m = h follows from the
i r+1−i
facts that c has no eigenvalue 1 and that the reflection representation is defined over the
reals.
In this note we will derive uniform expressions for the power sums r mn for any
i=1 i
n ∈ Z . Of course, for n = 0 the sum is r, and for n = 1 the symmetry m +m = h
>0 i r+1−i
P
1
shows that the sum is 1rh. We shall see that
2
r
mn = n!rTd (γ ,...,γ )
i n 1 n
i=1
X
where Td (γ ,...,γ ) denotes the nth Todd polynomial evaluated at γ ,...,γ (for n odd
n 1 n 1 n
Td (γ ,...,γ ) does not depend on γ , as follows from Proposition 3.1). The γ ’s can
n 1 n n i
be chosen to be polynomials in four parameters (details below) with integer coefficients.
This answers Panyushev’s question in [6].
2 Some history and preliminaries
For type A the exponents are just 1,2,...,r and one has Bernoulli’s formula
r
r
1
in = B (r +1)−B (1) (2.1)
n+1 n+1
n+1
i=1
X (cid:0) (cid:1)
where B (x) is the (n+1)st Bernoulli polynomial, defined by the expansion
n+1
∞ tn tetx
B (x) = .
n n! et −1
n=0
X
For general types uniform formulae for the power sums up to third power are listed in
the epilogue of [8]. Besides the Coxeter number h and the rank r they depend (for the
squares and the cubes) on a further parameter γ which is defined for the crystallographic
types with crystallographic root system Φ ( = Φ ∪Φ a decomposition into the sets of
+ −
positive and negative roots) by the formula (see [2, Ch. VI, § 1, no. 12])
hλ|ϕihµ|ϕi
= γhλ|µi (λ,µ ∈ span Φ) (2.2)
hϕ|ϕi2 R
ϕ∈Φ
X
where h | i denotes the Killing form on span Φ, which is the W-invariant (symmetric)
R
bilinear form characterized by
hλ|µi = hλ|ϕihµ|ϕi (λ,µ ∈ span Φ).
R
ϕ∈Φ
X
It turns out that γ = kgg∨ where k = hθ|θi/hθ |θ i ∈ {1,2,3} with θ,θ ∈ Φ the highest
s s s +
resp. highest short roots, and g = 1/hθ|θi ∈ Z is the dual Coxeter number of Φ whereas
>0
g∨ is the dual Coxeter number of the dual root system Φ∨. So γ = h2 if Φ is simply-laced.
For the noncrystallographic types γ = 2m2 −5m+6 for I (m) (the formula is also valid
2
for the crystallographic types, where m = 3,4,6); γ = 124 for type H ; and γ = 1116 for
3
type H .
4
The formulae from [8] read as follows:
r if n = 0,
r 1rh if n = 1,
mn = 2 (2.3)
i 1r(h2 +γ −h) if n = 2,
Xi=1 6
1rh(γ −h) if n = 3.
4
2
Remark 2.1 The power sum for the fourth powers is not of the form r times a function
depending only on h and γ, as a computation for the types A and D shows.
h−1 (h+2)/2
Panyushev recently gave the universal formula [6, Proposition 3.1]
1
ht(ϕ)2 = r(h+1)γ (2.4)
12
ϕX∈Φ+
for the sum of the heights squares of all positive roots. He then suspects [6, Remark 3.4]
that for the sum of the heights of all positive roots there is no similar formula in the
general case; however, for simply-laced root systems he mentions
1
ht(ϕ) = r(h2 +h) (2.5)
6
ϕX∈Φ+
and asks for which values of n there is a nice closed expression for ht(ϕ)n. Our
ϕ∈Φ+
result shows that there are universal formulae for all n ∈ Z . In fact, let (k ,...,k )
>0 1 h−1
P
be the partition dual to (m ,...,m ); then it is well-known (see, e.g., [4, Section 3.20])
r 1
that there are exactly k roots of height j in Φ . Hence
j +
r
ht(ϕ)n = 1n +2n +···+mn . (2.6)
i
ϕX∈Φ+ Xi=1(cid:0) (cid:1)
In particular, using (2.3) we recover (2.4) and have
r m2 +m 1
ht(ϕ) = i i = r h2 +γ +2h (2.7)
2 12
ϕX∈Φ+ Xi=1 (cid:0) (cid:1)
which generalizes (2.5) to all types.
Alternatively, using the symmetry m +m = h we can write as in [3, Proposition 2.1]
i r+1−i
r r r
h2 m −3h m2 +2 m3 = 0. (2.8)
i i i
i=1 i=1 i=1
X X X
Hence
ht(ϕ)2 (2=.6) r mi(mi +1)(2mi +1) = r mi3 + r mi2 + r mi
6 3 2 6
ϕX∈Φ+ Xi=1 Xi=1 Xi=1 Xi=1
(2=.8) −h2 r mi +h r mi2 + r mi2 + r mi
6 2 2 6
i=1 i=1 i=1 i=1
X X X X
r m (m +1) h+1 h2 −1 r
i i
= (h+1) − + m
i
2 2 6
Xi=1 (cid:16) (cid:17)Xi=1
(2.6) rh(h+2) rh
= (h+1) ht(ϕ)−(h+1)
=
12
ϕX∈Φ+ | {z2}
so that (2.7) is recovered from (2.4).
We shall stick to the exponents rather than the heights in order not to restrict our con-
siderations to the crystallographic types.
3
3 Power sums and Todd polynomials
The observation that (2.3) can be written as
r = 0!rTd if n = 0,
0
r 1rh = 1!rTd (h) if n = 1,
mn = 2 1 (3.1)
i 1r(h2 +γ −h) = 2!rTd (h,γ −h) if n = 2,
Xi=1 6 2
1rh(γ −h) = 3!rTd (h,γ −h,∗) if n = 3,
4 3
where Td = 1, Td (c ) = 1c , Td (c ,c ) = 1 (c2 +c ), and Td (c ,c ,c ) = 1 c c are
0 1 1 2 1 2 1 2 12 1 2 3 1 2 3 24 1 2
Todd polynomials (the general definition will be recalled in the proof of Theorem 3.3),
suggests the ansatz
r
mn = n!rTd (γ ,...,γ ). (3.2)
i n 1 n
i=1
X
From (3.1) and (3.2) we get
γ = h and γ = γ −h (3.3)
1 2
and are looking for solutions γ ,γ ,.... Note that the symmetry m +m = h implies
3 4 i r+1−i
the identities (for a,b ∈ Z )
>0
a r b r
a b
(−1)a−j hj ma+b−j = (−1)b−j hj ma+b−j (3.4)
j i j i
j=0 (cid:18) (cid:19) i=1 j=0 (cid:18) (cid:19) i=1
X X X X
that generalize (2.8), which is (3.4) for {a,b} = {1,2}.
Proposition 3.1 For a,b ∈ Z one has the identity
>0
a
a
(−1)a−j cj(a+b−j)!Td (c ,...,c )
j 1 a+b−j 1 a+b−j
j=0 (cid:18) (cid:19)
X
b
b
= (−1)b−j cj(a+b−j)!Td (c ,...,c ). (3.5)
j 1 a+b−j 1 a+b−j
j=0 (cid:18) (cid:19)
X
Proof. For instance, one verifies the formula (3.5) for a = 0 and all b ∈ Z by using a
>0
generating series and then proceeds by induction on a. (cid:3)
Strictly speaking we don’t need Proposition 3.1. But it is worth noting that it indicates
that we seem to be on the right track when using the ansatz (3.2).
Lemma 3.2 Let m 6 ··· 6 m ∈ Z be such that there are multisets V and V of
1 r >0 + −
positive integers satisfying
r q (1−qv)
qmi = v∈V+ . (3.6)
(1−qv)
i=1 Qv∈V−
X
Then Q
v = r v (3.7)
vY∈V+ vY∈V−
|V | = |V |. (3.8)
+ −
4
Proof. The equality (3.7) is clear from the q → 1 limit in (3.6); (3.8) follows since 1−qv
has exactly one factor 1−q and the polynomial on the left hand side in (3.6) has neither
a zero nor a pole at q = 1. Note also that m = 1 and m > 1 if r > 2. (cid:3)
1 2
Theorem 3.3 Let m 6 ··· 6 m ∈ Z be such that there are multisets V and V of
1 r >0 + −
positive integers satisfying
r q (1−qv)
qmi = v∈V+ . (3.6)
(1−qv)
i=1 Qv∈V−
X
Q
We fix (for simplicity) a positive integer p and define γ ( = 1),γ ,γ ,γ ,... by the gener-
0 1 2 3
ating series
∞ (1−vt) p 1+pt
γ tn = v∈V− . (3.9)
n
(1−vt) 1−pt
n=0 Qv∈V+ r
X
Then for n ∈ Z>0 Q
r
mn = n!rTd (γ ,...,γ ). (3.10)
i n 1 n
i=1
X
Proof. We consider the exponential generating series (withq := et) of bothsides in (3.10)
∞ r mn tn = r emit = r qmi (3=.6) q v∈V+(1−qv) (3.11)
i n! (1−qv)
Xn=0(cid:16)Xi=1 (cid:17) Xi=1 Xi=1 Qv∈V−
∞ tn ∞ Q ∞ x t
n!rTd (γ ,...,γ ) = r Td (γ ,...,γ )tn = r j (3.12)
n 1 n n! n 1 n 1−e−xjt
Xn=0(cid:16) (cid:17) Xn=0 Yj=1
where the last equality incorporates the definition of the Todd polynomials by means
of their generating series in t with coefficients in the elementary symmetric functions in
x ,x ,... so that
1 2
∞ ∞
(1+x t) = γ tn
j n
j=1 n=0
Y X
and hence by (3.9)
(1−pt) (1−vt)p ∞
v∈V+ (1+x t)p = 1
(1+pt) (1−vt)p j
Qv∈V− j=1
Y
(that is, the supersymmetric eleQmentary symmetric functions in −p, −v (p times, for
every v ∈ V ), x (p times), x (p times), ...; −p, v (p times, for every v ∈ V ) all vanish)
+ 1 2 −
so that the formal expansion
−pt 1−e−pt −vt p 1−evt p ∞ x t p
j
= 1
1−ept pt 1−evt −vt 1−e−xjt
(cid:16) (cid:17)(cid:16) (cid:17)vY∈V+(cid:16) (cid:17) vY∈V−(cid:16) (cid:17) (cid:16)Yj=1 (cid:17)
= e−pt
| {z }
or taking pth roots (look at t = 0 to choose the correct branch)
∞ x t 1−evt −vt
j = et .
1−e−xjt −vt 1−evt
Yj=1 vY∈V+(cid:16) (cid:17)vY∈V−(cid:16) (cid:17)
5
Therefore we can write the right hand side in (3.12) as (recall q = et)
∞ x t r v q (1−qv) q (1−qv)
r j = v∈V− · v∈V+ = v∈V+
1−e−xjt v (1−qv) (1−qv)
j=1 Qv∈V+ Qv∈V− Qv∈V−
Y
Q= 1 Q Q
| {z }
where we have used (3.8) |V | = |V | to cancel factors t and then (3.7) to simplify the
+ −
product. Thus the right hand side of (3.12) is identical to the right hand side of (3.11),
which proves (3.10). (cid:3)
Remark 3.4 Instead of the definition (3.9) for γ ,γ ,γ ,γ ,... one can define more gen-
0 1 2 3
erally
∞ γ tn = v∈V−(1−vt) K 1+πkt µk
n
(1−vt) 1−π t
Xn=0 Qv∈V+ kY=1(cid:16) k (cid:17)
with π ,...,π ∈ R and µ ,...,µQ∈ Q satisfying K π µ = 1 (and for general m
1 K 1 K k=1 k k 1
(with qm1 instead of q as first factor in the right hand side of (3.6)) just require that
K π µ = m ). P
k=1 k k 1
P
4 Root system considerations
To apply Theorem 3.3 in the context of root systems we need the following proposition.
Proposition 4.1 Let m 6 ··· 6 m be the exponents of an irreducible (crystallographic
1 r
(and reduced) or noncrystallographic) finite root system (of rank r). Then there are mul-
tisets V and V of positive integers such that
+ −
r q (1−qv)
qmi = v∈V+ . (3.6)
(1−qv)
i=1 Qv∈V−
X
Q
Furthermore, |V | 6 2 if V ∩V = ∅.
± + −
Proof. According to the first note added in proof in [7] I. G. Macdonald was acquainted
with the fact that (3.6) holds for all irreducible finite Coxeter groups.
The classification shows that the following three cases exhaust all possible types.
(1) For the types A , C /B , and types of rank 6 3 the sequence of exponents forms an
r r r
arithmetic progression 1,m ,...,1+(r −1)(m −1) (or just 1 if r = 1). Hence
2 2
r q if r = 1
qmi = q(1−qr(m2−1))
if r > 2
Xi=1 1−qm2−1
so that we can take V = V = ∅if r = 1 and V = {r(m −1)} and V = {m −1}
+ − + 2 − 2
if r > 2.
6
(2) For the types of rank 4 we have
4 q(1−q2(m2−1))(1−q2(h−m2−1))
qmi = q +qm2 +qh−m2 +qh−1 =
(1−qm2−1)(1−qh−m2−1)
i=1
X
so that we can take V = {2(m −1),2(h−m −1)} and V = {m −1,h−m −1}.
+ 2 2 − 2 2
(3) For the simply-laced types (ADE) the root system is the Weyl group orbit of the
highest root: Φ = Wθ. The stabilizer of θ is W , the reflection group generated
⊥θ
by those simple reflections in W that fix θ. The root system is thus isomorphic
as a W-set to W/W . We need the usual length function ℓ : W → Z defined
⊥θ >0
as ℓ(w) = k if w can be written as a product of k but not less than k simple
reflections. If ϕ = wθ is any positive root with w chosen such that ℓ(w) is minimal,
then ht(ϕ) = ht(θ) − ℓ(w) = h − 1 − ℓ(w). Since the reflection along a simple
root ψ maps ψ (of height 1) to −ψ (of height −1), we have similarly the equality
ht(ϕ) = ht(θ) − ℓ(w) − 1 = h − 2 − ℓ(w) if ϕ = wθ is any negative root with w
chosen such that ℓ(w) is minimal. So we have
qℓ(w) = qh−1−ht(ϕ) +qh−2+ht(ϕ)
wW⊥θX∈W/W⊥θ ϕX∈Φ+(cid:0) (cid:1)
ℓ(w)minimal
and since 1,...,m , 1,...,m , ..., 1,...,m enumerates ht(ϕ) as ϕ runs over Φ ,
1 2 r +
we can continue
r mi
= qh−1−j +qh−2+j
i=1 j=1
XX(cid:0) (cid:1)
and using the symmetry m +m = h we obtain
i r+1−i
r h−1 r 1−qh
= qmi−1+j = qmi−1 .
1−q
Xi=1 Xj=0 (cid:16)Xi=1 (cid:17)
Ontheother handbytheChevalley-Solomon identity forthePoincar´e series offinite
Coxeter groups (see, e.g., [4, Section 3.15]) we have
r 1−qmi+1 s 1−q
qℓ(w) =
1−q 1−qmei+1
wW⊥θX∈W/W⊥θ (cid:16)Yi=1 (cid:17)(cid:16)Yi=1 (cid:17)
ℓ(w)minimal
where m ,...,m liststheexponents ofalltheirreducible components ofW . Since
1 s ⊥θ
m +1 = h we finally get
r
e e
r q r−1(1−qmi+1)
qmi = i=1
(1−q)r−s−1 s (1−qmei+1)
i=1 Qi=1
X
and the following table finishes the proof. (QWe have left out the types A which
r
were already dealt with in case (1).)
7
type W exponents+1 type W⊥θ exponents+1 V+ V−
Dr (r >4) 2,4,...,2r−2,r A1+Dr−2 2,2,4,...,2r−6,r−2 {r,2r−4} {2,r−2}
E6 2,5,6,8,9,12 A5 2,3,4,5,6 {8,9} {3,4}
E7 2,6,8,10,12,14,18 D6 2,4,6,8,10,6 {12,14} {4,6}
E8 2,8,12,14,18,20,24,30 E7 2,6,8,10,12,14,18 {20,24} {6,10}
Multisets are needed for type D . (cid:3)
4
Note that for r > 2 (3.6) implies that m − 1 ∈ V . Furthermore, for all the crystallo-
2 −
graphic types except A and G , m −1 = d is the largest coefficient of the highest root
1 2 2
(when written as a linear combination of the simple roots). This observation extends to
the noncrystallographic types H and H if we define d = 4 and d = 10, respectively, as
3 4
suggested by the following folding procedure, D H and E H .
6 3 8 4
1 2 2 2 3 4 5
r r r r r r r
D (cid:8)(cid:8) E (cid:8)(cid:8)
6 r r(cid:8) r 8 r r (cid:8)r r
1 2 1 2 4 6 3
2 4 3 4 7 10 8
H r r r H r r r r
3 4
5 5
For I (m) we have m −1 = m−2, but the folding procedure gives d = m . In fact, for
2 2 2
m = 2k + 1 odd A I (2k + 1) with d = k (and the other coefficient is k, too). For
2k 2
(cid:4) (cid:5)
m = 2k even D I (2k) with d = k (and the other coefficient is k −1); alternatively
k+1 2
we can fold A I (2k) and also E I (12), E I (18), and E I (30).
2k−1 2 6 2 7 2 8 2
For A , C , B , I (m), and H one can append the same element(s) to both V and V to
r r r 2 3 + −
make all the above multisets V and V have cardinality 2.
+ −
The following proposition gives a uniform description of multisets V = {A,B} and
+
V = {α,β} satisfying (3.6) in terms of three parameters: the Coxeter number h, the
−
coefficient d, and ν := the number of times d occurs among the marks in the extended
Dynkin diagram minus 1, and extended to the noncrystallographic types as displayed in
thefollowingtable. Thetablealsoshowsthevaluesofγ (see(2.2)andthetextafterwards).
Someparametersβ (andfortypeA alsoα)areirrelevant andareleftunspecified. Clearly,
1
one can interchange A ↔ B and also α ↔ β.
type r h γ d A,B α,β ν
A 1 2 4 1 α,β α,β 1
1
A (r > 2) r r+1 (r +1)2 1 r,β 1,β r
r
C /B (r > 2) r 2r 4r2 +2r −2 2 2r,β 2,β r −2
r r
D (r > 4) r 2r −2 (2r−2)2 2 r,2(r −2) 2,r −2 r −4
r
E 6 12 144 3 8,9 3,4 0
6
E 7 18 324 4 12,14 4,6 0
7
E 8 30 900 6 20,24 6,10 0
8
F 4 12 162 4 8,12 4,6 0
4
G = I (6) 2 6 48 3 8,β 4,β 0
2 2
H = I (5) 2 5 31 2 6,β 3,β 1
2 2
H 3 10 124 4 12,β 4,β 0
3
H 4 30 1116 10 20,36 10,18 0
4
I (2k +1) (k > 3) 2 2k +1 8k2 −2k +3 k 4k −2,β 2k −1,β 1
2
I (2k) (k > 4) 2 2k 8k2 −10k +6 k 4k −4,β 2k −2,β 0
2
8
redefined parameters d and ν for I (2k +1) (k > 2)
2
type r h γ d A,B α,β ν
I (m) (m > 4) 2 m 2m2 −5m+6 m 2m−4,β m−2,β 0
2 2
The table shows that in the cases where β has a well-defined value (and α = m −1), this
2
value is m −1 except for D (r > 7), where β = m −1. With the redefinition of
3 r ⌊(r+1)/2⌋
d and ν for the types I (2k +1) (k > 2) the formula h = d(r +2+ν) is true in general,
2 2
and it is also true for H = I (5) with the original parameters d = 2 and ν = 1.
2 2
Proposition 4.2 The equality (3.6) in Proposition 4.1 holds if the multisets V are given
±
as
V = {d, 2d−2+ν} and
−
V = {4d−4+dν, h−d−(d−1)ν}
+
with d = m and ν = 0 for I (m) (m > 4); and for H = I (5) the original values d = 2
2 2 2 2
and ν = 1 also work.
The choice in Proposition 4.2 of the irrelevant parameters is thus α = β = 1 for type A
1
and as shown in the following table.
type A C /B G H with d = 2, ν = 1 H I (m) with d = m, ν = 0
r r r 2 2 3 2 2
β r r 3 2 6 m
2
Proof. Let us first look at those exceptional types for which d | h (including I (m)
2
(m > 5)). Here we have ν = 0 and the (multi)set of exponents is
h h
m ,...,m = 1+jd 0 6 j 6 −2 ∪ 2d−1+jd 0 6 j 6 −2
1 r
d d
n (cid:12) o n (cid:12) o
(see [3,(cid:8)Theorem 3.(cid:9)2 (i)] adding(cid:12)H and I (m)) so that (cid:12)
(cid:12) 4 2 (cid:12)
r hd−2 hd−2 q(1−q4d−4)(1−qh−d)
qmi = q1+jd +q2d−1+jd = q(1+q2d−2) qjd =
(1−qd)(1−q2d−2)
i=1 j=0 j=0
X X(cid:0) (cid:1) X
in agreement with the expressions for V (with ν = 0).
±
For the remaining types we use the following table.
type h d ν 4d−4+dν, h−d−(d−1)ν d, 2d−2+ν
A (r > 1) r+1 1 r r,r 1,r
r
C /B (r > 2) 2r 2 r −2 2r,r 2,r
r r
D (r > 4) 2r −2 2 r −4 2r −4,r 2,r−2
r
E 18 4 0 12,14 4,6
7
H 5 2 1 6,2 2,3
2
H 10 4 0 12,6 4,6
3
This is in agreement with the table before Proposition 4.2. (cid:3)
9
Remark 4.3 For the DE types one has V = a, b and V = b, ra , where the pa-
− 2 2 + 4
rameters a and b are as in Kostant’s article [5]. Note also that for those types a = d and
(cid:8) (cid:9) (cid:8) (cid:9) 2
b = h+2 −d. We can already look ahead and use (5.4) to obtain h = dr −4d+6; from
2 2
(5.5) and h2 = γ (still for the DE types) and using the equality h = dr −4d+6 we get
d(h−2r −6d+26) = 24.
5 Synthesis and further computations
Proposition 4.1 shows that Theorem 3.3 can be applied in the context of root systems
with V = {A,B} and V = {α,β} as in the table before Proposition 4.2.
+ −
Define γ ,γ ,γ ,γ ,... (depending on a parameter p) by the series expansion
0 1 2 3
∞
(1−αt)(1−βt) p 1+pt
γ tn = . (5.1)
n
(1−At)(1−Bt) 1−pt
n=0 r
X
The series expansions
∞ n
(1−αt)(1−βt)
= 1−(α+β)t+αβt2 AjBn−j tn (5.2)
(1−At)(1−Bt)
(cid:0) (cid:1)Xn=0(cid:16)Xj=0 (cid:17)
and
p 1+pt ∞ 1 ∞ −1 ∞
= p (pt)j p (−pt)k =: p tn (5.3)
n
1−pt j k
r j=0 (cid:18) (cid:19) ! k=0(cid:18) (cid:19) ! n=0
X X X
2p2 +4 4p2 +2 6p4 +20p2 +4
= 1+2t+2t2 + t3 + t4 + t5 +···
3 3 15
specializing for p = 1 and p = 2
∞
1+t
= 1+2 tn
1−t
n=1
X
∞
1+2t 2n
= (1+2t)t2n = 1+2t+2t2 +4t3 +6t4 +12t5 +···
1−2t n
r n=0(cid:18) (cid:19)
X
can be used to write down an explicit formula for γ defined in (5.1).
n
Note that the series expansion of (1+pt)/(1−pt) 1/p has integer coefficients if p = 2k
with k ∈ Z . In fact, for f(t) = 1+ ∞ a tn we let
>0 (cid:0) n=1 n (cid:1)
P ∞
Tf(t) := f(2t) = 1+ b tn.
n
n=1
p X
A comparison of coefficients shows that
n−1
1
b = 2n−1a − b b ,
n n j n−j
2
j=1
X
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