Higher Apery-like numbers arising from special values of the spectral zeta function for the non-commutative harmonic oscillator

0.23 MB·English

by  Kazufumi Kimoto

Higher Ap´ery-like numbers arising from
9 special values of the spectral zeta function
0
0 for the non-commutative harmonic oscillator
2

n
a Kazufumi Kimoto
J

0 January 20, 2009
2

]
T Abstract
N AgeneralizationoftheAp´ery-likenumbers,whichisusedtodescribethespecialvaluesζQ(2)andζQ(3)of
. thespectralzetafunctionforthenon-commutativeharmonicoscillator, areintroducedandstudied. Infact,
h
we give a recurrence relation for them, which shows a ladder structure among them. Further, we consider
t
a the‘rationalpart’ofthehigherAp´ery-likenumbers. Wediscussseveralkindsofcongruencerelationsamong
m them, which are regarded as an analogue of theones among Ap´ery numbers.
[

2 1 Introduction
v
8
The non-commutative harmonic oscillator is the system of differential equations defined by the operator
5
6 α 0 1 d2 1 0 1 d 1
.0 Q=Qα,β := 0 β −2dx2 + 2x2 + 1 −0 xdx + 2 , (1.1)
1 (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
0 whereαandβ arerealparameters. Inthispaper,wealwaysassumethatα>0,β >0andαβ >1. Underthese
9
conditions,onecanshowthatthe operatorQdefinesanunbounded, positive,self-adjointoperatoronthespace
0
: L2(R;C2) of C2-valued square integrable functions which has only a discrete spectrum, and the multiplicities
v
m(λ) of the eigenvalues λ Spec(Q) are uniformly bounded [22]. Hence, in this case, it is meaningful to define
i ∈
X its spectral zeta function ζ (s) = TrQ−s = m(λ)λ−s. This series converges absolutely if s > 1,
Q λ∈Spec(Q) ℜ
r andhencedefinesaholomorphicfunctiononthehalfplane s>1. Further,ζ (s)ismeromorphicallycontinued
a P ℜ Q
to the whole complex plane C which has ‘trivial zeros’ at s=0, 2, 4,... (see [7], [21]).
− −
The aim of this paper is to study the higher Ap´ery-like numbers J (n) defined by
k
(1 x4)(1 x4 x4) ndx dx dx
J (n):=2k − 1 − 2··· k 1 2··· k
k (1 x2 x2)2 1 x2 x2
Z[0,1]k(cid:18) − 1··· k (cid:19) − 1··· k
fork 2 andn 0,whichareageneralizationofthe Ap´ery-likenumbersJ (n)andJ (n)studiedin[11]. This
2 3
≥ ≥
object arises from the special values of the spectral zeta function ζ (s): In [8], the generating functions of the
Q
numbers J (n) and J (n) are used to describe the special values ζ (2) and ζ (3) of the spectral zeta function
2 3 Q Q
ζ (s). Similarly, the higher Ap´ery-like numbers J (n) are closely related to the special values ζ (k) (see 3.3).
Q k Q
§
We first show that J (n) satisfy three-term (inhomogeneous) recurrence relations, which is translated to
k
(inhomogeneous) singly confluent Heun differential equations for their generating functions. The point is that
theserelationsordifferentialequationsareconnectingJ (n)’sandJ (n)’s. Thisfactimplies thattherecould
k k−2
be a certainrelationbetweenζ (k) andζ (k 2). It wouldbe veryinteresting ifone canutilize these relations
Q Q
−
to understand a modular interpretation of ζ (4),ζ (6),... based on that of ζ (2) (see [12]). We also notice
Q Q Q
thatthese recurrencerelationsquite resembleto thoseforAp´ery numbers usedtoprovethe irrationalityofζ(2)
and ζ(3) (see [27]), and this is why we call J (n) the (higher) Ap´ery-like numbers.
k
1
2 K. Kimoto
By a suitable change of variable in the differential equation, we also obtain another kind of recurrence
relations, which allow us to define the rational part of the higher Ap´ery-like numbers (or normalized higher
Ap´ery-like numbers) J˜ (n). In fact, each J (n) is a linear combination of the Riemann zeta values ζ(k),ζ(k
k k
−
2),... and the coefficients are given by J˜ (n)’s. Since there are various kind of congruence relations satisfied
m
by Ap´ery numbers (see, e.g. [4], [5], [1]), it would be natural and interesting to find an analogue for our higher
Ap´ery-like numbers. Actually, we give several congruence relations among J˜(n) and J˜(n) in [12]. We add
2 3
such congruence relations among J˜ (n), and give some conjectural congruences.
k
2 Ap´ery numbers for ζ(2) and ζ(3)
As a quick reference for the readers, we recall the definitions and several properties on the original Ap´ery
numbers.
2.1 Ap´ery numbers for ζ(2)
Ap´ery numbers for ζ(2) are given by
n n 2 n+k n n 2 n+k n ( 1)m−1 k ( 1)n+m−1
A (n)= , B (n)= 2 − + − .
2 k=0(cid:18)k(cid:19) (cid:18) k (cid:19) 2 k=0(cid:18)k(cid:19) (cid:18) k (cid:19) m=1 m2 m=1m2 mn n+mm !
X X X X
(cid:0) (cid:1)(cid:0) (cid:1)
These numbers satisfy a recurrence relation of the same form
n2u(n) (11n2 11n+3)u(n 1) (n 1)2u(n 2)=0 (n 2) (2.1)
− − − − − − ≥
with initial conditions A (0) = 1,A (1) = 3 and B (0) = 0,B (1) = 5. The ratio B (n)/A (n) converges to
2 2 2 2 2 2
ζ(2), and this convergence is rapid enough to prove the irrationality of ζ(2). Consider the generating functions
∞ ∞
(t)= A (n)tn, (t)= B (n)tn, (t)= (t)ζ(2) (t).
2 2 2 2 2 2 2
A B R A −B
n=0 n=0
X X
It is proved that
L (t)=0, L (t)= 5, L (t)=5,
2 2 2 2 2 2
A B − R
where L is a differential operator given by
2
d2 d
L =t(t2+11t 1) +(3t2+22t 1) +(t+3).
2 − dt2 − dt
The function (t) is also expressed as follows:
2
R
1 1 dxdy
(t)= .
2
R 1 xy+txy(1 x)(1 y)
Z0 Z0 − − −
The family Q2 : 1 xy +txy(1 x)(1 y) = 0 of algebraic curves, which comes from the denominator of
t − − −
the integrand, is birationally equivalent to the universal family C2 of elliptic curves having rational 5-torsion.
t
Moreover, the differential equation L (t) = 0 is regarded as a Picard-Fuchs equation for this family, and
2 2
A
(t) is interpreted as a period of C2 (see [2]).
A2 t
Higher Ap´ery-like numbers arising from NCHO 3
2.2 Ap´ery numbers for ζ(3)
Ap´ery numbers for ζ(3) are given by
n n 2 n+k 2 n n 2 n+k 2 n 1 k ( 1)m−1
A (n)= , B (n)= + −
3 k=0(cid:18)k(cid:19) (cid:18) k (cid:19) 3 k=0(cid:18)k(cid:19) (cid:18) k (cid:19) m=1m3 m=12m3 mn n+mm !
X X X X
(cid:0) (cid:1)(cid:0) (cid:1)
These numbers satisfy a recurrence relation of the same form
n3u(n) (34n3 51n2+27n 5)u(n 1)+(n 1)3u(n 2)=0 (n 2)
− − − − − − ≥
with initial conditions A (0) = 1,A (1) = 5 and B (0) = 0,B (1) = 6. The ratio B (n)/A (n) converges to
3 3 3 3 3 3
ζ(3) rapidly enough to allow us to prove the irrationality of ζ(3). Consider the generating functions
∞ ∞
(t)= A (n)tn, (t)= B (n)tn, (t)= (t)ζ(3) (t).
3 3 3 3 3 3 3
A B R A −B
n=0 n=0
X X
It is proved that
L (t)=0, L (t)=5, L (t)= 5,
3 3 3 3 3 3
A B R −
where L is a differential operator given by
3
d3 d2 d
L =t2(t2 34t2+1) +t(6t2 153t+3) +(7t2 112t+1) +(t 5).
3 − dt3 − dt2 − dt −
The function (t) is also expressed as follows:
3
R
1 1 1 dxdydz
(t)= .
3
R 1 (1 xy)z txyz(1 x)(1 y)(1 z)
Z0 Z0 Z0 − − − − − −
The family Q3 : 1 (1 xy)z txyz(1 x)(1 y)(1 z) = 0 of algebraic surfaces coming from the denomi-
t − − − − − −
nator of the integrand is birationally equivalent to a certain family C3 of K3 surfaces with Picard number 19.
t
Furthermore, the differential equation L (t)=0 is regarded as a Picard-Fuchs equation for this family, and
3 3
A
(t) is interpreted as a period of C3 (see [3]).
A3 t
2.3 Congruence relations for Ap´ery numbers
Ap´ery numbers A (n) and A (n) have various kind of congruence properties. Here we pick up severalof them,
2 3
for which we will discuss an Ap´ery-like analogue later.
Proposition2.1. Letpbeaprimeandn=n +n p+ +n pk bethep-aryexpansionofn Z (0 n <p).
0 1 k ≥0 j
··· ∈ ≤
Then it holds that
k k
A (n) A (n ) (mod p), A (n) A (n ) (mod p).
2 2 j 3 3 j
≡ ≡
j=0 j=0
Y Y
Proposition 2.2 ([4, Theorems 1 and 2]). For all odd prime p, it holds that
A (mpr 1) A (mpr−1 1) (mod pr),
2 3
− ≡ −
A (mpr 1) A (mpr−1 1) (mod pr)
3 3
− ≡ −
for any m,r Z . These congruence relations hold modulo p3r if p 5 (known and referred to as a super-
>0
∈ ≥
congruence).
4 K. Kimoto
We denote by η(τ) the Dedekind eta function
∞
η(τ)=q1/24 (1 qn), q =e2πiτ ( τ >0). (2.2)
− ℑ
n=1
Y
Proposition 2.3 ([28, Theorem 13.1]). For any odd prime p and any m,r Z with m odd, it holds that
>0
∈
A (mpr−1) λ A (mpr−1−1)+( 1)(p−1)/2p2A (mpr−2−1) 0 (mod pr). (2.3)
2 2 − p 2 2 − 2 2 ≡
Here λ is defined by
n
∞ ∞
λ qn =η(4τ)6 =q (1 q4n)6.
n
−
n=1 n=1
X Y
Proposition 2.4 ([5, Theorem 4]). For any odd prime p and any m,r Z with m odd, it holds that
>0
∈
A (mpr−1) γ A (mpr−1−1)+p3A (mpr−2−1) 0 (mod pr). (2.4)
3 2 − p 3 2 3 2 ≡
Here γ is defined by
n
∞ ∞
γ qn =η(2τ)4η(4τ)4 =q (1 q2n)4(1 q4n)4.
n
− −
n=1 n=1
X Y
3 Ap´ery-like numbers for ζ (2) and ζ (3)
Q Q
We introduce the Ap´ery like numbers J (n) and J (n), and give a brief explanation on their basic properties
2 3
and the connection between the special values ζ (2),ζ (3) of the spectral zeta function ζ (s).
Q Q Q
3.1 Definition
We define the Ap´ery-like numbers for ζ (2) and ζ (3) by
Q Q
1 1 (1 x4)(1 x4) n dx dx
J (n):=4 − 1 − 2 1 2 ,
2 (1 x2x2)2 1 x2x2
Z0 Z0 (cid:18) − 1 2 (cid:19) − 1 2
1 1 1 (1 x4)(1 x4x4) ndx dx dx
J (n):=8 − 1 − 2 3 1 2 3.
3 (1 x2x2x2)2 1 x2x2x2
Z0 Z0 Z0 (cid:18) − 1 2 3 (cid:19) − 1 2 3
The sequences J (n) and J (n) satisfy the recurrence formula (Propositions 4.11 and 6.4 in [8])
2 3
{ } { }
4n2J (n) (8n2 8n+3)J (n 1)+4(n 1)2J (n 2)=0, (3.1)
2 2 2
− − − − −
2n(n 1)!
4n2J (n) (8n2 8n+3)J (n 1)+4(n 1)2J (n 2)= − (3.2)
3 3 3
− − − − − (2n 1)!!
−
with the initial conditions
9 21 1
J (0)=3ζ(2), J (1)= ζ(2); J (0)=7ζ(3), J (1)= ζ(3)+ .
2 2 3 3
4 4 2
It is notable that the left-hand sides of these relations have the same shape. Since the relations (3.1),(3.2) and
the one (2.1) for A (n) have quite close shapes, we call the numbers J (n) and J (n) the Ap´ery-like numbers.
2 2 3
Higher Ap´ery-like numbers arising from NCHO 5
3.2 Generating functions and their differential equations
The generating functions for J (n) and J (n) are defined by
2 3
∞ 1 1 1 x2x2
w (t):= J (n)tn =4 − 1 2 dx dx , (3.3)
2 2 (1 x2x2)2 t(1 x4)(1 x4) 1 2
n=0 Z0 Z0 − 1 2 − − 1 − 2
X
∞ 1 1 1 1 x2x2x2
w (t):= J (n)tn =8 − 1 2 3 dx dx dx . (3.4)
3 3 (1 x2x2x2)2 t(1 x4)(1 x4x4) 1 2 3
n=0 Z0 Z0 Z0 − 1 2 3 − − 1 − 2 3
X
By the recurrence relations (3.1) and (3.2), we get the differential equations
w (t)=0, (3.5)
H 2
D
1 3
w (t)= F 1,1; ;t , (3.6)
DH 3 22 1 2
(cid:18) (cid:19)
where denotes the singly confluent Heun differential operator given by
H
D
d2 d 3
=t(1 t)2 +(1 3t)(1 t) +t . (3.7)
DH − dt2 − − dt − 4
(3.5) is solved in [18] as
3ζ(2) 1 1 t
w (t)= F , ;1; .
2 2 1
1 t 2 2 t 1
− (cid:18) − (cid:19)
Here F (a,b;c;z) is the Gaussian hypergeometric function. Now it is immediate that
2 1
n 1 2 n
J (n)=3ζ(2) ( 1)j −2 . (3.8)
2
− j j
j=0 (cid:18) (cid:19)(cid:18) (cid:19)
X
Similarly, (3.6) is solved in [11] as
7ζ(3) 1 1 t ∞ n 1 2 n k−1 1 1 −2
w (t)= F , ;1; 2 ( 1)k −2 −2 tn.
3 1 t2 1 2 2 t 1 −  − k k (2j+1)3 j 
− (cid:18) − (cid:19) n=0 k=0 (cid:18) (cid:19)(cid:18) (cid:19)j=0 (cid:18) (cid:19)
X X X
 
Therefore it follows that
n 1 2 n n 1 2 n j−1 1 1 −2
J (n)=7ζ(3) ( 1)j −2 2 ( 1)j −2 −2 . (3.9)
3 − j j − − j j (2k+1)3 k
j=0 (cid:18) (cid:19)(cid:18) (cid:19) j=0 (cid:18) (cid:19)(cid:18) (cid:19)k=0 (cid:18) (cid:19)
X X X
Remark 3.1. The function
1 1 1 t
W (T)= F , ;1;T2 = (1 t)w (t) T2 =
2 2 1 2
2 2 3ζ(2) − t 1
(cid:18) (cid:19) (cid:18) − (cid:19)
satisfies the differential equation
d2 d
T(T2 1) +(3T2 1) +T W (T)=0,
− dT2 − dT 2
(cid:18) (cid:19)
which can be regarded as a Picard-Fuchs equation for the universal family of elliptic curves having rational
4-torsion [12]. This is an analogue of the result [2] for the Ap´ery numbers for ζ(2) (see also Section 2.1). It is
natural to ask whether there is such a modular interpretation for w (t) (or “W (T)”). We have not obtained
3 3
an answer to this question so far.
♦
6 K. Kimoto
3.3 Connection to the special values of ζ (s)
Q
We also introduce another kind of generating functions for J (n) as
k
∞ 1 1 1 dx dx
g (z):= −2 J (n)zn =4 1 2 ,
2 2
nX=0(cid:18) n (cid:19) Z0 Z0 (1−x21x22)2+z(1−x41)(1−x42)
∞ 1 1 1 p1 dx dx dx
g (z):= −2 J (n)zn =8 1 2 3 .
3 2
nX=0(cid:18) n (cid:19) Z0 Z0 Z0 (1−x21x22x23)2+z(1−x41)(1−x42x43)
The special values of ζ (s) at s=2,3 are given as followps.
Q
Theorem 3.2 (Ichinose-Wakayama [8]). If αβ >2 (i.e. 0<1/(1 αβ)<1), then
−
α+β 2 1 α β 2 1
ζ (2)=2 ζ(2, )+ − g ,
Q 2
2 αβ(αβ−1)! 2 (cid:18)α+β(cid:19) (cid:18)αβ−1(cid:19)!
p α+β 3 1 α β 2 1
ζ (3)=2 ζ(3, )+3 − g ,
Q 3
2 αβ(αβ−1)! 2 (cid:18)α+β(cid:19) (cid:18)αβ−1(cid:19)!
where ζ(s,x)= ∞ (n+x)−s ispthe Hurwitz zeta function.
n=0
Remark 3.3. WePcan determine the functions g (x) and g (x) as follows:
2 3
g (x)=J (0)g (x), g (x)=J (0)g (x)+g (x),
2 2 2 3 3 2 3
where
e e e
1 1 1 x 2 1 3 2
g (x):= F , ;1; = F , ;1; x ,
2 2 1 2 1
√1+x 4 4 1+x 4 4 −
(cid:18) (cid:19) (cid:18) (cid:19)
ge (x):= −2 ∞ ( 1)n −12 3 x nn−1 1 −12 −2.
3 √1+x − n 1+x (2j+1)3 j
n=1 (cid:18) (cid:19) (cid:18) (cid:19) j=0 (cid:18) (cid:19)
X X
e
See [18] and [11] for detailed calculation.
♦
4 Higher Ap´ery-like numbers
Looking at the definition of J (n) and J (n), it is natural to introduce the numbers J (n) by
2 3 k
(1 x4)(1 x4 x4) ndx dx dx
J (n):=2k − 1 − 2··· k 1 2··· k.
k (1 x2 x2)2 1 x2 x2
Z[0,1]k(cid:18) − 1··· k (cid:19) − 1··· k
We refer to J (n) as higher Ap´ery-like numbers. In fact, the generating function
k
∞ 1 dx dx ...dx
g (z):= −2 J (n)zn =2k 1 2 k (4.1)
k k
nX=0(cid:18) n (cid:19) Z[0,1]k (1−x21x22...x2k)2+z(1−x41)(1−x42...x4k)
and its further generalizations are used to describe pthe ‘higher’ special values ζ (k) (k 4) like Theorem 3.2
Q
≥
(see Remark 4.1 below).
It is immediate that J (0)=(2k 1)ζ(k). Further, as we mentioned in [11], the formula
k
−
3⌊k/2⌋−1 1 1 1 ( 1)k
J (1)= ζ k 2m, + − − (4.2)
k 4 4m − 2 2k−1
m=0 (cid:18) (cid:19)
X
Higher Ap´ery-like numbers arising from NCHO 7
holds (see 6.2 for the calculation). It is directly verified that
§
4J (1) 3J (0)=J (1) (k 4).
k k k−2
− ≥
Remark 4.1. We can calculate that
4 2
α+β α β 1
ζ (4)=2 ζ(4,1/2)+4 − g
Q 4
2 αβ(αβ−1)! (cid:18)α+β(cid:19) (cid:18)αβ−1(cid:19)
p α β 2 16dx dx dx dx
1 2 3 4
+2 −
(cid:18)α+β(cid:19) Z[0,1]4 (1−x21x22x23x24)2+γ1(1−x41x42)(1−x43x44)
4
α β p 16dx1dx2dx3dx4
+ − ,
(cid:18)α+β(cid:19) Z[0,1]4 (1−x21x22x23x24)2+γ1(1−x41x42)(1−x43x44)+γ2(1−x41)(1−x42)(1−x43)(1−x44)!
p
where γ =1/(αβ 1) and γ =αβ/(αβ 1)2.
1 2
− − ♦
Similar to the case of J (n) and J (n), the higher Ap´ery-like numbers J (n) also satisfy a three-term
2 3 k
recurrence relation as follows.
Theorem 4.2. The numbers J (n) satisfy the recurrence relations
k
4n2J (n) (8n2 8n+3)J (n 1)+4(n 1)2J (n 2)=J (n) (4.3)
k k k k−2
− − − − −
for n 2 and k 4.
≥ ≥
We give the proof of Theorem 4.2 in 5. It is remarkable that the left-hand side of (4.3) has a common
§
shape with those of (3.1) and (3.2), and (4.3) gives a ‘vertical’ relation among J (n)’s, i.e. it connects J (n)’s
k k
and J (n)’s.
k−2
Example 4.3. First severalterms of J (n) are given by
4
45 9 615 807
J (0)=15ζ(4), J (1)= ζ(4)+ ζ(2), J (2)= ζ(4)+ ζ(2),
4 4 4
4 16 64 1024
2205 3745 129735 1044135
J (3)= ζ(4)+ ζ(2), J (4)= ζ(4)+ ζ(2),...
4 4
256 4096 16384 1048576
We also see that
9
4J (1) 3J (0)= ζ(2)=J (1),
4 4 2
− 4
123
16J (2) 19J (1)+4J (0)= ζ(2)=J (2),
4 4 4 2
− 64
441
36J (3) 51J (2)+16J (1)= ζ(2)=J (3),
4 4 4 2
− 256
25947
64J (4) 99J (3)+36J (2)= ζ(2)=J (4).
4 4 4 2
− 16384
♦
Define another kind of generating function for J (n) by
k
∞ 1 1 1 1 x2 x2
w (t):= J (n)tn =2k − 1··· k dx dx dx . (4.4)
k k ··· (1 x2 x2)2 (1 x4)(1 x4 x4)t 1 2··· k
n=0 Z0 Z0 Z0 − 1··· k − − 1 − 2··· k
X
Theorem 4.2 readily implies the
8 K. Kimoto
Corollary 4.4. The differential equation
w (t) w (0)
k−2 k−2
w (t)= − (4.5)
DH k 4t
holds for k 4. Here is the differential operator given in (3.7).
H
≥ D
Put
( 1)n 1/2
J (n):=0, J (n):= − , J (n):= − J (n) (k 2).
0 1 k k
n n ≥
(cid:18) (cid:19)
By Theorem 4.2, we have
8n3J (n) (1 2n)(8n2 8n+3)J (n 1)+2(n 1)(1 2n)(3 2n)J (n 2)=2nJ (n)
k k k k−2
− − − − − − − −
for k 2 and n 1. Hence, if we put
≥ ≥
d3 d2 d
:=8z2(1+z)2 +24z(1+z)(1+2z) +2(4+27z+27z2) +3(1+2z), (4.6)
DW dz3 dz2 dz
then we have the following (See also [11, Proposition A.3]).
Corollary 4.5. The differential equations
g (z)=0,
W 2
D
2
g (z)= ,
DW 3 −1+z
d g (z) g (0)
k−2 k−2
g (z)=2z − (k 4)
DW k dz z ≥
(cid:18) (cid:19)
hold.
5 Proof of Theorem 4.2
5.1 Setting the stage
Assume k 2. We notice that
≥
∞ ∞ ∞e−(t1+···+tk)/2(1 e−2t1)n(1 e−2(t2+···+tk))n
J (n)= − − dt dt
k Z0 Z0 ···Z0 (1−e−(t1+···+tk))2n+1 1··· k
∞ e−u/2 u tk−2
= du (1 e−2t)n(1 e−2u+2t)ndt
(1 e−u)2n+1 (k 2)! − −
Z0 − Z0 −
for each n 0. Let us introduce
≥
u tk−2
I(k) =I(k) (u):= (1 e−2t)n(1 e−2u+2t)mdt
n,m n,m (k 2)! − −
Z0 −
for n,m 0. We also put
≥
1 1
I(k) (u):= (I(k) (u)+I(k) (u)), I(k) (u):= (I(k) (u) I(k) (u)).
n,m 2 n,m m,n n,m 2 n,m − m,n
I(k) (u) is symmetric in n and m if k =2 so that I(2) (u)e=0, but I(k) (u)=0 in general.
n,m n,m n,m
6
e e
Higher Ap´ery-like numbers arising from NCHO 9
It is convenient to set I(k) (u)=0 when k<2. We see that
n,m
1 ∞ enu
J (n)= I(k)(u)du.
k 22n+1 (sinhu)2n+1 n,n
Z0 2
Thus we also set J (n) = 0 if k < 2. Under these convention, the following discussion for J (n) is reduced to
k k
the one given by Ichinose and Wakayama [8] when k =2,3.
For later use, we define
a(k)(u):=I(k)(u)=I(k)(u) (n 0),
n n,n n,n ≥
1
b(k)(u):= I(k) (u)+I(k) (u) =I(k) (n 1),
n 2 n,n−1 n−1,n n,n−1 ≥
1(cid:16) (cid:17)
˜b(k)(u):= I(k) (u) I(k) (u) =I(k) (n 1),
n 2 n,n−1 − n−1,n n,n−1 ≥
(cid:16) (cid:17) (k)(u)
(k)(u):=enua(k)(u), (k)(u):= Ane (n 0),
An n Bn (sinhu)2n+1 ≥
2
so that
1 ∞
J (n)= (k)(u)du.
k 22n+1 Bn
Z0
5.2 Recurrence formulas for I(k) (u)
n,m
Integration by parts implies
u d tk−2
I(k−1) = (1 e−2t)n(1 e−2u+2t)mdt
n,m dt(k 2)! − −
Z0 (cid:18) − (cid:19)
u tk−2 d
= (1 e−2t)n (1 e−2u+2t)mdt (5.1)
− (k 2)! dt − −
Z0 − (cid:18) (cid:19)
u tk−2 d
(1 e−2t)n (1 e−2u+2t)m dt.
− (k 2)! − dt −
Z0 − (cid:18) (cid:19)
when n,m 1. Since
≥
d
(1 e−2t)n =2ne−2t(1 e−2t)n−1
dt − −
=2n (1 e−2t)n−1 (1 e−2t)n ,
− − −
d
(1 e−2u+2t)m = 2(cid:0)me−2u+2t(1 e−2u+2t)m−1 (cid:1)
dt − − −
= 2m (1 e−2u+2t)m−1 (1 e−2u+2t)m
− − − −
for n,m 1, we obtain the (cid:0) (cid:1)
≥
Lemma 5.1. The following three relations hold:
1
I(k−1) =(n m)I(k) nI(k) +mI(k) (n,m 1), (5.2)
2 n,m − n,m− n−1,m n,m−1 ≥
1
nI(k) (2n 1)I(k) +(n 1)I(k) me−2uI(k) = I(k−1) I(k−1) (n 2,m 1), (5.3)
n,m− − n−1,m − n−2,m− n−1,m−1 2 n,m − n−1,m ≥ ≥
(cid:16) (cid:17)
1
mI(k) (2m 1)I(k) +(m 1)I(k) ne−2uI(k) = I(k−1) I(k−1) (n 1,m 2). (5.4)
n,m− − n,m−1 − n,m−2− n−1,m−1 2 n,m−1− n,m ≥ ≥
(cid:16) (cid:17)
10 K. Kimoto
Plugging (5.2) into (5.3), we get
I(k) (I(k) +I(k) )+(1 e−2u)I(k) =0 (n 1,m 1), (5.5)
n,m− n−1,m n,m−1 − n−1,m−1 ≥ ≥
which is a generalization of (4.14) in [8]. In particular, if we let n=m in (5.5), then we have
I(k) 2I(k) +(1 e−2u)I(k) =0. (5.6)
n,n− n,n−1 − n−1,n−1
Letting m=n 1 (or n=m 1 and exchanging m by n) in (5.5), we also have another specialization
− −
I(k) (I(k) +I(k) )+(1 e−2u)I(k) =0 (n 2),
n,n−1− n−1,n−1 n,n−2 − n−1,n−2 ≥
I(k) (I(k) +I(k) )+(1 e−2u)I(k) =0 (n 2).
n−1,n− n−2,n n−1,n−1 − n−2,n−1 ≥
Adding these equations, we get
I(k) =b(k)(u) a(k) (u)+(1 e−2u)b(k) (u) (n 2). (5.7)
n,n−2 n − n−1 − n−1 ≥
By specializing m=n in (5.3) and (5.4), we have
1
nI(k) (2n 1)I(k) +(n 1)I(k) ne−2uI(k) = (I(k−1) I(k−1)), (5.8)
n,n− − n−1,n − n−2,n− n−1,n−1 2 n,n − n−1,n
1
nI(k) (2n 1)I(k) +(n 1)I(k) ne−2uI(k) = (I(k−1) I(k−1)) (5.9)
n,n− − n,n−1 − n,n−2− n−1,n−1 2 n,n−1− n,n
for n 2. Similarly, specializing m=n 1 in (5.3) and n=m 1 in (5.4) (and exchanging m by n), we have
≥ − −
1
nI(k) (2n 1)I(k) +(n 1)I(k) (n 1)e−2uI(k) = (I(k−1) I(k−1) ),
n,n−1− − n−1,n−1 − n−2,n−1− − n−1,n−2 2 n,n−1− n−1,n−1
1
nI(k) (2n 1)I(k) +(n 1)I(k) (n 1)e−2uI(k) = (I(k−1) I(k−1))
n−1,n− − n−1,n−1 − n−1,n−2− − n−2,n−1 2 n−1,n−1− n−1,n
for n 2. Adding each pair of relations, we obtain
≥
2nI(k) 2(2n 1)I(k) +2(n 1)I(k) 2ne−2uI(k) =I(k−1), (5.10)
n,n− − n,n−1 − n,n−2− n−1,n−1 n,n−1
2nI(k) 2(2n 1)I(k) +2(n 1)(1 e−2u)I(k) =I(k−1). (5.11)
n,n−1− − n−1,n−1 − − n−1,n−2 en,n−1
The formulas (5.6), (5.10) and (5.11) are rewritten as follows. e
Lemma 5.2. The equations
a(k)(u)+(1 e−2u)a(k) (u)=2b(k)(u), (5.12)
n − n−1 n
1
na(k)(u) (2n 1)b(k)(u)+(n 1)I(k) ne−2ua(k) (u)= ˜b(k−1)(u), (5.13)
n − − n − n,n−2− n−1 2 n
1
nb(k)(u) (2n 1)a(k) (u)+(n 1)(1 e−2u)b(k) (u)= ˜b(k−1)(u) (5.14)
n − − n−1 − − n−1 2 n
hold.
As a corollary, we also get
Lemma 5.3. The equation
na(k)(u) (2n 1)(1+e−2u)a(k) (u)+(n 1)(1 e−2u)2a(k) (u)=˜b(k−1)(u) (5.15)
n − − n−1 − − n−2 n
holds.