Table Of Content´
On Painleve Related Functions Arising in
Random Matrix Theory
Leonard N. Choup
1
1 Department of Mathematical Sciences
0
2 University Alabama in Huntsville
n Huntsville, AL 35899, USA
a
J email: [email protected]
7
2
January 28, 2011
]
R
P
Abstract
.
h
at In deriving large n probability distribution function of the rightmost eigenvalue from
m the classical Random Matrix Theory Ensembles, one is faced with que question of finding
[ large n asymptotic of certain coupled set of functions. This paper presents some of these
functions in a new light.
1
v
8
1 1 Introduction
2
5
1. In the study of Edgeworth type expansions for the limiting distribution of the right-
0 most eigenvalue from Gaussian Random Matrix Ensembles, we run into finding large
1
n expansions of many key functions. For the Tracy-Widom distribution derivation,
1
: one needs the large n limits of these functions, and they can all be express in terms
v
i of the couple pair q and p, where q is the Hastings-McLeod solution to Painleve´ II
X
equation behaving at infinity as the Airy function. The frequency of these functions
r
a in the study of the largest eigenvalue of Gaussian and Laguerre Random Matrix En-
semble points to the necessity of a study of these functions in their own right. We
hope this will shed a light into understanding some derivations related to this aspect
of Random Matrix Theory and related field making use of such functions.
If one try to read through a proof of an expansion relating various asymptotic
functions, it’s easy to get lost in translation. But if the related functions are well
known, the reader will probably have a different experience and therefore a better
understanding of the techniques and tools used for the derivation.
We present in this paper the derivations of those functions arising in the study
of the largest eigenvalue for the Gaussian Ensemble of Random Matrix Theory in a
hope of achieving our goal set above.
Before stating our results, there is a need to define our functions.
1
For a Gaussian ensemble of n n matrices, the probability density that the eigen-
×
values lie in an infinitesimal intervals about the points x < ... < x is given
1 n
by
n
β
P (x , ,x ) = C exp x2 x x β. (1.1)
n,β 1 ··· n nβ −2 j | j − k|
!
1 j<k
X Y
Where β = 1 corresponds to the Gaussian Orthogonal Ensemble (GOE ),β = 2
n
corresponds to the Gaussian Unitary Ensemble (GUE ),and β = 4 for the Gaussian
n
Symplectic Ensemble (GSE ).
n
′
(1.1) can also be represented as a determinant involving the variables x s. For the
i
simplest case β = 2, we have
1 1
Pn,2(x1, ,xn) = (det[ϕj−1(xi)])2 = det[Kn,2(xi,xj)]i,j=1,···,n (1.2)
··· n! n!
with
n−1
nϕn(x)ϕn−1(y) ϕn−1(x)ϕn(y)
K (x,y) = ϕ (x)ϕ (y) = − (1.3)
n,2 k k
2 x y
k=0 r −
X
and
1
ϕ (x) = H (x)e−x2/2 with H (x) the Hermite polinomials
k (2kk!√π)1/2 k k
obtained by orthogonalizing the sequence xk, k = 0, ,n 1 with respect to
e−x2 over R. Using this representation, it can b{e shown th·a·t·the−pro}bability distribu-
tion function of the largest eigenvalue λ is given by the Fredholm determinant of
max
the operator with kernel K acting on the set (t ),
n,2
∞
F (t) = P(λ < t) = det(I K ). (1.4)
n,2 max n,2
−
In finding the Edgeworth type expansion of F , one needs large n expansion
n,2
of (1.3) or what it amount to, the large n expansion of ϕ . In [3], we derived the
n
following expression.
Let the rescaling function τ be defined by,
−1 −1
τ(x) = 2(n+c)+2 2n 6x, (1.5)
then p
ϕ (τ(x)) = n61 Ai(X)+ (2c−1) Ai′(X)n−13+ (10c2 10c+ 3)XAi(X)
n
2 − 2
(cid:26) (cid:20)
−2
+ X2Ai′(X) n 3 +O(n−1)Ai(X) (1.6)
20
(cid:21) (cid:27)
and
ϕn−1(τ(x)) = n16 Ai(X)+ (2c+1) Ai′(X)n−31+ (10c2+10c+ 3)XAi(X)
2 2
(cid:26) (cid:20)
2
−2
+ X2Ai′(X) n 3 +O(n−1)Ai(X) (1.7)
20
(cid:21) (cid:27)
Ai being the Airy function. These two functions enable us to obtain the following
expansion of the GUE kernel.
K (τ(X),τ(Y))dτ(X) = τ′K (τ(X),τ(Y))dX = K (X,Y) cAi(X)Ai(Y)n−31+
n,2 n Ai
−
(cid:26)
1
(X +Y)Ai′(X)Ai′(Y) (X2 +XY +Y2)Ai(X)Ai(Y)+
20 −
(cid:20)
−20c2 +3(Ai′(X)Ai(Y)+Ai(X)Ai′(Y)) n−23 +O(n−1)E(X,Y) dX. (1.8)
2
(cid:21) (cid:27)
In deriving the finite but large n probability distribution function of the largest eigen-
value using (1.8), and representation (1.4) we have to factor out of (1.8) the constant
term (with respect to n) to obtain the representation
F (τ(t)) = det (I K (X,Y)) I +(I K (X,Y))−1 cAi(X)Ai(Y)n−31
n,2 Ai Ai
− · − −
(cid:18) (cid:26) (cid:20)
1
(X +Y)Ai′(X)Ai′(Y) (X2 +XY +Y2)Ai(X)Ai(Y)+
20 −
(cid:20)
−20c2 +3(Ai′(X)Ai(Y)+Ai(X)Ai′(Y)) n−23 +O(n−1)E(X,Y) . (1.9)
2
(cid:21) (cid:21)(cid:27)(cid:19)
This Fredholm determinant is computed over the set (t, ). Thus to complete the
∞
determination of F (τ(t)) we need to determine the action of the integral operator
n,2
(I K ) on xiAi(x) and xiAi(x) where i = 0,1, . These are the special functions
Ai
− ···
in the GUE case, and they are independent of n, they are well known in the literature
(seeforexample[2,3,4,20,19, 17,21,23,18,22]). Forthesenindependent functions,
we just redefine them here and then introduce their n dependent counterparts.
′ ′ ∞
Ai(X) Ai (Y) Ai(Y) Ai (X)
K (X,Y) = − = Ai(X+Z) Ai(Y+Z)dZ. (1.10)
Ai
X Y
− Z0
ρ(X,Y;s) = (I K )−1(X,Y;s), R(X,Y;s) = ρ(X,Y;s) K (X,Y) (1.11)
Ai Ai
− ·
this last product is operator multiplication.
Q (x;s) = ((I K )−1, xiAi), (1.12)
i Ai
−
P (x;s) = ((I K )−1, xiAi′), (1.13)
i Ai
−
q (s) = Q (s;s), p (s) = P (s;s), q (s) := q(s) p (s) := p(s) (1.14)
i i i i 0 0
3
u (s) = (Q ,Ai), v (s) = (P ,Ai), u (s) := u(s), v (s) := v(s) (1.15)
i i i i 0 0
′ ′
v˜(s) = (Q ,Ai ), w (t) = (P ,Ai ), w (s) := w(s), and v˜ (s) := v˜(s). (1.16)
i i i i 0 0
Here ( , ) denotes the inner product on L2(s, ) and i = 0,1,2, . These are
· · ∞ ···
all well known functions, this paper is concerned with the n dependent counterparts
whose definitions are similar in nature. the changes needed here are on the kernel
definition. The operator kernel is of the same form as (1.8)
ϕ(x)ψ(y) ψ(x)ϕ(y)
K (x,y) = − (1.17)
n
x y
−
with
n n
ϕ(x) = 4 ϕn(x) and ψ(x) = 4 ϕn−1(x)
2 2
r r
Relating this to the previous set of function are the functions ϕ and ψ, they are Ai
′
and Ai . We have the following functions,
∞
ρ (x,y;t) := (I K )−1(x,y;t), R (x,y;t) := ρ (x,z;t)K (z,y;t)dz (1.18)
n n n n n
−
Zt
these are kernels of integral operators on (t )
∞
∞ ∞
Q (x;t) := ρ (x,y;t)yiϕ(y)dy, P (x;t) := ρ (x,y;t)yiψ(y)dy (1.19)
n,i n n,i n
Zt Zt
or
Qn(x;t) := (ρn,ϕ)(t ∞) Pn(x;t) := (ρn,ψ)(t, ∞).
And the other functions are
q (t) = Q (t;t), p (t) = P (t;t), q (t) := q (t) p (s) := p (t) (1.20)
n,i n,i n,i n,i n,0 n n,0 n
u (t) = (Q ,ϕ), v (t) = (P ,ϕ), u (t) := u (t), v (t) := v (t) (1.21)
n,i n,i n,i n,i n,0 n n,0 n
v˜ (t) = (Q ,ψ), w (t) = (P ,ψ), w (t) := w (t), andv˜ (t) := v˜ (t). (1.22)
n,i n,i n,i n,i n,0 n n,0 n
Here ( , ) denotes the inner product on L2(t, ) and i = 0,1,2, .
· · ∞ ···
Wewillliketopointoutthefollowingambiguityinthesedefinitions, then-independent
functions have a subscript i whereas the n dependent ones have the subscript n. We
were not able to find a suitable representations of the set of functions depending on
the matrix ensemble of n n matrices, but the choice of keeping with the original
×
Tracy and widom notation was made in part to help the reader go through the topic
without too much confusion. Thus whenever we use the subscript n we will refer to
the large size on the underlying matrix ensemble and when i is used it refers to the
exponent of the variable x appearing in the definition of that specific function and i
takes values from 0,1,2, . One exception is when we will use a second subscript to
···
distinguish between the 3 beta ensembles β = 1,2,4, in this case we will remind the
reader of the significance of those values.
4
In deriving the probability distribution function of the largest eigenvalue F (t)
n,1
for the orthogonal ensemble, and F (t) for the symplectic ensemble, we encounter
n,4
new sets on functions obeying the same set of relations.
If we define ε to be the integral operator with kernel ε(x,y) = 1sign(x y) then
2 −
∞
Q (x;t) := ρ (x,y;t)ε(ϕ)(y)dy, q (t) := Q (t;t) (1.23)
n,ε n n,ε n,ε
Zt
∞
P (x;t) := ρ (x,y;t)ε(ψ)(y)dy, p (t) := P (t;t). (1.24)
n,ε n n,ε n,ε
Zt
In a similar way we define
∞ ∞
u (t) := Q (x;t)ϕ(x)dx, v (t) := P (x;t)ϕ(x)dx. (1.25)
n,ε n,ε n,ε n,ε
Zt Zt
∞ ∞
v˜ (t) := Q (x;t)ψ(x)dx, and w (t) = P (x;t)ψ(x)dx. (1.26)
n,ε n,ε n,ε n,ε
Zt Zt
And finally we also have for the Gaussian Orthogonal Ensemble
t t t
(t) := R (x,t;t)dx, (t) := P (x;t)dx, (t) := Q (x;t)dx
n,1 n n,1 n n,1 n
R P Q
−∞ −∞ −∞
Z Z Z
(1.27)
(Note here that the second subscript here refers to the beta being 1 for the orthogonal
ensemble and has nothing to do with the previous discussion on i and n.) For the
Gaussian Symplectic Ensemble we have,
∞ ∞
(t) := ε(x,t)R (x,t;t)dx, (t) = ε(x t)P (x;t)dx, and
n,4 n n,4 n
R P −
−∞ −∞
Z Z
(1.28)
∞
(t) := ε(x t)Q (x;t)dx,
n,4 n
Q −
−∞
Z
and the 4 refers to beta being 4 for the Gaussian Symplectic Ensemble.
We have the large n expansion of most of these functions from previous work.
What is new in this paper are the large n expansion of Q , P this can be used
n,i n,i
to derive an expansion for u , v , v˜ , w . We also have closed formula for
n,i n,i n,i n,i
u , v˜ , q , , , , , , and .
n,ε n,ε n,ε n,1 n,1 n,1 n,4 n,4 n,4
Q P R Q P R
In the second section we will give a brief justification of Q and P follow in the
n,i n,i
third section with the justification of these last 9 functions. Again the motivation for
the derivation of these functions is due to their appearance in the Edgeworth type
expansion of the largest eigenvalue probability distribution function for the Gaussian
Orthogonal and Symplectic Ensembles.
5
2 Epsilon independent functions
Building on (2.5), (1.7) and (1.8) we find that
∞
Q (x) := ((I K )−1(x,y;t),yiϕ(y)) = (I K )−1(x,y;t)yiϕ(y)dy
n,i n,2 n,2
− −
Zt
∞
P (x) := ((I K )−1(x,y;t),yiψ(y)) = (I K )−1(x,y;t)yiϕ(y)dy
n,i n,2 n,2
− −
Zt
therefore we need to find ρ (x,y;t) = (I K )−1(x,y;t) in order to find an expres-
n n,2
−
sions for these two functions. But
(I K )−1(τ(X),τ(Y);τ(t)) = I +(I K )−1(X,Y;t) cAi(X)Ai(Y)n−31
n,2 Ai
− − −
(cid:26) (cid:20)
1
(X +Y)Ai′(X)Ai′(Y) (X2 +XY +Y2)Ai(X)Ai(Y)+
20 −
(cid:20)
−20c2 +3(Ai′(X)Ai(Y)+Ai(X)Ai′(Y)) n−32 +O(n−1)E(X,Y) −1 (I K (X,Y))−1
Ai
2 · −
(cid:21) (cid:21)(cid:27)
(2.1)
−1 1 ′
= I + cQ(X) Ai(Y)n 3 (P (X)+YP(X))Ai(Y)
1
− 20 −
(cid:26) (cid:20)
(Q (X)+YQ (X)+Y2Q(X)) Ai(Y)+ −20c2 +3(P(X)Ai(Y)+Q(X)Ai′(Y)) n−23
2 1
2
(cid:21)
−1
1
+O( )E(X,Y) (I K (X,Y))−1
Ai
n · −
(cid:21)(cid:27)
1
−1 ′
= I cQ(X) Ai(Y)n 3 + (P (X)+YP(X))Ai(Y)
1
− 20 −
(cid:26) (cid:20)
20c2 +3
(Q (X)+YQ (X)+Y2Q(X)) Ai(Y)+ − (P(X)Ai(Y)+Q(X)Ai′(Y))
2 1
2
+20c2Q(X;s)u(s)Ai(Y) n−32 +O(1)E(X,Y) (I K (X,Y))−1 = ρ (X,Y;t)
Ai n
n · −
(cid:21) (cid:21)(cid:27)
Note that with this representation of ρ all the Q and P will have no term inde-
n,i n,i
pendent of n, but only Q := Q and P := P , in [3] we find that
n,0 n n,0 n
1 2c 1 −1
Q (τ(X);τ(s)) = n6 Q(X;s)+ − P(X;s) cQ(X;s)u(s) n 3
n
2 −
(cid:20) (cid:20) (cid:21)
3 3
+ (10c2 10c+ )Q (X;s)+P (X;s)+( 30c2 +10c+ )Q(X;s)v(s)
1 2
− 2 − 2
(cid:20)
+P (X;s)v(s)+P(X;s)v (s) Q (X;s)u(s) Q (X;s)u (s) Q(X;s)u (s)
1 1 2 1 1 2
− − −
6
−2
+ ( 10c2 + 3)P(X;s)u(s)+20c2Q(X;s)u2(s) n 3 +O(n−1)E (X;s) , (2.2)
q
− 2 20
(cid:21) (cid:21)
and
1 2c+1 −1
P (τ(X);τ(s)) = n6 Q(X;s)+ P(X;s) cQ(X;s)u(s) n 3
n
2 −
(cid:20) (cid:20) (cid:21)
3 3
+ (10c2 +10c+ )Q (X;s)+P (X;s)+( 30c2 10c+ )Q(X;s)v(s)
1 2
2 − − 2
(cid:20)
+P (X;s)v(s)+P(X;s)v (s) Q (X;s)u(s) Q (X;s)u (s) Q(X;s)u (s)
1 1 2 1 1 2
− − −
−2
+ ( 10c2 + 3)P(X;s)u(s)+20c2Q(X;s)u2(s) n 3 +O(n−1)E (X;s) . (2.3)
p
− 2 20
(cid:21) (cid:21)
Using
Qn,i(τ(X),τ(s)) = (ρn(τ(X),τ(Y),τ(s)),(τ(Y))iϕ(τ(Y)))(τ(s) ∞) =
i i i−k
i! 22(n+c) 2
k!(i k)! 2knk2 (ρn(τ(X),τ(Y);τ(s)),Ykϕ(τ(Y)))(τ(s) ∞).
k=0 −
X
and
Xkϕ(τ(X)) = n61 XkAi(X)+ (2c−1)XkAi′(X)n−31+ (10c2 10c+ 3)Xk+1Ai(X)
2 − 2
(cid:26) (cid:20)
−2
+ Xk+2Ai′(X) n 3 +O(n−1)Ai(X) (2.4)
20
(cid:21) (cid:27)
we find that
ρ(X,Y;s) ϕ(τ(X)) = n61 Q (X;s)+ (2c−1)P (X)n−13+ (10c2 10c+ 3)Q (X)
k k k+1
· 2 − 2
(cid:26) (cid:20)
−2
+ P (X) n 3 +O(n−1)Q (X) . (2.5)
k+2 k
20
(cid:21) (cid:27)
Combining this with the action of the first factor on the right of (2.1) gives the
following expression for (ρ (τ(X),τ(Y);τ(s)),Xkϕ(τ(X))
n
n16 Q (X;s)+ 2c−1P (X;s) cu (s)Q(X;s) n−31 + (10c 20c2)v˜ (s)Q(X;s)+
k k k k
2 − −
(cid:26) (cid:20) (cid:21)
(cid:2)
3
(10c2 10c+ )Qk+1(X;s)+Pk+2(X;s)+P1(X;s)vk(s)+P(X)(Y Ai′(Y),Qk(Y))(s ∞)
− 2
uk(s)Q2(X;s) Q1(X;s)(Y Ai(Y),Qk(Y;s))(s ∞) Q(X;s)(Y2Ai(Y),Qk(Y;s))(s ∞)
− − −
20c2 +3 20c2 +3 n−23
+− P(X;s)u (s)+ − Q(X;s)v (s)+20c2Q(X;s)u(s)u (s)
k k k
2 2 20
(cid:21)
7
1
+O( )E(X;s) .
n
(cid:27)
To simplify the inner product in this last expression, we use the following recurrence
relation derived in [21]
Q (X;s) = XkQ(X;s) (v Q u P ) to have
k − i+j=k−1;i,j>0 j i − j i
P ∞ ∞
(Qk(X;s),XAi(X))(s ∞) = X Ai(X)ρ(X,Y;s)YkAi(Y)dY dX
Zs Zs
= (Q (X;s),XkAi(X)) = (XQ(X;s)+u(s)P(X;s) v(s)Q(X;s),XkAi(X)) =
1
−
u (s)+u(s)v˜ (s) v(s)u (s)
k+1 k k
−
and
(Qk(X;s),X2Ai(X))(s ∞) = (Q2(X;s),XkAi(X))(s ∞) =
= (X2Q(X;s) v(s)(XQ(X;s)+u(s)P(X;s) v(s)Q(X;s)) u(s)(XP(X;s)
− − −
w(s)Q(X;s)+v(s)P(X;s)) v (s)Q(X;s)+u (s)P(X;s),XkAi(X))
1 1
− −
= u (s) v(s)u v(s)u(s)v˜ (s)+v(s)2u (s) u(s)v˜ (s)
k+2 k+1 k k k+1
− − −
+u(s)w(s)u (s) u(s)v(s)v˜ (s) v (s)u (s)+u (s)v˜ (s).
k k 1 k 1 k
− −
we also have
(Qk(X;s),XAi′(X))(s ∞) = (P1(X;s),XkAi(X)) =
v˜ (s)+v(s)v˜ (s) w(s)u (s).
k+1 k k
−
We therefore have
i i! 22i−k(n+c)i−2k
Q (τ(X),τ(s)) = . Q (X;s)+
n,i k!(i k)! nk2−61 k
k=0 − (cid:26)
X
2c 1
− P (X;s) cu (s)Q(X;s) n−31 + (10c 20c2)v˜ (s)Q(X;s)+
k k k
2 − −
(cid:20) (cid:21) (cid:20)
3
(10c2 10c+ )Q (X;s)+P (X;s)+P (X;s)v (s)+P(X) v˜ (s)+v(s)v˜ (s) w(s)u (s)
k+1 k+2 1 k k+1 k k
− 2 −
(cid:18) (cid:19)
u (s)Q (X;s) Q (X;s) u (s)+u(s)v˜ (s) v(s)u (s)
k 2 1 k+1 k k
− − −
(cid:18) (cid:19)
Q(X;s) u (s) v(s)u v(s)u(s)v˜ (s)+v(s)2u (s) u(s)v˜ (s)
k+2 k+1 k k k+1
− − − −
(cid:18)
+u(s)w(s)u (s) u(s)v(s)v˜ (s) v (s)u (s)+u (s)v˜ (s)
k k 1 k 1 k
− −
(cid:19)
20c2 +3 20c2 +3 n−23
+− P(X;s)u (s)+ − Q(X;s)v (s)+20c2Q(X;s)u(s)u (s)
k k k
2 2 20
(cid:21)
8
1
+O( )E(X;s) . (2.6)
n
(cid:27)
In a similar way we have
Pn,i(τ(X),τ(s)) = (ρn(τ(X),τ(Y),τ(s)),(τ(Y))iψ(τ(Y)))(τ(s) ∞) =
i i i−k
i! 22(n+c) 2
k!(i k)! 2knk2 (ρn(τ(X),τ(Y);τ(s)),Ykψ(τ(Y)))(τ(s) ∞).
k=0 −
X
And (ρ (τ(X),τ(Y);τ(s)),Xkψ(τ(X)) is equal to
n
n61 Q (X;s)+ 2c+1P (X;s) cu (s)Q(X;s) n−31 + (10c+20c2)v˜ (s)Q(X;s)+
k k k k
2 − −
(cid:26) (cid:20) (cid:21)
(cid:2)
3
(10c2+10c+ )Qk+1(X;s)+Pk+2(X;s)+P1(X;s)vk(s)+P(X)(Y Ai′(Y),Qk(Y))(s ∞)
2
uk(s)Q2(X;s) Q1(X;s)(Y Ai(Y),Qk(Y;s))(s ∞) Q(X;s)(Y2Ai(Y),Qk(Y;s))(s ∞)
− − −
20c2 +3 20c2 +3 n−23
+− P(X;s)u (s)+ − Q(X;s)v (s)+20c2Q(X;s)u(s)u (s)
k k k
2 2 20
(cid:21)
1
+O( )E(X;s) .
n
(cid:27)
this therefore gives
i i! 22i−k(n+c)i−2k
P (τ(X),τ(s)) =
n,i k!(i k)! nk2−61 ×
k=0 −
X
Q (X;s)+ 2c+1P (X;s) cu (s)Q(X;s) n−31 + (10c+20c2)v˜ (s)Q(X;s)+
k k k k
2 − −
(cid:26) (cid:20) (cid:21)
(cid:2)
3
(10c2 +10c+ )Q (X;s)+P (X;s)+P (X;s)v (s)+
k+1 k+2 1 k
2
P(X) v˜ (s)+v(s)v˜ (s) w(s)u (s)
k+1 k k
−
(cid:18) (cid:19)
u (s)Q (X;s) Q (X;s) u (s)+u(s)v˜ (s) v(s)u (s)
k 2 1 k+1 k k
− − −
(cid:18) (cid:19)
Q(X;s) u (s) v(s)u v(s)u(s)v˜ (s)+v(s)2u (s) u(s)v˜ (s)
k+2 k+1 k k k+1
− − − −
(cid:18)
+u(s)w(s)u (s) u(s)v(s)v˜ (s) v (s)u (s)+u (s)v˜ (s)
k k 1 k 1 k
− −
(cid:19)
20c2 +3 20c2 +3 n−23
+− P(X;s)u (s)+ − Q(X;s)v (s)+20c2Q(X;s)u(s)u (s)
k k k
2 2 20
(cid:21)
9
1
+O( )E(X;s) . (2.7)
n
(cid:27)
When we set i to zero we recover Q (X;s) and P (X;s).
n n
We see immediately that these two series representations of Q and P are not
n,i n,i
−1
in terms of n 3 when i is not zero. We can use (2.5), (1.7), (2.6) and (2.9), to derive
an expansion for u , v , v˜ and w from their representations
n,i n,i n,i n,i
un,i(t) = (Qn,i(x;t),ϕ(x))(t ∞) vn,i(t) = (Pn,i(x;t),ϕ(x))(t ∞)
v˜n,i(t) = (Qn,i(x;t),ψ(x))(t ∞) and wn,i(t) = (Pn,i(x;t),ψ(x))(t ∞).
We would like to note that (2.6) and (2.9) are the new quantities in this section, as
additional corollary the derivation of q (t) and p (t).
n,i n,i
i i! 22i−k(n+c)i−2k
q (τ(s)) = . q (s)+
n,i k!(i k)! nk2−16 k
k=0 − (cid:26)
X
2c−1p (s) cu (s)q(s) n−13 + (10c 20c2)v˜ (s)q(s)+
k k k
2 − −
(cid:20) (cid:21) (cid:20)
3
(10c2 10c+ )q (s)+p (s)+p (s)v (s)+p(s) v˜ (s)+v(s)v˜ (s) w(s)u (s)
k+1 k+2 1 k k+1 k k
− 2 −
(cid:18) (cid:19)
u (s)q (s) q (s) u (s)+u(s)v˜ (s) v(s)u (s)
k 2 1 k+1 k k
− − −
(cid:18) (cid:19)
q(s) u (s) v(s)u v(s)u(s)v˜ (s)+v(s)2u (s) u(s)v˜ (s)
k+2 k+1 k k k+1
− − − −
(cid:18)
+u(s)w(s)u (s) u(s)v(s)v˜ (s) v (s)u (s)+u (s)v˜ (s)
k k 1 k 1 k
− −
(cid:19)
20c2 +3 20c2 +3 n−23
+− p(s)u (s)+ − Q(X;s)v (s)+20c2q(s)u(s)u (s)
k k k
2 2 20
(cid:21)
1
+O( )e (s) , (2.8)
q
n
(cid:27)
i i! 22i−k(n+c)i−2k
p (τ(s)) =
n,i k!(i k)! nk2−61 ×
k=0 −
X
q (s)+ 2c+1p (s) cu (s)q(s) n−31 + (10c+20c2)v˜ (s)q(s)+
k k k k
2 − −
(cid:26) (cid:20) (cid:21)
(cid:2)
3
(10c2 +10c+ )q (s)+p (s)+p (s)v (s)+
k+1 k+2 1 k
2
p(s) v˜ (s)+v(s)v˜ (s) w(s)u (s)
k+1 k k
−
(cid:18) (cid:19)
10
