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bo-graduation/nltk-book/pattern-master/examples/03-en/texts/1701.00100.txt

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arXiv:1701.00100v1 [math.CA] 31 Dec 2016
, x ( ) ln-1 x (
) x i , i = -1, R, = 0 ( ). . , .
On properties of the coefficients of the complicated and exotic expansions of the solutions of the sixth Painlev<65>e
equation
I. V. Goryuchkina
It is known, that among the formal solutions of the sixth Painlev<65>e equation there met series with integer power exponents of the independent variable x with coefficients in form of formal Laurent series (with finite main parts) in log-1 x (complicated expansions), or in x i , where
i = -1, R, = 0 (exotic expansions). These coefficients can be computed consecutively. Here we research analytic properties of the series, that are the coefficients of the complicated and exotic expansions of the solutions of the sixth Painlev<65>e equation.
1. .
(y )2 1 1
1
11
1
y=
+
+
-y +
+
+
2 y y-1 y-x
x x-1 y-x
y(y - 1)(y - x)
x x - 1 x(x - 1)
+ x2(x - 1)2
a
+
by2
+
c (y
-
1)2
+
d
(y
-
x)2
,
(1)
a, b, c, d <20> , x y <20> ,
y = dy/dx. x = 0, x =
1 x = , ,
1) x = z, y = z/w, 2) x = 1/z, y = 1/w, 3) x = 1 - z, y = 1 - w,
(. [1]). . x = 0 (1),
2
, .
x , ln-1 x(- ) x i ( R, i = -1) ( ) , . [2]. , x , ln-1 x x i ( = 0). , , (1) , .
y = k(x) xk,
(2)
k=0
k(x) <20>
k(x) =
ckj j,
j=0
ckj C, Z,
= ln-1 x = x i ( = 0).
(3)
-
, (2), <20> , [2].
, [3] () , . , [3] , ( ) , . -,
3
. [3] . , [3] . , . ( [3], ) , [3], , . , y = 0(x) + u , 0(x) <20> . ( ) , , u = 1(x)x. , (2), . .
, k(x) (2) (1), , , . , , . . (. . ), . .
k(x) .
() 0(x) x i , = 0 (x = 0, 1, ) . [4] , (. .
4
). , .. k(x) .
(. [5], [6]) , ( , ), . .
: k(x) (2) (1).
, , , , [3] . : [3] , , .
2. . -
, .
(x) x = 0 r R {},
ln |(x)|
lim
= r,
(4)
x0 ln |x|
xD
D <20> ,
. (x) x =
r R {},
ln |(x)|
lim
= r,
(5)
x xD
ln |x|
5
D <20> , .
(x), r R {} , , r.
(2) .
g(x, u, u , . . . , u(n)) = 0,
(6)
(x) xq1uq20(xu )q21 . . . (xnu(n))q2n,
(7)
(x) <20> -
(.. ), q1 C, q21, . . . , q2n Z+, (n)(x) 0, (n)(x) -n.
(6) u =
= k(x)xk,
(8)
k=0
k(x) <20> , k R, (kn)(x) 0, k(n)(x) -n, , k+1 > k.
(7) (6) -
( ) (q1, q2),
q2 = q20 + . . . + q2n. [3]
(x) xq1q20(x )q21 . . . (xn(n))q2n
q1 + q20. {Qi = (q1i , q2i ), i = 0, . . . , m} <20> -
(6), . . ,
, R <20> (1, 0). Qi, R = ci R. c = min ci.
i=0,...,m
(7) (6) -
, Qi, R = c R, (. [7]), g^(x, u0, . . . , un),
g^(x, u0, . . . , un) = 0
(9)
6
<EFBFBD> .
1. (6) u = , (8), (9) ()
u = ^, ^ = 0(x).
(10)
. 0 = 0, (6)
u = x0v.
(11)
G(x, v, v , . . . , v(n)) = 0,
(12)
v = , = x0 , .
, (11)
(6)
(x) xq1uq20(xu )q21 . . . (xnu(n))q2n
(x) xq1+q20vq~20(xv )q~21 . . . (xnv(n))q~2n,
(13)
q~20, . . . , q~2n Z+, q~20 + . . . + q~2n = q2. , (11) g^(x, u, u . . . , u(n)) xc P0(x, v, v . . . , v(n)), c <20> q1 +q20, P0(x, v, v . . . , v(n)) <20> v, v . . . , v(n) , (6) (13) q1 + q20 > c. (12)
xc [P0(x, v0, . . . , vn) + x1P1(x, v0, . . . , vn) + . . . + xtPt(x, v0, . . . , vn)] = 0,
vj = xjv(j), P0(x, v0, . . . , vn), . . . , Pt(x, v0, . . . , vn) <20> v0, . . . , vn , 1, . . . , n R+, 1, . . . , n = 0. xc,
P0(x, v0, . . . , vn) + x1P1(x, v0, . . . , vn) + . . . + xtPt(x, v0, . . . , vn) = 0, (14)
G(x, v0, . . . , vn) = 0. (14) 1 <20> 3 [3] , p G(x, 0, . . . , n) 0 (
7
, Pj x, 0, . . . , n = 0 p Pj(x, 0, . . . , n) = 0, Pj x, 0, . . . , n ).
u = ,
= ~ k0(x1, . . . , xN ) k0(x)
k0=0
(14)
u = ^ + w, ^ = ~ 0(x1, . . . , xN ) 0(x).
(15)
(15) (14), ( , (14) v0, . . . , vn) ,
P0(x, ^ 0, . . . , ^ n) +
P0(x,
^ 0, . v0
.
.
,
^ n)
w0
+
.
.
.
+
P0(x,
^ 0, . vn
.
.
,
^ n)
wn
+
+ . . . + x1P1(x, ^ 0, . . . , ^ n) + <20> <20> <20> = 0, ^ j = xj ^ (j), wj = xj w(j). (16)
(16), -
, P0(x, ^ 0, . . . , ^ n), p(P0(x, ^ 0, . . . , ^ n)) , - ( P0(x, ^ 0, . . . , ^ n) = 0).
(16) . , p(wj) > p(^ j) = 0, P0(x, ^ 0, . . . , ^ n), . . . , Pt(x, ^ 0, . . . , ^ n) v0, . . . , vn ,
, 2 3 [3], n > <20> <20> <20> > 1 > 0. -
(14), (16) , . . P0(x, ^ 0, . . . , ^ n) = 0. -
P0(x, v0, . . . , vn) = 0 (11) x c, ,
g^(x, ^0, . . . , ^n) = 0, ^j = xj ^(j), ^ = x0 ^ . 2
3. . (1) . x2(x - 1)2y(y - 1)(y - x), .
2x2(x - 1)2y(y - 1)(y - x)y - x2(x - 1)2(3y2 - 2xy - 2y + x)y 2+, (17)
8
+2xy(x - 1)(y - 1)(2xy - x2 - y)y - 2y6a + 4a(x + 1)y5-
-2 (a + d)x2 + (4a + b + c - d)x + (a - c) y4+
+4x ((a + b + c + d)x + (a + b - c - d)) y3-
-2 (b + c)x3 + (a + 4b - c + d)x2 + (b - d)x y2 + 4bx2(x + 1)y - 2bx3 = 0,
, (1), . [2]. [2] , 0(x)
(2) <20> , = ln-1 x = x i . x , . , , (2) .
(17)
y = 0(x) + xu.
(18)
L 0, 0, <20> 0, U + xM x, 0, 0, <20> 0, U + H x, 0, 0, <20> 0 = 0, (19)
U = (u0, u1, u2), uj = xju(j),
0
=
0(x),
0
=
x d0(x) , dx
<EFBFBD> 0
=
x2
d20(x) dx2
,
L 0, 0, <20> 0, U = 220 (0 - 1) u2 +
20 320 - 30 0 - 30 + 2 0 u1 -
(20)
2 6a50 - 10a40 + 4a30 - 4c30 - 30 - 320<32> 0 + 30 20 + 20 + 20<32> 0 - 20 u0,
M x, 0, 0, <20> 0, U H x, 0, 0, <20> 0 . (19)
u = k+1(x) xk.
(21)
k=0
9
4. . (19) c (21), . (19) u0, u1, u2 , L 0, 0, <20> 0, U ) xM (x, 0, 0, <20> 0, U ). x = 0 , L 0, 0, <20> 0, U . , L
, u0, u1 u2 , M <20> .
[2] a = c, a, c = 0 (2) k(x), ln-1 x
2(c - a)
0(x)
=
(c
-
a)2(ln x
+
C )2
-
. 2a
(22)
(21) (22) (19)
k=1
Lk(0, 0, <20> 0, k, k, <20> k) - Nk(0, 0, . . . , <20> 0, k-1, k-1, <20> k-1)
j
=
j (x),
j
=
x dj (x) , dx
<EFBFBD> j
=
x2
d2j (x) dx2
,
Lk 0, 0, <20> 0, k, k, <20> k =
xk,
(23) (24)
= L 0, 0, <20> 0, k, k + (k - 1)k, <20> k + 2(k - 1) k + (k - 1)(k - 2)k ,
Nk 0, 0, <20> 0, . . . , k-1, k-1, <20> k-1 . , , k(x) -
k
Lk 0, 0, <20> 0, k, k, <20> k = Nk 0, 0, <20> 0, . . . , k-1, k-1, <20> k-1 . (25)
(25) x ,
= ( + C)-1 = (ln x + C)-1, C C.
(26)
, (26) ,
x dy = -2 dy ,
dx
d
x2
d2y dx2
=
4
d2y d2
+ (23
+
2) dy . d
10
(-2a2 + a2 - 2ac + c2)4
(-c + a)64
.
Lk
,
^k(),
d^k() d
,
d2^k() d2
= Nk(),
(27)
Lk
,
^k(),
d^k() d
,
d2^k() d2
=
4P2()
d2^k() d2
+
2P1()
d^k() d
+
P0()^k(),
^k() = k(x), P2 P0 <20> , P1 <20> , P2(0), P1(0), P0(0) = 0, Nk() <20> .
,
= 0 . ,
a2()2
d2 d2
+
d a1() d
+
a0(),
a2(), a1(), a0() <20> ,
a2() a1(), a0().
, k(x), ln-1(x) c , . ,
-
(21) -
(25) ( 0(x) (22)). (27) , ..
Lk
,
^k(),
d^k() d
,
d2^k() d2
= -8k2^k() + . . . ,
<20> ( ), . k(x) <20> ln-1 x.
a = c = 0 (2) c
1
0(x)
=
2a
(ln x
+
, C)
C C.
(28)
,
11
(2) c (22). .
.
1. k(x) (k 1) (2) c (22) (28) ln-1 x,
.
[8] [9].
, . , , . , ( ) k(x) ln-1 x , , .
5. .
(21) (19) k(x) xi
-4C0(2a - 2c + C1)
, (29)
x 2c-2a-C1(C12 + 8C1a + 16a2 - 16ac) - 2C1C0 + C02x- 2c-2a-C1
C0, C1 <20> , C0 = 0, 2c-2a-C1 R, 2c - 2a - C1 < 0, = sgn(Im 2c - 2a - C1).
(., , [2]) B0 , B1 , B2 , B6 B7 . , ( C0 C1) (2) -
(29) (1). ,
<EFBFBD> B0
0(x)
=
2c
- C3 2a
cos2(ln(C2x) C3-2c/2)
1 +
sin2(ln(C2x)C3-2c/2) ,
(30)
a = 0, C0 =
C32
+
4C3a
+
4a2
-
16ac ,
C2 2c-C3
C1 = C3 - 2a,
C32 +
4C3a + 4a2 - 16ac = 0, C3 = 2c, C2 = 0, 2c - C3 R, 2c - C3 < 0,
12
<20> 2at2 + (C3 - 2a)t + 2c - C3 = 0, = sgn(Im 2c - C3);
<EFBFBD> B1
1 - c/a
0(x)
= 1
-
,
C2x
2c-
2a
(31)
a = c = 0, C0 = 8 a( c - a) C2, C1 = 4 a( c - a), C2 = 0,
Re( 2c - 2a) = 0, = sgn(Im( 2c - 2a));
<EFBFBD> B2
1 + c/a
0(x)
=
1
-C2x 2c+ 2a
,
(32)
a = c = 0, C0 = -8 a( c + a) C2, C1 = -4 a( c + a),
C2 = 0, Re( 2c + 2a) = 0, = sgn(Im( 2c + 2a));
<EFBFBD> B6
1
0(x)
=
1
+
, C2x 2a
(33)
a = 0, c = 0, C0 = 8aC2, C1 = -4a, C2 = 0, = sgn(Im 2a);
<EFBFBD> B7
0(x)
=
2c - C1 C1
1
sin2(ln(C2x) C1-2c/2)
a = 0,C0 = -C1/C2 2c-C1, C2 = 0, 2c - C1 R, = sgn(Im 2c - C1).
(34) 2c - C1 < 0,
( C0 C1) (1), (2) (29). , (29) Cxi.
1(x). , . (21) (19), , x.
L1 0, 0, <20> 0, 1, 1, <20> 1 + N1 0, 0, <20> 0 = 0,
(35)
0, 0, <20> 0, 1, 1, <20> 1 (23),
L1 0, 0, <20> 0, 1, 1, <20> 1 = 220(0 - 1)<29> 1 + 20(320 - 30 0 - 30 + 2 0) 1
13
-2(6a50 - 10a40 + 4a30 - 4c30 + 30 - 320<32> 0 + 30 20 + 20 + 20<32> 0 - 20)1, N1 0, 0, <20> 0 = 4a50 - 2(4a + b + c - d)40 + 4(a + b - c - d)30 - 630 0
-430<33> 0 + 620 20 - 2(b - d)20 + 620 0 + 220<32> 0 - 20 20 + 20<32> 0 - 20.
(29) (35) x = C = Cxi, R, = 0. , ,
dy
dy
x = i ,
dx
d
x2
d2y dx2
=
-22
d2y d2
-
(
+
dy i) ,
d
. . -
. -
,
. (29), 0, 0, <20> 0 -
Cxi = Cxi
42
0
=
A2
+
B
+
, 1
A = 4 + 4(a + c)2 + 4(a - c)2, B = 22 - 4(a - c),
0
=
4 i 3(A2 - 1) -(A2 + B + 1)2
,
(36)
<EFBFBD> 0
=
43(A2(i
-
)4
+
AB(i + )3 + 6A2 (A2 + B + 1)3
-
B(i
-
)
-
i
-
) .
(35) y = 1(x)
(A2 + B + 1)6
- 16 4 2
8
p2j
j+2
d2y d2
+
p1j
j+1
dy d
+
p0j
j y
+ tj j = 0,
(37)
j=0
p2j, p1j, p0j, tj C, p20 p28 = 0. (37) ,
y = C1 y1() + C2 y2() + y3(),
C1, C2 <20> , y1(), y2(), y3() <20> .
, (35) = Cxi <20> (A2 + B + 1)6,
14
, -16 4 2 (37), d2y
d2
-
(A2
+ B 84
+
1)6
20(0
-
1).
(38)
(38) <20> , 5 = 0, a1, a2, a3, a4 . , = 0 <20> 2, = a1 = a2 <20> 1, a1 a2 <20> (A-42)2+B+1 = 0, a3 a4 <20> 3, a3 a4 <20> A2+B+1 = 0. (37) = 0, , a1, a2, a3, a4. p20 p28 = 0, (37) , . (37) = + aj, , (37) . , , . [10]. , , . , (37)
y = CiFi( - aj)( - aj)i ln<6C>i ( - aj)+
i=1,2
+ F3( - aj)( - aj)3 ln<6C>3 ( - aj),
(39)
F1(), F2(), F3() C{}, 1, 2, 3 C, <20>1, <20>2, <20>3 Z. ,
. , [11].
(37) . , y3() <20> , y1() y2() <20> , = 0 = .
2. 1(x) (21) (19) (29) Cxi.
15
, 1(x) = y3(Cxi), y3() <20> (37).
2. -
(37) , ,
(39), , -
. -
1,
(37), -
. -
-.
, ( -
), ,
( ), -
(- -
)
,
(37).
(37) , -
. - (37) <20>
[(0, 0), (0, 1), (8, 1), (8, 0)]. -
(
) ( -
),
<EFBFBD> ( -
) (
).
() 0
. 0 -
-222
d2y d2
+
2(
+
dy 2i)
d
-
2(
+
i)2y
=0
-
222
d2y d2
+
2(
+
dy 2i)
d
-
2(
+
i)2y
+
(
+
i)2
-
1
+
2b
-
2d
= 0,
<20> . , (0, 1), <20> [(0, 1), (0, 0)]. -
16
y
=
(C1
+
C2
ln
)1+
i
+
(
+
i)2 + 2(
1 + 2b + i)2
-
2d ,
(40)
<20>
y
=
(C1
+
C2
ln
)1+
i
,
C1, C2 <20> . , , (40) , .
[7], , (40)
y = C1
a1kk + C2 ln
a2kk
i
+
a3kk,
k=0
k=0
k=0
(41)
a1k, a2k, a3k C, a10 = a20 = 1,
( + i)2 + 1 + 2b - 2d
a30 =
2( + i)2
,
(37). -
A48
-222
d2y d2
-
2(3
-
dy 2i)
d
-
2(
-
i)2y
=0
A48
-222
d2y d2
-
2(3
-
dy 2i)
d
-
2(
-
i)2y
-
(
-
i)2
-
1
+
2b
-
2d
= 0,
<20> . (8, 1), <20> [(8, 1), (8, 0)].
y
=
(C1
+
C2
ln
)-1+
i
+
(
-
i)2 + 2(
1 + 2b - i)2
-
2d ,
(42)
<20>
y
=
(C1
+
C2
ln
)-1+
i
,
C1, C2 <20> . ,
17
, (42) , .
(42)
y=
C1 b1k + C2 ln b2k
i
+
b3k ,
k
k
k
k=0
k=0
k=0
(43)
b1k, b2k, b3k C, b10 = b20 = 1,
( - i)2 + 1 + 2b - 2d
b30 =
2( - i)2
,
(37). (37) a1 a2,
(A - 42)2 + B + 1 = 0. (37) = + aj, j = 1, 2. . , , .
8
P2j
j +1
d2y d 2
+
P1j
j
dy d
+
P0j
jy
+
Tj
j
= 0,
(44)
j=0
P2j, P1j, P0j, Tj C, P20 = 0. - (44) <20> (-1, 1), (0, 0), (8, 0), (8, 1). , , = 0, . . , (-1, 1) [(-1, 1), (0, 0)] . -y + y = 0 -y + y = , C, (44), y = C1 = 0 y = , C1 <20> . C1, C2 (C2 C)
y = C1 c1kk + C22 c2kk + c3kk,
k=0
k=0
k=0
(45)
c1k, c2k, c3k C, c10 = c20 = 1, c30 = , (44).
18
(37) a3 a4, A2 +B+ 1 = 0. (37) = + aj, j = 3, 4. . , , .
8
S2j
j +3
d2y d 2
+
S1j
j +2
dy d
+
S0j
j +2y
+
Kj
j
= 0,
(46)
j=0
S2j, S1j, S0j, Kj C, S20 = 0. - <20> (1, 1), (0, 0), (8, 0), (8, 1). -
, ,
-
= 0, . . , (1, 1)
[(1, 1), (0, 0)] -
.
2(y + 3y ) = 0 2(y + 3y ) = , C,
(44),
y
=
C1 2
,
C1
=
0
y=
,
C1
<EFBFBD> . -
C1, C2 (C2 C)
y
=
C1 2
d1kk + C2
d2k k
+
1
d3k k ,
k=0
k=0
k=0
(47)
d1k, d2k, d3k C, d10 = d20 = 1, d30 = , (44). , 0, , a1, a2, a3, a4
(37) . , y = C1y1() + C2y2() + y3()
i
i
y = C1f1() + C2 ln f2() + f3(),
f1(), f2(), f3() <20> , . . y1() =
i
i
C1f1() y2() = C2 ln f2() <20> , -
,
.
19
y3() = f3() <20> , . , 1(x) = y3(Cxi). 2
k(x) (21) (19) (29).
Lk(0, 0, <20> 0, k, k, <20> k) - Nk(0, 0, . . . , <20> 0, k-1, k-1, <20> k-1) xk,
k=1
j, j, <20> j (23), Lk 0, 0, <20> 0, k, k, <20> k -
(24), Nk 0, 0, <20> 0, . . . , k-1, k-1, <20> k-1 .
, (21) (19), k(x), k N
Lk(0, 0, <20> 0, k, k, <20> k) = Nk(0, 0, . . . , <20> 0, k-1, k-1, <20> k-1).
(48)
, (48) k, , .
x = C = Cxi, R, = 0, k(x) = ^k(), k N, (48). , (48)
Q2()
2
d2^k() d2
+
Q1k()
d^k() d
+
Q0k()^k()
=
Nk(),
(49)
Q2() = -22 20(0 - 1),
Q1k() = 2 i (i + 2k - 3)Q2() + i Q1(),
Q0k() = (k - 2)(k - 1)Q2() + (k - 1)Q1() + Q0(), Q1() = 20(320 - 30 0 - 30 + 2 0),
Q0() = -2(6a50-10a40+4a30-4c30+30-320<32> 0+30 20+20+20<32> 0- 20)1, 0, 0, <20> 0 (36), Nk() <20> .
3. (49) ^k() ( ) .
20
3. k (49).
^k() = rk kjj,
(50)
j=0
rk Z, kj C. k = 1 1. k = 2. (49) <20>
0 1, . .
N2 R2 A2jj, R2 Z,
j=0
A2j C. Q2, Q1k, Q2k, , ^2() =
r2 2jj (49),
j=0
8 B2a2
(2i
-
)242
+
O(3)
2jr2+j = R2
A2j j .
j=0
j=0
(51)
a = 0, B = 0, 2i - = 0, R. r2, R2, (51) . 2j . k = 3, N3 <20> , ( ) 0 1, ( ) 2. (50) (49)
8 B2a2
(ki
-
k
+
)242
+
O(3)
kjrk+j = Rk
Akj j ,
j=0
j=0
(52)
Rk Z, Akj C. k = 2. , ^3() . , , (49) . 2
4. k(x) = ^k() (21) (19) (29) = 0 ( ) .
4 , (49) . 2
21
5. (49) 0, , a1, a2, a3, a4 C, .
5. 0, , a1, a2, a3, a4 C. . (49) k = 2 . , (49) k = 2 , . k = 3, (49) <20> ( ), . , , k. 2
5 , (49) = 0, , a1, a2, a3, a4 C. , (19) (21) (29), Cxi = 0, , a1, a2, a3, a4.
, ^k() = a1, a2, a3, a4 C (49).
, 3(x) 4(x) (21) (29) (19) Cxi. , . .
. k(x) = ^k() <20> = Cxi .
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