Специальные функции. Мицик М.Ф. - 15 стр.

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1
−
k

121
...,,,
−k
ξ
ξ
ξ
 (-1;1). 

( )
xf
k 1−


[ ][ ] [ ][ ]
1;,;...,,;,;1
112211 −−−
−
kkk
ξ
ξ
ξ
ξ
ξ
ξ
, 
( ) ( )
xfxf
kk
=
′
−1
. 
( )
xf
k
 (-1;1) 
k
.
, 
( )
xf
k
, 
( )
xf
k 1−
,

1
±
=
x

0
≥
−
kn
.
, 
(
)
(
)
xPnxf
n
n
n
!2=
, .
 
:
( )
021
2
≡−++− ω
ω
xz
dz
d
zzx
,

( )
zx;ω
 (37). 
 (38), , 
:
( )
( ) ( ) ( )
.1,1,021
0
1
0
2
≤<≡−++−
∑∑
∞
=
−
∞
=
xzzxPxzzxnPzzx
n
n
n
n
n
n


z
, :
( ) ( ) ( ) ( ) ( )
.0121
11
=++−+
−+
xnPxPxnxPn
nnn
(44)

( )
021
2
≡++− ω
ω
z
dx
d
zzx
:
( ) ( ) ( ) ( )
.02
11
=−
′
+
′
−
′
−+
xPxPxPxxP
nnnn
(45)
 (44) (45), 
:
( ) ( ) ( ) ( )
...,2,1,0,1
1
=+=
′
−
′
+
nxPnxPxxP
nnn
(46)
( ) ( ) ( )
...,3,2,1,
1
==
′
−
′
−
nxnPxPxPx
nnn
(47)
( ) ( ) ( ) ( )
...,3,2,1,12
11
=+=
′
−
′
−+
nxPnxPxP
nnn
(48)
.
k −1                   ξ1 , ξ 2 , ..., ξ k −1                                                          (-1;1).
               f k −1 ( x )
                     [− 1; ξ1 ], [ξ1; ξ 2 ], ..., [ξ k −2 ; ξ k −1 ], [ξ k −1;1],
 f k′−1 (x ) = f k ( x )                                                                          .              f k (x )
                                (-1;1)                       k                                .
                                             ,               f k (x ) ,                                               f k −1 ( x ) ,
                           x = ±1                                                n − k ≥ 0.
                   ,
 f n ( x ) = 2 n! Pn (x ) ,
               n
                                                                                                         .


                                             :
                                (1 − 2 x z + z ) ddzω + (z − x )ω ≡ 0 ,
                                                         2



     ω ( x; z )                                                       (37).
                                             (38),                      ,
                                 :

(1 − 2 x z + z )∑ nP (x ) z
                          ∞                                                ∞
                   2
                                     n
                                                 n −1
                                                        + (z − x )∑ Pn ( x ) z n ≡ 0 ,                z < 1 , x ≤ 1.
                         n =0                                             n =0


               z                                                          ,              :
                   ( n + 1) Pn+1 ( x ) − ( 2n + 1) x Pn ( x ) + nPn−1 (x ) = 0.                                             (44)


                                (1 − 2 x z + z ) ddxω + z ω ≡ 0
                                                         2



                                                                                              :
                   Pn′+1 (x ) − 2 x Pn′(x ) + Pn′−1 (x ) − Pn ( x ) = 0.                                                    (45)
                                                   (44)          (45),
                                         :
                   Pn′+1 (x ) − x Pn′(x ) = (n + 1)Pn (x ), n = 0, 1, 2, ...                                                (46)
                   x Pn′(x ) − Pn′−1 (x ) = nPn (x ), n = 1, 2, 3, ...                                                      (47)
                    Pn′+1 ( x ) − Pn′−1 (x ) = (2n + 1)Pn ( x ), n = 1, 2, 3, ...                                           (48)

                                                                                                         .

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