Специальные функции. Мицик М.Ф. - 16 стр.

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, 
( )
xP
n
,
 (46)
n

1
−
n

 (47)
( )
xP
n 1−
′
. 
( )
( ) ( ) ( )
.1
1
2
xnxPxnPxPx
nnn
−=
′
−
−
(49)
 (49) 
x
 (47)
( )
xP
n 1−
′
,
:
( )
( ) ( ) ( )
...,2,1,0,011
2
==++
′
′
− nxPnnxPx
nn
(50)
 (50) 

( )
( )
,01
2
=+
− xu
dx
du
x
dx
d
λ
(51)

.1+= nnλ

1
±
=
x
. 
( )
1+= nnλ
, 
[ ]
1;1−
 (). : 
λ
 (51) 
[ ]
1;1−
?
.
 2.

[ ]
1;1−

( )
...,2,1,0,1 =+= nnnλ
.
-

[ ]
1;1−
:
( ) ( )
mndxxPxP
mn
≠=
∫
−
,0
1
1
(52)
, 
( )
xP
n

( )
xP
m

:
( )
( )
( ) ( )
,011
2
=++
− xPnn
dx
xdP
x
dx
d
n
n
(53)
( )
( )
( ) ( )
,011
2
=++
− xPmm
dx
xdP
x
dx
d
n
m
(54)
 (53) 
( )
xP
m
, (54) – 
( )
xP
n

, :
                          ,                                                                                                 Pn ( x ) ,
              (46) n         n −1
                 (47) Pn′−1 (x ) .
                  (                  )
                   1 − x 2 Pn′(x ) = nPn−1 (x ) − nxPn ( x ) .                                                                   (49)
                                                     (49)        x                                   (47) Pn′−1 (x ) ,
         :
                                       ′
                      (      n      
                                         )
                   1 − x 2 P ′ (x ) + n (n + 1)P (x ) = 0 , n = 0, 1, 2, ...
                                                n                                                                              (50)
                 (50)

                                  d 
                                 dx 
                                       1 − x     (
                                             2 du 

                                               dx 
                                                            )
                                                     + λ u(x ) = 0 ,                                                             (51)

                               λ = n(n + 1) .
                   x = ±1                                                                                           .
λ = n(n + 1) ,                                                                                                   [− 1;1]
         (                                                  ).                            :                             λ
              (51)                                                                        [− 1;1]                  ?
                                                                                                             .
                              2.
         [− 1;1]                                                         λ = n(n + 1), n = 0, 1, 2, ...

                                                                                                         .

                                                                                                                                 -
                                    [− 1;1] :
                                    1

                                    ∫ P (x) P (x)dx = 0 ,
                                    −1
                                             n         m                        n≠m                                              (52)

                                                                     ,             Pn ( x )   Pm ( x )
             :
                            2 dPn ( x ) 
                d 
                dx 
                      1 − x    (  dx 
                                                 )
                                          + n (n + 1)Pn (x ) = 0 ,                                                               (53)

                            2 dPm ( x ) 
                d 
                dx 
                      1 − x    (  dx 
                                                 )
                                          + m (m + 1)Pn (x ) = 0 ,                                                               (54)

             (53)      Pm ( x ) , (54) –       Pn ( x )
                                ,                      :

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