Специальные функции. Мицик М.Ф. - 8 стр.

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 
( )
...,2,1,2 == kkm
,     
:
( )
...,2,1,
2
2
22
2
=
+
=
−
k
kk
a
a
k
k
ν
(22)

( )
,
12
2
0
2
+
−=
ν
a
a
( )
( )( )
,
2122222
4
0
2
2
4
++
=
+
−=
ννν
aa
a
( )
( )( )( )
,
321322332
6
0
2
4
6
+++⋅
−=
+
−=
νννν
a
a
a
.  k
:
( )
( )( )( )
....,2,1,
....21!2
1
2
0
2
=
+++
−= k
kk
a
a
k
k
k
ννν
, :
( )
( )
( )( )( )
.
....21!
2
1
0 0
2
0
2
2
∑∑
∞
=
∞
=
+++
−
==
m m
k
k
k
k
kk
x
a
xxaxxy
ννν
νν
(23)
.
 (23) 
( )
.
12
1
0
+Γ
=
ν
ν
a
(24)
.
1

0
a

 (24) 
ν
, 
(
)
xJ
ν
:
( )
( )
( )
∑
∞
=
++Γ
−
=
0
2
1!
2
1
2
k
k
k
kk
x
x
xJ
ν
ν
ν (25)

0
>
ν
,      
 (25), 
ν

ν
−
:
                m = 2k , (k = 1, 2, ...) ,
                                    :
                                            a2 k − 2
                         a2 k =                        , k = 1, 2, ...
                                        2 2 k (k + ν )
                                                                                      (22)


                                           a0
                         a2 = −                   ,
                                        2 (ν + 1)
                                            2


                     a2                a0
                  a4 = −   = 4                   ,
                2 2(ν + 2) 2 2(ν + 1)(ν + 2)
                            2


              a                      a0
    a6 = − 2 4       =− 6                            ,
          2 3(ν + 3)    2 2 ⋅ 3(ν + 1)(ν + 2)(ν + 3)
                                            .                                                  k
                                        :
                  a2 k = (− 1)
                                                a0
                                                                   , k = 1, 2,....
                                k

                                    2 k!(ν + 1)(ν + 2)....(ν + k )
                                        2k


                     ,                                                                     :
                                                                                 2k
                                              x
                ∞           ∞
                                     (− 1) a0 
                                                                    k


    y( x) = x ∑ a2k x = x ∑
             ν       2k  ν                    2            .
               m =0        m=0 k! (ν + 1)(
                                         ν + 2 )....(ν + k )                          (23)


                                                                             .

                                                         (23)
                                                     1
                                        a0 =                .
                                                2ν Γ(ν + 1)
                                                                                      (24)

        1   .                                                                           a0
  (24)                                                                           ν,   Jν ( x )
                                                :
                                                                        2k
                                                      x
                                         ν ∞ (− 1)  
                                                                k

                                     x              2
                         Jν ( x ) =   ∑
                                     2  k =0 k!Γ(k +ν + 1)
                                                                                      (25)
                    ν >0,
(25),                 ν             −ν :

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