Классические ортогональные полиномы. Балакин А.Б. - 51 стр.

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T
n
(x)
a = −1 , b = 1 , X(x) = 1 − x
2
, W (x) = (1 − x
2
)
−
1
2
,
K
n
= (−1)
n
2
n
Γ(n +
1
2
)
Γ(
1
2
)
, λ
n
= n
2
.
U
n
(x)
a = −1 , b = 1 , X(x) = 1 − x
2
, W (x) = (1 − x
2
)
1
2
,
K
n
= (−1)
n
2
n+1
Γ(n +
3
2
)
(n + 1)Γ(
1
2
)
, λ
n
= n(n + 2) .
T
n
(x)
(1 − x
2
)
d
2
dx
2
T
n
(x) − x
d
dx
T
n
(x) + n
2
T
n
(x) = 0 .
U
n
(x)
(1 − x
2
)
d
2
dx
2
U
n
(x) − 3x
d
dx
U
n
(x) + n(n + 2) T
n
(x) = 0 .
T
n
(x)
T
n
(x) = (−1)
n
Γ(
1
2
)
2
n
Γ(n +
1
2
)
(1 − x
2
)
1
2
d
n
dx
n
·
(1 − x
2
)
n−
1
2
¸
.
U
n
(x)
U
n
(x) = (−1)
n
(n + 1)Γ(
1
2
)
2
n+1
Γ(n +
3
2
)
(1 − x
2
)
−
1
2
d
n
dx
n
·
(1 − x
2
)
n+
1
2
¸
.
T
n
(x) =
n
2
[n/2]
X
m=0
(−1)
m
(n − m − 1)!
m!(n − 2m)!
(2x)
n−2m
,
U
n
(x) =
[n/2]
X
m=0
(−1)
m
(n − m)!
m!(n − 2m)!
(2x)
n−2m
.
T
n
(x) U
n
(x)
T
n
(−x) = (−1)
n
T
n
(x) .
Ñòàíäàðòèçàöèÿ Tn (x)
                                                                             1
     a = −1 ,       b = 1,       X(x) = 1 − x2 ,          W (x) = (1 − x2 )− 2 ,
                                Γ(n + 21 )
                                      n n
                    Kn = (−1) 2            ,              λn = n2 .                (201)
                                  Γ( 12 )
Ñòàíäàðòèçàöèÿ Un (x)
                                                                             1
      a = −1 ,       b = 1,       X(x) = 1 − x2 ,         W (x) = (1 − x2 ) 2 ,

                             n n+1      Γ(n + 32 )
           Kn = (−1) 2                                ,   λn = n(n + 2) .          (202)
                                       (n + 1)Γ( 12 )
Äèôôåðåíöèàëüíîå óðàâíåíèå äëÿ Tn (x)
                  d2           d
                     2
         (1 − x ) 2 Tn (x) − x Tn (x) + n2 Tn (x) = 0 .                            (203)
                 dx           dx
Äèôôåðåíöèàëüíîå óðàâíåíèå äëÿ Un (x)
            d2               d
     (1 − x2 ) U n (x) − 3x    Un (x) + n(n + 2) Tn (x) = 0 .                      (204)
           dx2              dx
Ôîðìóëà Ðîäðèãà äëÿ Tn (x)
                      Γ( 21 )
                         n             2 12 d
                                             n ·
                                                        2 n− 12
                                                                ¸
     Tn (x) = (−1) n            (1 − x  )        (1 − x  )        .                (205)
                   2 Γ(n + 12 )            dxn
Ôîðìóëà Ðîäðèãà äëÿ Un (x)
                  (n + 1)Γ( 21 )
                     n                 2 − 12 d
                                               n ·
                                                         2 n+ 21
                                                                 ¸
   Un (x) = (−1) n+1             (1 − x )          (1 − x )        .               (206)
                 2 Γ(n + 32 )                dxn
ßâíîå ïðåäñòàâëåíèå ïîëèíîìîâ ×åáûøåâà
                          n [n/2]
                             X          (n − m − 1)!
                 Tn (x) =         (−1)m              (2x)n−2m ,                    (207)
                          2 m=0         m!(n − 2m)!
                              [n/2]
                             (n − m)!
                               X
                  Un (x) =            (−1)m
                                        (2x)n−2m .                                 (208)
                   m=0      m!(n − 2m)!
Ñâîéñòâà ÷åòíîñòè (àíàëîãè÷íû äëÿ Tn (x) è Un (x))

                              Tn (−x) = (−1)n Tn (x) .                             (209)

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