Polinomlar ketma-ketligi
Yilda matematika , Yakobi polinomlari (vaqti-vaqti bilan chaqiriladi gipergeometrik polinomlar ) P (a , β ) n x )klassik ortogonal polinomlar . Ular vaznga nisbatan ortogonaldir (1 − x )a x )β oraliqda [−1, 1] . The Gegenbauer polinomlari va shuning uchun ham Legendre , Zernike va Chebyshev polinomlari , Jakobi polinomlarining alohida holatlari.[1]
Jacobi polinomlari tomonidan kiritilgan Karl Gustav Yakob Jakobi .
Ta'riflar
Gipergeometrik funktsiya orqali Yakobi polinomlari gipergeometrik funktsiya quyidagicha:[2]
P n ( a , β ) ( z ) = ( a + 1 ) n n ! 2 F 1 ( − n , 1 + a + β + n ; a + 1 ; 1 2 ( 1 − z ) ) , { displaystyle P_ {n} ^ {( alfa, beta)} (z) = { frac {( alfa +1) _ {n}} {n!}} , {} _ {2} F_ {1} chap (-n, 1 + alfa + beta + n; alfa +1; { tfrac {1} {2}} (1-z) o'ng),} qayerda ( a + 1 ) n { displaystyle ( alfa +1) _ {n}} Pochhammerning ramzi (ko'tarilayotgan faktorial uchun). Bunday holda, gipergeometrik funktsiya uchun ketma-ketlik cheklangan, shuning uchun quyidagi ekvivalent ifoda olinadi:
P n ( a , β ) ( z ) = Γ ( a + n + 1 ) n ! Γ ( a + β + n + 1 ) ∑ m = 0 n ( n m ) Γ ( a + β + n + m + 1 ) Γ ( a + m + 1 ) ( z − 1 2 ) m . { displaystyle P_ {n} ^ {( alfa, beta)} (z) = { frac { Gamma ( alfa + n + 1)} {n! , Gamma ( alfa + beta + n + 1)}} sum _ {m = 0} ^ {n} {n m} { frac { Gamma ( alfa + beta + n + m + 1)}} { Gamma ( alfa) ni tanlang + m + 1)}} chap ({ frac {z-1} {2}} o'ng) ^ {m}.} Rodrigesning formulasi Ekvivalent ta'rifi tomonidan berilgan Rodrigesning formulasi :[1] [3]
P n ( a , β ) ( z ) = ( − 1 ) n 2 n n ! ( 1 − z ) − a ( 1 + z ) − β d n d z n { ( 1 − z ) a ( 1 + z ) β ( 1 − z 2 ) n } . { displaystyle P_ {n} ^ {( alfa, beta)} (z) = { frac {(-1) ^ {n}} {2 ^ {n} n!}} (1-z) ^ {- alfa} (1 + z) ^ {- beta} { frac {d ^ {n}} {dz ^ {n}}} left {(1-z) ^ { alpha} (1 + z) ^ { beta} chap (1-z ^ {2} o'ng) ^ {n} o'ng }.} Agar a = β = 0 { displaystyle alpha = beta = 0} Legendre polinomlari :
P n ( z ) = 1 2 n n ! d n d z n ( z 2 − 1 ) n . { displaystyle P_ {n} (z) = { frac {1} {2 ^ {n} n!}} { frac {d ^ {n}} {dz ^ {n}}} (z ^ {2 } -1) ^ {n} ;.} Haqiqiy argument uchun muqobil ifoda Haqiqatdan x muqobil ravishda Jakobi polinomini quyidagicha yozish mumkin
P n ( a , β ) ( x ) = ∑ s = 0 n ( n + a n − s ) ( n + β s ) ( x − 1 2 ) s ( x + 1 2 ) n − s { displaystyle P_ {n} ^ {( alfa, beta)} (x) = sum _ {s = 0} ^ {n} {n + alpha ns}} {n + beta s} ni tanlang} chap ({ frac {x-1} {2}} o'ng) ^ {s} chap ({ frac {x + 1} {2}} o'ng) ^ {ns}} va butun son uchun n
( z n ) = { Γ ( z + 1 ) Γ ( n + 1 ) Γ ( z − n + 1 ) n ≥ 0 0 n < 0 { displaystyle {z select n} = { begin {case} { frac { Gamma (z + 1)} { Gamma (n + 1) Gamma (z-n + 1)}} & n geq 0 0 & n <0 end {case}}} qayerda Γ (z ) bo'ladi Gamma funktsiyasi .
Maxsus holatda to'rtta miqdor n , n + a , n + β n + a + β
P n ( a , β ) ( x ) = ( n + a ) ! ( n + β ) ! ∑ s = 0 n 1 s ! ( n + a − s ) ! ( β + s ) ! ( n − s ) ! ( x − 1 2 ) n − s ( x + 1 2 ) s . { displaystyle P_ {n} ^ {( alfa, beta)} (x) = (n + alfa)! (n + beta)! sum _ {s = 0} ^ {n} { frac {1 } {s! (n + alfa -s)! ( beta + s)! (ns)!}} chap ({ frac {x-1} {2}} right) ^ {ns} left ( { frac {x + 1} {2}} right) ^ {s}.} (1 )
Yig‘indagi qiymatlar butun songa teng bo‘ladi s buning uchun faktoriallarning dalillari salbiy emas.
Maxsus holatlar P 0 ( a , β ) ( z ) = 1 , { displaystyle P_ {0} ^ {( alfa, beta)} (z) = 1,} P 1 ( a , β ) ( z ) = ( a + 1 ) + ( a + β + 2 ) z − 1 2 , { displaystyle P_ {1} ^ {( alfa, beta)} (z) = ( alfa +1) + ( alfa + beta +2) { frac {z-1} {2}}, } P 2 ( a , β ) ( z ) = ( a + 1 ) ( a + 2 ) 2 + ( a + 2 ) ( a + β + 3 ) z − 1 2 + ( a + β + 3 ) ( a + β + 4 ) 2 ( z − 1 2 ) 2 , . . . { displaystyle P_ {2} ^ {( alfa, beta)} (z) = { frac {( alfa +1) ( alfa +2)} {2}} + ( alfa +2) { alfa + beta +3) { frac {z-1} {2}} + { frac {( alfa + beta +3) ( alfa + beta +4)} {2}} chap ({ frac {z-1} {2}} o'ng) ^ {2}, ...} Asosiy xususiyatlar
Ortogonallik Yakobi polinomlari ortogonallik shartini qondiradi
∫ − 1 1 ( 1 − x ) a ( 1 + x ) β P m ( a , β ) ( x ) P n ( a , β ) ( x ) d x = 2 a + β + 1 2 n + a + β + 1 Γ ( n + a + 1 ) Γ ( n + β + 1 ) Γ ( n + a + β + 1 ) n ! δ n m , a , β > − 1. { displaystyle int _ {- 1} ^ {1} (1-x) ^ { alpha} (1 + x) ^ { beta} P_ {m} ^ {( alfa, beta)} (x ) P_ {n} ^ {( alfa, beta)} (x) , dx = { frac {2 ^ { alpha + beta +1}} {2n + alfa + beta +1}} { frac { Gamma (n + alfa +1) Gamma (n + beta +1)} { Gamma (n + alfa + beta +1) n!}} delta _ {nm}, qquad alpha , beta> -1.} Belgilanganidek, ularning vazni bo'yicha birlik normasi yo'q. Buni yuqoridagi tenglamaning o'ng tomonining kvadrat ildiziga bo'linib, qachon tuzatish mumkin n = m { displaystyle n = m}
Garchi u ortonormal asosga ega bo'lmasa-da, ba'zida soddaligi tufayli alternativ normallashtirishga ustunlik beriladi:
P n ( a , β ) ( 1 ) = ( n + a n ) . { displaystyle P_ {n} ^ {( alfa, beta)} (1) = {n + alfa n} ni tanlang.} Simmetriya munosabati Polinomlar simmetriya munosabatiga ega
P n ( a , β ) ( − z ) = ( − 1 ) n P n ( β , a ) ( z ) ; { displaystyle P_ {n} ^ {( alfa, beta)} (- z) = (- 1) ^ {n} P_ {n} ^ {( beta, alfa)} (z);} shuning uchun boshqa terminal qiymati
P n ( a , β ) ( − 1 ) = ( − 1 ) n ( n + β n ) . { displaystyle P_ {n} ^ {( alfa, beta)} (- 1) = (- 1) ^ {n} {n + beta n} ni tanlang.} Hosilalari The k aniq ifodaning th hosilasi olib keladi
d k d z k P n ( a , β ) ( z ) = Γ ( a + β + n + 1 + k ) 2 k Γ ( a + β + n + 1 ) P n − k ( a + k , β + k ) ( z ) . { displaystyle { frac {d ^ {k}} {dz ^ {k}}} P_ {n} ^ {( alfa, beta)} (z) = { frac { Gamma ( alfa + ) beta + n + 1 + k)} {2 ^ {k} Gamma ( alfa + beta + n + 1)}} P_ {nk} ^ {( alfa + k, beta + k)} (z ).} Differentsial tenglama Yakobiy polinom P (a , β ) n chiziqli bir hil differentsial tenglama [1]
( 1 − x 2 ) y ″ + ( β − a − ( a + β + 2 ) x ) y ′ + n ( n + a + β + 1 ) y = 0. { displaystyle chap (1-x ^ {2} o'ng) y '' + ( beta - alfa - ( alfa + beta +2) x) y '+ n (n + alfa + beta + 1) y = 0.} Takrorlanish munosabatlari The takrorlanish munosabati yakobi polinomlari uchun belgilangan a ,β bu:[1]
2 n ( n + a + β ) ( 2 n + a + β − 2 ) P n ( a , β ) ( z ) = ( 2 n + a + β − 1 ) { ( 2 n + a + β ) ( 2 n + a + β − 2 ) z + a 2 − β 2 } P n − 1 ( a , β ) ( z ) − 2 ( n + a − 1 ) ( n + β − 1 ) ( 2 n + a + β ) P n − 2 ( a , β ) ( z ) , { displaystyle { begin {aligned} & 2n (n + alfa + beta) (2n + alfa + beta -2) P_ {n} ^ {( alfa, beta)} (z) & qquad = (2n + alfa + beta -1) { Big {} (2n + alfa + beta) (2n + alfa + beta -2) z + alfa ^ {2} - beta ^ {2} { Big }} P_ {n-1} ^ {( alfa, beta)} (z) -2 (n + alfa -1) (n + beta -1) (2n + alfa + beta) P_ { n-2} ^ {( alfa, beta)} (z), end {hizalanmış}}} uchun n = 2, 3, ....
Yakobi polinomlarini gipergeometrik funktsiya nuqtai nazaridan tavsiflash mumkin bo'lganligi sababli, gipergeometrik funktsiyani takrorlashlari Jakobi polinomlarining ekvivalent takrorlanishlarini beradi. Xususan, Gaussning tutashgan munosabatlari o'zliklariga mos keladi
( z − 1 ) d d z P n ( a , β ) ( z ) = 1 2 ( z − 1 ) ( 1 + a + β + n ) P n − 1 ( a + 1 , β + 1 ) = n P n ( a , β ) − ( a + n ) P n − 1 ( a , β + 1 ) = ( 1 + a + β + n ) ( P n ( a , β + 1 ) − P n ( a , β ) ) = ( a + n ) P n ( a − 1 , β + 1 ) − a P n ( a , β ) = 2 ( n + 1 ) P n + 1 ( a , β − 1 ) − ( z ( 1 + a + β + n ) + a + 1 + n − β ) P n ( a , β ) 1 + z = ( 2 β + n + n z ) P n ( a , β ) − 2 ( β + n ) P n ( a , β − 1 ) 1 + z = 1 − z 1 + z ( β P n ( a , β ) − ( β + n ) P n ( a + 1 , β − 1 ) ) . { displaystyle { begin {aligned} (z-1) { frac {d} {dz}} P_ {n} ^ {( alpha, beta)} (z) & = { frac {1} { 2}} (z-1) (1+ alfa + beta + n) P_ {n-1} ^ {( alfa +1, beta +1)} & = nP_ {n} ^ {( alfa, beta)} - ( alfa + n) P_ {n-1} ^ {( alfa, beta +1)} & = (1+ alfa + beta + n) left ( P_ {n} ^ {( alfa, beta +1)} - P_ {n} ^ {( alfa, beta)} right) & = ( alfa + n) P_ {n} ^ { ( alfa -1, beta +1)} - alfa P_ {n} ^ {( alfa, beta)} & = { frac {2 (n + 1) P_ {n + 1} ^ {( alfa, beta -1)} - chap (z (1+ alfa + beta + n) + alfa + 1 + n- beta o'ng) P_ {n} ^ {( alfa, beta)}} {1 + z}} & = { frac {(2 beta + n + nz) P_ {n} ^ {( alfa, beta)} - 2 ( beta + n) P_ {n} ^ {( alfa, beta -1)}} {1 + z}} & = { frac {1-z} {1 + z}} chap ( beta P_ {n} ^ {( alfa, beta)} - ((beta + n) P_ {n} ^ {( alfa +1, beta -1)} o'ng) ,. end {hizalangan}}} Yaratuvchi funktsiya The ishlab chiqarish funktsiyasi Jacobi polinomlari tomonidan berilgan
∑ n = 0 ∞ P n ( a , β ) ( z ) t n = 2 a + β R − 1 ( 1 − t + R ) − a ( 1 + t + R ) − β , { displaystyle sum _ {n = 0} ^ { infty} P_ {n} ^ {( alfa, beta)} (z) t ^ {n} = 2 ^ { alfa + beta} R ^ {-1} (1-t + R) ^ {- alfa} (1 + t + R) ^ {- beta},} qayerda
R = R ( z , t ) = ( 1 − 2 z t + t 2 ) 1 2 , { displaystyle R = R (z, t) = chap (1-2zt + t ^ {2} o'ng) ^ { frac {1} {2}} ~,} va filial kvadrat ildiz shunday tanlangan R (z , 0) = 1.[1]
Jakobi polinomlarining asimptotikasi
Uchun x ning ichki qismida [−1, 1] , ning asimptotikasi P (a , β ) n n Darbux formulasi bilan berilgan[1]
P n ( a , β ) ( cos θ ) = n − 1 2 k ( θ ) cos ( N θ + γ ) + O ( n − 3 2 ) , { displaystyle P_ {n} ^ {( alfa, beta)} ( cos theta) = n ^ {- { frac {1} {2}}} k ( theta) cos (N theta) + gamma) + O chap (n ^ {- { frac {3} {2}}} o'ng),} qayerda
k ( θ ) = π − 1 2 gunoh − a − 1 2 θ 2 cos − β − 1 2 θ 2 , N = n + 1 2 ( a + β + 1 ) , γ = − π 2 ( a + 1 2 ) , { displaystyle { begin {aligned} k ( theta) & = pi ^ {- { frac {1} {2}}} sin ^ {- alpha - { frac {1} {2}} } { tfrac { theta} {2}} cos ^ {- beta - { frac {1} {2}}} { tfrac { theta} {2}}, N & = n + { tfrac {1} {2}} ( alfa + beta +1), gamma & = - { tfrac { pi} {2}} left ( alpha + { tfrac {1} {2 }} right), end {hizalangan}}} va "O "muddat [ε, π -ε] har ε> 0 uchun.
± 1 nuqtalari yaqinidagi Jakobi polinomlarining asimptotikasi Mehler-Geyn formulasi
lim n → ∞ n − a P n ( a , β ) ( cos ( z n ) ) = ( z 2 ) − a J a ( z ) lim n → ∞ n − β P n ( a , β ) ( cos ( π − z n ) ) = ( z 2 ) − β J β ( z ) { displaystyle { begin {aligned} lim _ {n to infty} n ^ {- alpha} P_ {n} ^ {( alfa, beta)}} chap ( cos left ({ tfrac {z} {n}} right) right) & = left ({ tfrac {z} {2}} right) ^ {- alpha} J _ { alpha} (z) lim _ {n to infty} n ^ {- beta} P_ {n} ^ {( alfa, beta)} chap ( cos left ( pi - { tfrac {z} {n}} o'ng) o'ng) va = chap ({ tfrac {z} {2}} o'ng) ^ {- beta} J _ { beta} (z) end {hizalangan}}} bu erda chegaralar bir xil z chegaralangan holda domen .
Tashqaridagi asimptotiklar [−1, 1] aniqroq emas.
Ilovalar
Wigner d-matritsasi Ifoda (1 Wigner d-matritsasi d j m ’,m π ) jakobi polinomlari bo'yicha:[4]
d m ′ m j ( ϕ ) = [ ( j + m ) ! ( j − m ) ! ( j + m ′ ) ! ( j − m ′ ) ! ] 1 2 ( gunoh ϕ 2 ) m − m ′ ( cos ϕ 2 ) m + m ′ P j − m ( m − m ′ , m + m ′ ) ( cos ϕ ) . { displaystyle d_ {m'm} ^ {j} ( phi) = chap [{ frac {(j + m)! (jm)!} {(j + m ')! (j-m') !}} o'ng] ^ { frac {1} {2}} chap ( sin { tfrac { phi} {2}} o'ng) ^ {m-m '} chap ( cos { tfrac { phi} {2}} o'ng) ^ {m + m '} P_ {jm} ^ {(m-m', m + m ')} ( cos phi).} Shuningdek qarang
Izohlar
^ a b v d e f Cheze, Gábor (1939). "IV. Jakobi polinomlari.". Ortogonal polinomlar XXIII . Amerika matematik jamiyati. ISBN 978-0-8218-1023-1 JANOB 0372517 . ^ Abramovits, Milton ; Stegun, Irene Ann , eds. (1983) [1964 yil iyun]. "22-bob" . Matematik funktsiyalar uchun formulalar, grafikalar va matematik jadvallar bilan qo'llanma 55 (To'qqizinchi o'ninchi asl nashrning tuzatishlar bilan qo'shimcha tuzatishlar bilan qayta nashr etilishi (1972 yil dekabr); birinchi nashr). Vashington Kolumbiyasi; Nyu-York: Amerika Qo'shma Shtatlari Savdo vazirligi, Milliy standartlar byurosi; Dover nashrlari. p. 561. ISBN 978-0-486-61272-0 LCCN 64-60036 . JANOB 0167642 . LCCN 65-12253 .^ P.K. Suetin (2001) [1994], "Jacobi_polynomials" , Matematika entsiklopediyasi EMS Press ^ Biedenharn, L.C .; Louck, JD (1981). Kvant fizikasidagi burchakli momentum . O'qish: Addison-Uesli. Qo'shimcha o'qish
Endryus, Jorj E.; Askey, Richard; Roy, Ranjan (1999), Maxsus funktsiyalar , Matematika entsiklopediyasi va uning qo'llanilishi, 71 , Kembrij universiteti matbuoti , ISBN 978-0-521-62321-6 JANOB 1688958 , ISBN 978-0-521-78988-2 Koornwinder, Tom X.; Vong, Roderik S. S.; Koekoek, Roelof; Svartov, René F. (2010), "Ortogonal polinomlar" , yilda Olver, Frank V. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Klark, Charlz V. (tahr.), NIST Matematik funktsiyalar bo'yicha qo'llanma ISBN 978-0-521-19225-5 JANOB 2723248 Tashqi havolalar
![d _ {{m'm}} ^ {j} ( phi) = left [{ frac {(j + m)! (jm)!} {(j + m ')! (j-m')! }} o'ng] ^ {{{ frac {1} {2}}}} chap ( sin { tfrac { phi} {2}} o'ng) ^ {{m-m '}} chap ( cos { tfrac { phi} {2}} o'ng) ^ {{m + m '}} P _ {{jm}} ^ {{(m-m', m + m ')}} ( cos phi).](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c078b36a1cb81f94eacdd3ffd60597f378e5847)