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1: 18.3 Definitions
§18.3 Definitions
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …
Name p n ( x ) ( a , b ) w ( x ) h n k n k ~ n / k n Constraints
Jacobi P n ( α , β ) ( x ) ( 1 , 1 ) ( 1 x ) α ( 1 + x ) β 𝒜 n ( n + α + β + 1 ) n 2 n n ! n ( α β ) 2 n + α + β α , β > 1
For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). … For 1 β > α > 1 a finite system of Jacobi polynomials P n ( α , β ) ( x ) is orthogonal on ( 1 , ) with weight function w ( x ) = ( x 1 ) α ( x + 1 ) β . For ν and N > 1 2 a finite system of Jacobi polynomials P n ( N 1 + i ν , N 1 i ν ) ( i x ) (called pseudo Jacobi polynomials or Routh–Romanovski polynomials) is orthogonal on ( , ) with w ( x ) = ( 1 + x 2 ) N 1 e 2 ν arctan x . …
2: 18.7 Interrelations and Limit Relations
18.7.3 T n ( x ) = P n ( 1 2 , 1 2 ) ( x ) / P n ( 1 2 , 1 2 ) ( 1 ) ,
18.7.4 U n ( x ) = C n ( 1 ) ( x ) = ( n + 1 ) P n ( 1 2 , 1 2 ) ( x ) / P n ( 1 2 , 1 2 ) ( 1 ) ,
18.7.13 P 2 n ( α , α ) ( x ) P 2 n ( α , α ) ( 1 ) = P n ( α , 1 2 ) ( 2 x 2 1 ) P n ( α , 1 2 ) ( 1 ) ,
18.7.14 P 2 n + 1 ( α , α ) ( x ) P 2 n + 1 ( α , α ) ( 1 ) = x P n ( α , 1 2 ) ( 2 x 2 1 ) P n ( α , 1 2 ) ( 1 ) .
18.7.23 lim α α 1 2 n P n ( α , α ) ( α 1 2 x ) = H n ( x ) 2 n n ! .
3: 18.14 Inequalities
Jacobi
Jacobi
Let R n ( x ) = P n ( α , β ) ( x ) / P n ( α , β ) ( 1 ) . …
Jacobi
Szegő–Szász Inequality
4: 18.6 Symmetry, Special Values, and Limits to Monomials
Table 18.6.1: Classical OP’s: symmetry and special values.
p n ( x ) p n ( x ) p n ( 1 ) p 2 n ( 0 ) p 2 n + 1 ( 0 )
P n ( α , β ) ( x ) ( 1 ) n P n ( β , α ) ( x ) ( α + 1 ) n / n !
P n ( α , α ) ( x ) ( 1 ) n P n ( α , α ) ( x ) ( α + 1 ) n / n ! ( 1 4 ) n ( n + α + 1 ) n / n ! ( 1 4 ) n ( n + α + 1 ) n + 1 / n !
§18.6(ii) Limits to Monomials
18.6.2 lim α P n ( α , β ) ( x ) P n ( α , β ) ( 1 ) = ( 1 + x 2 ) n ,
18.6.3 lim β P n ( α , β ) ( x ) P n ( α , β ) ( 1 ) = ( 1 x 2 ) n ,
5: 18.4 Graphics
See accompanying text
Figure 18.4.1: Jacobi polynomials P n ( 1.5 , 0.5 ) ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify
See accompanying text
Figure 18.4.2: Jacobi polynomials P n ( 1.25 , 0.75 ) ( x ) , n = 7 , 8 . This illustrates inequalities for extrema of a Jacobi polynomial; see (18.14.16). … Magnify
6: 18.9 Recurrence Relations and Derivatives
For p n ( x ) = P n ( α , β ) ( x ) , … For p n ( x ) = P n ( α , β ) ( x ) , …
18.9.3 P n ( α , β 1 ) ( x ) P n ( α 1 , β ) ( x ) = P n 1 ( α , β ) ( x ) ,
18.9.4 ( 1 x ) P n ( α + 1 , β ) ( x ) + ( 1 + x ) P n ( α , β + 1 ) ( x ) = 2 P n ( α , β ) ( x ) ,
Jacobi
7: 18.10 Integral Representations
18.10.1 P n ( α , α ) ( cos θ ) P n ( α , α ) ( 1 ) = C n ( α + 1 2 ) ( cos θ ) C n ( α + 1 2 ) ( 1 ) = 2 α + 1 2 Γ ( α + 1 ) π 1 2 Γ ( α + 1 2 ) ( sin θ ) 2 α 0 θ cos ( ( n + α + 1 2 ) ϕ ) ( cos ϕ cos θ ) α + 1 2 d ϕ , 0 < θ < π , α > 1 2 .
Generalizations of (18.10.1) for P n ( α , β ) are given in Gasper (1975, (6),(8)) and Koornwinder (1975a, (5.7),(5.8)). …
Jacobi
18.10.4 P n ( α , α ) ( cos θ ) P n ( α , α ) ( 1 ) = C n ( α + 1 2 ) ( cos θ ) C n ( α + 1 2 ) ( 1 ) = Γ ( α + 1 ) π 1 2 Γ ( α + 1 2 ) 0 π ( cos θ + i sin θ cos ϕ ) n ( sin ϕ ) 2 α d ϕ , α > 1 2 .
Table 18.10.1: Classical OP’s: contour integral representations (18.10.8).
p n ( x ) g 0 ( x ) g 1 ( z , x ) g 2 ( z , x ) c Conditions
P n ( α , β ) ( x ) ( 1 x ) α ( 1 + x ) β z 2 1 2 ( z x ) ( 1 z ) α ( 1 + z ) β x ± 1 outside C .
8: 18.12 Generating Functions
Jacobi
18.12.1 2 α + β R ( 1 + R z ) α ( 1 + R + z ) β = n = 0 P n ( α , β ) ( x ) z n , R = 1 2 x z + z 2 , | z | < 1 ,
18.12.3 ( 1 + z ) α β 1 F 1 2 ( 1 2 ( α + β + 1 ) , 1 2 ( α + β + 2 ) β + 1 ; 2 ( x + 1 ) z ( 1 + z ) 2 ) = n = 0 ( α + β + 1 ) n ( β + 1 ) n P n ( α , β ) ( x ) z n , | z | < 1 ,
18.12.3_5 1 + z ( 1 2 x z + z 2 ) β + 3 2 = n = 0 ( 2 β + 2 ) n ( β + 1 ) n P n ( β + 1 , β ) ( x ) z n , | z | < 1 ,
18.12.4 ( 1 2 x z + z 2 ) λ = n = 0 C n ( λ ) ( x ) z n = n = 0 ( 2 λ ) n ( λ + 1 2 ) n P n ( λ 1 2 , λ 1 2 ) ( x ) z n , | z | < 1 .
9: 18.8 Differential Equations
Table 18.8.1: Classical OP’s: differential equations A ( x ) f ′′ ( x ) + B ( x ) f ( x ) + C ( x ) f ( x ) + λ n f ( x ) = 0 .
# f ( x ) A ( x ) B ( x ) C ( x ) λ n
1 P n ( α , β ) ( x ) 1 x 2 β α ( α + β + 2 ) x 0 n ( n + α + β + 1 )
2 ( sin 1 2 x ) α + 1 2 ( cos 1 2 x ) β + 1 2 × P n ( α , β ) ( cos x ) 1 0 1 4 α 2 4 sin 2 1 2 x + 1 4 β 2 4 cos 2 1 2 x ( n + 1 2 ( α + β + 1 ) ) 2
3 ( sin x ) α + 1 2 P n ( α , α ) ( cos x ) 1 0 ( 1 4 α 2 ) / sin 2 x ( n + α + 1 2 ) 2
10: 18.1 Notation
  • Jacobi: P n ( α , β ) ( x ) .

  • Big q -Jacobi: P n ( x ; a , b , c ; q ) .

  • Little q -Jacobi: p n ( x ; a , b ; q ) .

  • Nor do we consider the shifted Jacobi polynomials:
    18.1.2 G n ( p , q , x ) = n ! ( n + p ) n P n ( p q , q 1 ) ( 2 x 1 ) ,