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1: 18.3 Definitions
§18.3 Definitions
This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre.
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, 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 exact values of the coefficients of the Jacobi polynomials P n ( α , β ) ( x ) , the ultraspherical polynomials C n ( λ ) ( x ) , the Chebyshev polynomials T n ( x ) and U n ( x ) , the Legendre polynomials P n ( x ) , the Laguerre polynomials L n ( x ) , and the Hermite polynomials H n ( x ) , see Abramowitz and Stegun (1964, pp. 793–801). … However, most of these formulas can be obtained by specialization of formulas for Jacobi polynomials, via (18.7.4)–(18.7.6). …
2: 18.14 Inequalities
Jacobi
Jacobi
Let R n ( x ) = P n ( α , β ) ( x ) / P n ( α , β ) ( 1 ) . …
Jacobi
Szegő–Szász Inequality
3: 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 ) ,
18.7.5 V n ( x ) = ( 2 n + 1 ) P n ( 1 2 , - 1 2 ) ( x ) / P n ( 1 2 , - 1 2 ) ( 1 ) ,
18.7.6 W n ( x ) = 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 ) .
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: 18.9 Recurrence Relations and Derivatives
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
18.9.17 ( 2 n + α + β ) ( 1 - x 2 ) d d x P n ( α , β ) ( x ) = n ( α - β - ( 2 n + α + β ) x ) P n ( α , β ) ( x ) + 2 ( n + α ) ( n + β ) P n - 1 ( α , β ) ( x ) ,
6: 18.30 Associated OP’s
Associated Jacobi Polynomials
18.30.4 P n ( α , β ) ( x ; c ) = p n ( x ; c ) , n = 0 , 1 , ,
18.30.5 ( - 1 ) n ( α + β + c + 1 ) n n ! P n ( α , β ) ( x ; c ) ( α + β + 2 c + 1 ) n ( β + c + 1 ) n = = 0 n ( - n ) ( n + α + β + 2 c + 1 ) ( c + 1 ) ( β + c + 1 ) ( 1 2 x + 1 2 ) F 3 4 ( - n , n + + α + β + 2 c + 1 , β + c , c β + + c + 1 , + c + 1 , α + β + 2 c ; 1 ) ,
For corresponding corecursive associated Jacobi polynomials see Letessier (1995). …
18.30.6 P n ( x ; c ) = P n ( 0 , 0 ) ( x ; c ) , n = 0 , 1 , .
7: 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
8: 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
P n ( α , β ) ( x ) 1 - x 2 β - α - ( α + β + 2 ) x 0 n ( n + α + β + 1 )
( 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
( 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
9: 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 .
Jacobi
18.10.3 P n ( α , β ) ( cos θ ) P n ( α , β ) ( 1 ) = 2 Γ ( α + 1 ) π 1 2 Γ ( α - β ) Γ ( β + 1 2 ) 0 1 0 π ( ( cos 1 2 θ ) 2 - r 2 ( sin 1 2 θ ) 2 + i r sin θ cos ϕ ) n ( 1 - r 2 ) α - β - 1 r 2 β + 1 ( sin ϕ ) 2 β d ϕ d r , α > β > - 1 2 .
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 .
10: 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 .
18.12.2 ( 1 2 ( 1 - x ) z ) - 1 2 α J α ( 2 ( 1 - x ) z ) ( 1 2 ( 1 + x ) z ) - 1 2 β I β ( 2 ( 1 + x ) z ) = n = 0 P n ( α , β ) ( x ) Γ ( n + α + 1 ) Γ ( n + β + 1 ) z n .
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 ,
18.12.4 ( 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 .