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1: 18.3 Definitions
§18.3 Definitions
Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§18.2(i) and 18.2(iii)). …
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
Laguerre L n ( α ) ( x ) ( 0 , ) e - x x α Γ ( n + α + 1 ) / n ! ( - 1 ) n / n ! - n ( n + α ) α > - 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). …
2: 18.4 Graphics
See accompanying text
Figure 18.4.5: Laguerre polynomials L n ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify
See accompanying text
Figure 18.4.6: Laguerre polynomials L 3 ( α ) ( x ) , α = 0 , 1 , 2 , 3 , 4 . Magnify
See accompanying text
Figure 18.4.8: Laguerre polynomials L 3 ( α ) ( x ) , 0 α 3 , 0 x 10 . Magnify 3D Help
See accompanying text
Figure 18.4.9: Laguerre polynomials L 4 ( α ) ( x ) , 0 α 3 , 0 x 10 . Magnify 3D Help
3: 18.14 Inequalities
Laguerre
Laguerre
18.14.12 ( L n ( α ) ( x ) ) 2 L n - 1 ( α ) ( x ) L n + 1 ( α ) ( x ) , 0 x < , α 0 .
Laguerre
18.14.24 | L n ( α ) ( x n , 0 ) | < | L n ( α ) ( x n , 1 ) | < < | L n ( α ) ( x n , n - 1 ) | .
4: 18.41 Tables
For P n ( x ) ( = P n ( x ) ) see §14.33. Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates T n ( x ) , U n ( x ) , L n ( x ) , and H n ( x ) for n = 0 ( 1 ) 12 . The ranges of x are 0.2 ( .2 ) 1 for T n ( x ) and U n ( x ) , and 0.5 , 1 , 3 , 5 , 10 for L n ( x ) and H n ( x ) . … For P n ( x ) , L n ( x ) , and H n ( x ) see §3.5(v). …
5: 18.9 Recurrence Relations and Derivatives
18.9.13 L n ( α ) ( x ) = L n ( α + 1 ) ( x ) - L n - 1 ( α + 1 ) ( x ) ,
18.9.14 x L n ( α + 1 ) ( x ) = - ( n + 1 ) L n + 1 ( α ) ( x ) + ( n + α + 1 ) L n ( α ) ( x ) .
Laguerre
18.9.23 d d x L n ( α ) ( x ) = - L n - 1 ( α + 1 ) ( x ) ,
18.9.24 d d x ( e - x x α L n ( α ) ( x ) ) = ( n + 1 ) e - x x α - 1 L n + 1 ( α - 1 ) ( x ) .
6: 18.17 Integrals
Laguerre
Laguerre
Laguerre
Laguerre
Laguerre
7: 18.6 Symmetry, Special Values, and Limits to Monomials
Laguerre
18.6.1 L n ( α ) ( 0 ) = ( α + 1 ) n n ! .
§18.6(ii) Limits to Monomials
18.6.5 lim α L n ( α ) ( α x ) L n ( α ) ( 0 ) = ( 1 - x ) n .
8: 18.18 Sums
18.18.10 L n ( α 1 + + α r + r - 1 ) ( x 1 + + x r ) = m 1 + + m r = n L m 1 ( α 1 ) ( x 1 ) L m r ( α r ) ( x r ) .
18.18.12 L n ( α ) ( λ x ) L n ( α ) ( 0 ) = = 0 n ( n ) λ ( 1 - λ ) n - L ( α ) ( x ) L ( α ) ( 0 ) .
Laguerre
18.18.37 = 0 n L ( α ) ( x ) = L n ( α + 1 ) ( x ) ,
18.18.38 = 0 n L ( α ) ( x ) L n - ( β ) ( y ) = L n ( α + β + 1 ) ( x + y ) .
9: 18.1 Notation
Classical OP’s
  • Laguerre: L n ( α ) ( x ) and L n ( x ) = L n ( 0 ) ( x ) . ( L n ( α ) ( x ) with α 0 is also called Generalized Laguerre.)

  • q -Laguerre: L n ( α ) ( x ; q ) .

  • 10: 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
    L n ( α ) ( x ) x α + 1 - x 0 n
    e - 1 2 x 2 x α + 1 2 L n ( α ) ( x 2 ) 1 0 - x 2 + ( 1 4 - α 2 ) x - 2 4 n + 2 α + 2