The file OrthogonalPolynomials.mth defines functions for generating orthogonal polynomials.  These polynomials are described in Chapter 22 of Abramowitz and Stegun [1965].  The function definitions in the file are automatically loaded when any of its functions are first used.


CHEBYCHEV_T(n, x) simplifies to the nth Chebychev polynomial of the first kind Tn(x) based on the recursion

T0(x) = 1

T1(x) = x

T(n+1)(x) = 2·x·Tn(x) - T(n-1)(x)

For example,

CHEBYCHEV_T(6, x)

simplifies to T6(x)

    6       4       2     
32·x  - 48·x  + 18·x  - 1 


CHEBYCHEV_T_LIST(n, x) simplifies to a vector of the first n Chebychev polynomials of the first kind.  For example,

CHEBYCHEV_T_LIST(7, x)

simplifies to a vector of T0(x) through T6(x).


CHEBYCHEV_U(n, x) simplifies to the nth Chebychev polynomial of the second kind Un(x) based on the recursion

U0(x) = 1

U1(x) = 2·x

U(n+1)(x) = 2·x·Un(x) - U(n-1)(x)

For example,

CHEBYCHEV_U(6, x)

simplifies to U6(x)

    6       4       2     
64·x  - 80·x  + 24·x  - 1 


CHEBYCHEV_U_LIST(n, x) simplifies to a vector of the first n Chebychev polynomials of the second kind.  For example,

CHEBYCHEV_U_LIST(7, x)

simplifies to a vector of U0(x) through U6(x).


LEGENDRE_P(n, x) simplifies to the nth Legendre polynomial Pn(x) based on the recursion

P0(x) = 1

P1(x) = x

(n+1)·P(n+1)(x) = (2·n+1)·x·Pn(x) - n·P(n-1)(x)

For example,

LEGENDRE_P(6, x)

simplifies to P6(x)

      6        4        2      
 231·x  - 315·x  + 105·x  - 5  
—————————————————————————————— 
              16               


LEGENDRE_P_LIST(n, x) simplifies to a vector of the first n Legendre polynomials.  For example,

LEGENDRE_P_LIST(7, x)

simplifies to a vector of P0(x) through P6(x).


ASSOCIATED_LEGENDRE_P(n, m, x) simplifies to the nth associated Legendre polynomial Pnm(x).  For example,

ASSOCIATED_LEGENDRE_P(4, 2, x)

simplifies to P4(2)(x)

          2      2       
 15·(1 - x )·(7·x  - 1)  
———————————————————————— 
            2            


HERMITE_H(n, x) simplifies to the nth Hermite polynomial Hn(x) based on the recursion

H0(x) = 1

H1(x) = 2·x

H(n+1)(x) = 2·x·Hn(x) - 2·n·H(n-1)(x)

For example,

HERMITE_H(6, x)

simplifies to H6(x)

    6        4        2       
64·x  - 480·x  + 720·x  - 120 


HERMITE_H_LIST(n, x) simplifies to a vector of the first n Hermite polynomials.  For example,

HERMITE_H_LIST(7, x)

simplifies to a vector of H0(x) through H6(x).


HERMITE_HE(n, x) simplifies to the nth associated Hermite polynomial Hen(x) based on the recursion

He0(x) = 1

He1(x) = x

He(n+1)(x) = x·Hen(x) - n·He(n-1)(x)

For example,

HERMITE_HE(6, x)

simplifies to He6(x)

 6       4       2      
x  - 15·x  + 45·x  - 15 


HERMITE_HE_LIST(n, x) simplifies to a vector of the first n associated Hermite polynomials.  For example,

HERMITE_HE_LIST(7, x)

simplifies to a vector of He0(x) through He6(x).


WEBER_D(n, x) simplifies to Weber's nth parabolic cylinder function Dn(x).  For example,

WEBER_D(6, x)

simplifies to

    2                             
 - x /4   6       4       2       
e      ·(x  - 15·x  + 45·x  - 15) 


LAGUERRE_L(n, x) simplifies to the nth Laguerre polynomial Ln(x) based on the recursion

L0(x) = 1

L1(x) = -x + 1

(n+1)·L(n+1)(x) = (2·n+1-x)·Ln(x) - n·L(n-1)(x)

For example,

LAGUERRE_L(4, x)

simplifies to L4(x)

  4        3                   
 x      2·x        2           
———— - —————— + 3·x  - 4·x + 1 
 24       3                    


LAGUERRE_L_LIST(n, x) simplifies to a vector of the first n Laguerre polynomials.  For example,

LAGUERRE_L_LIST(5, x)

simplifies to a vector of L0(x) through L4(x).


GENERALIZED_LAGUERRE(n, α, x) simplifies to the nth generalized Laguerre polynomial Ln(α)(x) based on the recursion

L0(α)(x) = 1

L1(α)(x) = -x + α + 1

(n+1)·L(n+1)(α)(x) = (2·n+α+1-x)·Ln(α)(x) - (n+α)·L(n-1)(α)(x)

For example,

GENERALIZED_LAGUERRE(4, 2, x)

simplifies to L4(2)(x)

  4              2              
 x      3    15·x               
———— - x  + ——————— - 20·x + 15 
 24            2                


GENERALIZED_LAGUERRE_LIST(n, α, x) simplifies to a vector of the first n generalized α Laguerre polynomials.  For example,

GENERALIZED_LAGUERRE_LIST(5, 2, x)

simplifies to a vector of L0(2)(x) through L4(2)(x).


JACOBI_P(n, α, β, x) simplifies to the nth Jacobi polynomial Pn(α, β)(x).  For example,

JACOBI_P(4, 2, 1, x)

simplifies to P4(2,1)(x)

        4       3       2             
 5·(33·x  + 12·x  - 18·x  - 4·x + 1)  
————————————————————————————————————— 
                  8                   


JACOBI_P_LIST(n, α, β, x) simplifies to a vector of the first n Jacobi (α, β) polynomials.  For example,

JACOBI_P_LIST(5, 2, 1, x)

simplifies to a vector of P0(2,1)(x) through P4(2,1)(x).


GEGENBAUER_C(n, α, x) simplifies to the nth Gegenbauer ultraspherical polynomial Cnα(x).  For example,

GEGENBAUER_C(4, 2, x)

simplifies to C4(2)(x)

    4       2     
80·x  - 48·x  + 3 


GEGENBAUER_C_LIST(n, α, x) simplifies to a vector of the first n Gegenbauer ultraspherical α polynomials.  For example,

GEGENBAUER_C_LIST(5, 2, x)

simplifies to a vector of C0(2)(x) through C4(2)(x).


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