65.2 Functions and Variables for orthogonal polynomials

Function: assoc_legendre_p (n, m, x) ¶

The associated Legendre function of the first kind of degree n and order m.

Reference: Abramowitz and Stegun, equations 22.5.37, page 779, 8.6.6 (second equation), page 334, and 8.2.5, page 333.

Function: assoc_legendre_q (n, m, x) ¶

The associated Legendre function of the second kind of degree n and order m.

Reference: Abramowitz and Stegun, equation 8.5.3 and 8.1.8.

Function: chebyshev_t (n, x) ¶

The Chebyshev function of the first kind.

Reference: Abramowitz and Stegun, equation 22.5.47, page 779.

Function: chebyshev_u (n, x) ¶

The Chebyshev function of the second kind.

Reference: Abramowitz and Stegun, equation 22.5.48, page 779.

Function: gen_laguerre (n, a, x) ¶

The generalized Laguerre polynomial of degree n.

Reference: Abramowitz and Stegun, equation 22.5.54, page 780.

Function: hermite (n, x) ¶

The Hermite polynomial.

Reference: Abramowitz and Stegun, equation 22.5.55, page 780.

Function: intervalp (e) ¶

Return true if the input is an interval and return false if it isn’t.

Function: jacobi_p (n, a, b, x) ¶

The Jacobi polynomial.

The Jacobi polynomials are actually defined for all a and b; however, the Jacobi polynomial weight (1 - x)^a (1 + x)^b isn’t integrable for a <= -1 or b <= -1.

Reference: Abramowitz and Stegun, equation 22.5.42, page 779.

Function: laguerre (n, x) ¶

The Laguerre polynomial.

Reference: Abramowitz and Stegun, equations 22.5.16 and 22.5.54, page 780.

Function: legendre_p (n, x) ¶

The Legendre polynomial of the first kind.

Reference: Abramowitz and Stegun, equations 22.5.50 and 22.5.51, page 779.

Function: legendre_q (n, x) ¶

The Legendre polynomial of the first kind.

Reference: Abramowitz and Stegun, equations 8.5.3 and 8.1.8.

Function: orthopoly_recur (f, args) ¶

Returns a recursion relation for the orthogonal function family f with arguments args. The recursion is with respect to the polynomial degree.

(%i1) orthopoly_recur (legendre_p, [n, x]);

                (2 n - 1) P     (x) x + (1 - n) P     (x)
                           n - 1                 n - 2
(%o1)   P (x) = -----------------------------------------
         n                          n

The second argument to orthopoly_recur must be a list with the correct number of arguments for the function f; if it isn’t, Maxima signals an error.

(%i1) orthopoly_recur (jacobi_p, [n, x]);

Function jacobi_p needs 4 arguments, instead it received 2
 -- an error.  Quitting.  To debug this try debugmode(true);

Additionally, when f isn’t the name of one of the families of orthogonal polynomials, an error is signalled.

(%i1) orthopoly_recur (foo, [n, x]);

A recursion relation for foo isn't known to Maxima
 -- an error.  Quitting.  To debug this try debugmode(true);

Variable: orthopoly_returns_intervals ¶

Default value: true

When orthopoly_returns_intervals is true, floating point results are returned in the form interval (c, r), where c is the center of an interval and r is its radius. The center can be a complex number; in that case, the interval is a disk in the complex plane.

Function: orthopoly_weight (f, args) ¶

Returns a three element list; the first element is the formula of the weight for the orthogonal polynomial family f with arguments given by the list args; the second and third elements give the lower and upper endpoints of the interval of orthogonality. For example,

(%i1) w : orthopoly_weight (hermite, [n, x]);
                            2
                         - x
(%o1)                 [%e    , - inf, inf]
(%i2) integrate(w[1]*hermite(3, x)*hermite(2, x), x, w[2], w[3]);
(%o2)                           0

The main variable of f must be a symbol; if it isn’t, Maxima signals an error.

Function: pochhammer (n, x) ¶

The Pochhammer symbol. For nonnegative integers n with n <= pochhammer_max_index, the expression pochhammer (x, n) evaluates to the product x (x + 1) (x + 2) ... (x + n - 1) when n > 0 and to 1 when n = 0. For negative n, pochhammer (x, n) is defined as (-1)^n / pochhammer (1 - x, -n). Thus

(%i1) pochhammer (x, 3);
(%o1)                   x (x + 1) (x + 2)
(%i2) pochhammer (x, -3);
                                 1
(%o2)               - -----------------------
                      (1 - x) (2 - x) (3 - x)

To convert a Pochhammer symbol into a quotient of gamma functions, (see Abramowitz and Stegun, equation 6.1.22) use makegamma, for example

(%i1) makegamma (pochhammer (x, n));
                          gamma(x + n)
(%o1)                     ------------
                            gamma(x)

When n exceeds pochhammer_max_index or when n is symbolic, pochhammer returns a noun form.

(%i1) pochhammer (x, n);
(%o1)                         (x)
                                 n

Variable: pochhammer_max_index ¶

Default value: 100

pochhammer (n, x) expands to a product if and only if n <= pochhammer_max_index.

Examples:

(%i1) pochhammer (x, 3), pochhammer_max_index : 3;
(%o1)                   x (x + 1) (x + 2)
(%i2) pochhammer (x, 4), pochhammer_max_index : 3;
(%o2)                         (x)
                                 4

Reference: Abramowitz and Stegun, equation 6.1.16, page 256.

Function: spherical_bessel_j (n, x) ¶

The spherical Bessel function of the first kind.

Reference: Abramowitz and Stegun, equations 10.1.8, page 437 and 10.1.15, page 439.

Function: spherical_bessel_y (n, x) ¶

The spherical Bessel function of the second kind.

Reference: Abramowitz and Stegun, equations 10.1.9, page 437 and 10.1.15, page 439.

Function: spherical_hankel1 (n, x) ¶

The spherical Hankel function of the first kind.

Reference: Abramowitz and Stegun, equation 10.1.36, page 439.

Function: spherical_hankel2 (n, x) ¶

The spherical Hankel function of the second kind.

Reference: Abramowitz and Stegun, equation 10.1.17, page 439.

Function: spherical_harmonic (n, m, x, y) ¶

The spherical harmonic function.

Reference: Merzbacher 9.64.

Function: unit_step (x) ¶

The left-continuous unit step function; thus unit_step (x) vanishes for x <= 0 and equals 1 for x > 0.

If you want a unit step function that takes on the value 1/2 at zero, use (1 + signum (x))/2.

Function: ultraspherical (n, a, x) ¶

The ultraspherical polynomial (also known as the Gegenbauer polynomial).

Reference: Abramowitz and Stegun, equation 22.5.46, page 779.