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The specfun
package contains Maxima
code for the evaluation of all orthogonal polynomials listed in
Chapter 22 of Abramowitz and Stegun. These include Chebyshev,
Laguerre, Hermite, Jacobi, Legendre, and ultraspherical (Gegenbauer)
polynomials. Additionally, specfun
contains code for
spherical Bessel, spherical Hankel, and spherical harmonic functions.
The specfun
package is not part of Maxima proper;
it is loaded at request of the user via load
or automatically via the autoload
system.
The following table lists each function in specfun
,
its Maxima name, restrictions on its arguments, and a
reference to the algorithm specfun
uses to evaluate it.
With few exceptions, specfun
follows the conventions of
Abramowitz and Stegun.
In all cases, m and n must be integers.
A&S refers to Abramowitz and Stegun, Handbook of Mathematical Functions (10th printing, December 1972), G&R to Gradshteyn and Ryzhik, Table of Integrals, Series, and Products (1980 corrected and enlarged edition), and Merzbacher to Quantum Mechanics (second edition, 1970).
Function Chebyshev T Chebyshev U generalized Laguerre Laguerre Hermite Jacobi associated Legendre P associated Legendre Q Legendre P Legendre Q spherical Hankel 1st spherical Hankel 2nd spherical Bessel J spherical Bessel Y spherical harmonic ultraspherical (Gegenbauer)Maxima Name  Restrictions  Reference(s)

chebyshev_t(n, x)  n > 1  A&S 22.5.31

chebyshev_u(n, x)  n > 1  A&S 22.5.32

gen_laguerre(n,a,x)  n > 1  A&S page 789

laguerre(n,x)  n > 1  A&S 22.5.67

hermite(n,x)  n > 1  A&S 22.4.40, 22.5.41

jacobi_p(n,a,b,x)  n > 1, a, b > 1  A&S page 789

assoc_legendre_p(n,m,x)  n > 1  A&S 22.5.37, 8.6.6, 8.2.5

assoc_legendre_q(n,m,x)  n > 1, m > 1  G & R 8.706

legendre_p(n,m,x)  n > 1  A&S 22.5.35

legendre_q(n,m,x)  n > 1  A&S 8.6.19

spherical_hankel1(n, x)  n > 1  A&S 10.1.36

spherical_hankel2(n, x)  n > 1  A&S 10.1.17

spherical_bessel_j(n,x)  n > 1  A&S 10.1.8, 10.1.15

spherical_bessel_y(n,x)  n > 1  A&S 10.1.9, 10.1.15

spherical_harmonic(n,m,x,y)  n > 1, m <= n  Merzbacher 9.64

ultraspherical(n,a,x)  n > 1  A&S 22.5.27 
specfun
package is primarily intended for symbolic
computation. It is hoped that it gives accurate floating point
results as well; however, no claims are made that the algorithms
are well suited for numerical evaluation. Some effort, however,
has been made to provide
good numerical performance.
When all arguments, except for the order, are floats (but not bigfloats),
many functions in specfun
call a float modedeclared version of the
Jacobi function. This greatly speeds floating point evaluation
of the orthogonal polynomials.
specfun
handles most domain errors by returning an unevaluated
function.
No simplification rules (based on recursion relations) are defined for unevaluated functions.
It is possible for an expression
involving sums of unevaluated special functions to vanish, yet
Maxima is unable to reduce it to zero.
load ("specfun")
loads the specfun
package.
Alternatively, setup_autoload
causes the package to be loaded
when one of the specfun
functions appears in an expression.
setup_autoload
may appear at the command line or in the maximainit.mac
file.
See setup_autoload
.
An example use of specfun
is
(%i1) load ("specfun")$ (%i2) [hermite (0, x), hermite (1, x), hermite (2, x)]; 2 (%o2) [1, 2 x,  2 (1  2 x )] (%i3) diff (hermite (n, x), x); (%o3) 2 n hermite(n  1, x)
jacobi_p
, ultraspherical
, chebyshev_t
,
chebyshev_u
, and legendre_p
),
return a series representation when the order is a symbolic integer.
The series representation is not
used by specfun
for any computations,
but it may be simplified by Maxima automatically,
or it may be possible to use the series to evaluate the function
through further manipulations.
For example:
(%i1) load ("specfun")$ (%i2) legendre_p (n, x); (%o2) legendre_p(n, x) (%i3) ultraspherical (n, 3/2, 2); genfact(3, n,  1) jacobi_p(n, 1, 1, 2) (%o3)  genfact(2, n,  1) (%i4) declare (n, integer)$ (%i5) legendre_p (n, x); n  1 ==== \ n  i% (%o5) ( > binomial(n, i%) binomial(n, n  i%) (x  1) / ==== i% = 1 i% n n n (x + 1) + (x + 1) + (x  1) )/2 (%i6) ultraspherical (n, 3/2, 2); n  1 ==== \ i% (%o6) genfact(3, n,  1) ( > 3 binomial(n + 1, i%) / ==== i% = 1 n binomial(n + 1, n  i%) + (n + 1) 3 + n + 1) n /(genfact(2, n,  1) 2 )
sumhack
to true provides a fix. For example:
(%i1) load ("specfun")$ (%i2) declare (n, integer)$ (%i3) e: legendre_p(n,x)$ (%i4) ev (e, sum, n=0); Lower bound to sum: 1 is greater than the upper bound:  1  an error. Quitting. To debug this try debugmode(true); (%i5) ev (e, sum, n=0, sumhack=true); (%o5) 1
specfun
have a gradef
property;
derivatives with respect to the order or other function parameters
are undefined, and an attempt to compute such a derivative yields an error message.
The specfun
package and its documentation were written by
Barton Willis of the University of Nebraska at Kearney. It is
released under the terms of the General Public License (GPL).
Send bug reports and comments on this package to
willisb@unk.edu. In your report, please include
the Maxima version, as reported by build_info()
,
and the specfun
version, as reported by get ('specfun, 'version)
.
load ("specfun")
loads this function.
See assoc_legendre_q, legendre_p, and legendre_q.
Returns the associated Legendre function of the second kind for integers n > 1 and m > 1.
Reference: Gradshteyn and Ryzhik 8.706 page 1000.
load ("specfun")
loads this function.
See also assoc_legendre_p, legendre_p, and legendre_q.
Returns the Chebyshev function of the first kind for integers n > 1.
Reference: A&S 22.5.31 page 778 and A&S 6.1.22 page 256.
load ("specfun")
loads this function.
See also chebyshev_u.
Returns the Chebyshev function of the second kind for integers n > 1.
Reference: A&S, 22.8.3 page 783 and A&S 6.1.22 page 256.
load ("specfun")
loads this function.
See also chebyshev_t.
Returns the generalized Laguerre polynomial for integers n > 1.
load ("specfun")
loads this function.
Reference: table on page 789 in A&S.
Returns the Hermite polynomial for integers n > 1.
load ("specfun")
loads this function.
Reference: A&S 22.5.40 and 22.5.41, page 779.
Returns the Jacobi polynomial for integers n > 1 and a and b symbolic or a > 1 and b > 1. (The Jacobi polynomials are actually defined for all a and b ; however, the Jacobi polynomial weight (1x)^a(1+x)^b isn't integrable for a <= 1 or b <= 1.)
When a, b, and x are floats (but not bfloats) specfun calls a special modedeclared version of jacobi_p. For numerical values, the modedeclared version is much faster than the other version. Many functions in specfun are computed as a special case of the Jacobi polynomials; they also enjoy the speed boost from the modedeclared version of jacobi.
If n has been declared to be an integer, jacobi_p (n, a, b, x) returns a summation representation for the Jacobi function. Because Maxima simplifies 0^0 to 0 in a sum, two terms of the sum are added outside the summation.
load ("specfun")
loads this function.
Reference: table on page 789 in A&S.
Returns the Laguerre polynomial for integers n > 1.
Reference: A&S 22.5.16, page 778 and A&S page 789.
load ("specfun")
loads this function.
See also gen_laguerre.
Returns the Legendre polynomial of the first kind for integers n > 1.
Reference: A&S 22.5.35 page 779.
load ("specfun")
loads this function.
See legendre_q.
Returns the Legendre polynomial of the first kind for integers n > 1.
Reference: A&S 8.6.19 page 334.
load ("specfun")
loads this function.
See also legendre_p.
Returns the spherical Bessel function of the first kind for integers n > 1.
Reference: A&S 10.1.8 page 437 and A&S 10.1.15 page 439.
load ("specfun")
loads this function.
See also spherical_hankel1, spherical_hankel2, and spherical_bessel_y.
Returns the spherical Bessel function of the second kind for integers n > 1.
Reference: A&S 10.1.9 page 437 and 10.1.15 page 439.
load ("specfun")
loads this function.
See also spherical_hankel1, spherical_hankel2, and spherical_bessel_y.
Returns the spherical hankel function of the first kind for integers n > 1.
Reference: A&S 10.1.36 page 439.
load ("specfun")
loads this function.
See also spherical_hankel2, spherical_bessel_j, and spherical_bessel_y.
Returns the spherical hankel function of the second kind for integers n > 1.
Reference: A&S 10.1.17 page 439.
load ("specfun")
loads this function.
See also spherical_hankel1, spherical_bessel_j, and spherical_bessel_y.
Returns the spherical harmonic function for integers n > 1 and  m  <= n .
Reference: Merzbacher 9.64.
load ("specfun")
loads this function.
See also assoc_legendre_p.
Returns the ultraspherical polynomials for integers n > 1. The ultraspherical polynomials are also known as Gegenbauer polynomials.
Reference: A&S 22.5.27
load ("specfun")
loads this function.
See also jacobi_p.
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