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Orthogonal Polynomials

Introduction to Orthogonal Polynomials

The specfun package, located in the share directory, 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 following table lists each function in specfun, its Maxima name, restrictions on its arguments ( m and n must be integers), and a reference to the algorithm specfun uses to evaluate it. With few exceptions, specfun follows the conventions of Abramowitz and Stegun. Before you use specfun, check that specfun's conventions match your expectations.

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 (2ed, 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
The 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 bfloats), 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 attempt has been made to define simplification rules (based on recursion relations) for unevaluated functions. Users should be aware that it is possible for an expression involving sums of unevaluated special functions to vanish, yet Maxima is unable to reduce it to zero. Be careful. To access functions in specfun, you must first load specfun.o. Alternatively, you may append autoload statements to your init.lsp file (located in your working directory). To autoload the hermite function, for example, append 
   (defprop |$hermite| #"specfun.o" autoload) 
   (add2lnc '|$hermite| $props)
to your init.lsp file. An example use of specfun is 
(c1) load("specfun.o")$
(c2) [hermite(0,x),hermite(1,x),hermite(2,x)];
(d2) [1,2*x,-2*(1-2*x^2)]
(c3) diff(hermite(n,x),x);
(d3) 2*n*hermite(n-1,x)
When using the compiled version of specfun, be especially careful to use the correct number of function arguments; calling them with too few arguments may generate a fatal error messages. For example 
(c1) load("specfun")$
/* chebyshev_t requires two arguments. */
(c2) chebyshev_t(8);
Error: Caught fatal error [memory may be damaged]
Fast links are on: do (si::use-fast-links nil) for debugging
Error signalled by MMAPCAR.
Broken at SIMPLIFY.  Type :H for Help.
Maxima code translated into Lisp handles such errors more gracefully. If specfun.LISP is installed on your machine, the same computation results in a clear error message. For example 
(c1) load("specfun.LISP")$
(c2) chebyshev_t(8);
Error: Expected 2 args but received 1 args
Fast links are on: do (si::use-fast-links nil) for debugging
Error signalled by MACSYMA-TOP-LEVEL.
Broken at |$CHEBYSHEV_T|.  Type :H for Help.
Generally, compiled code runs faster than translated code; however, translated code may be better for program development. For some functions, when the order is symbolic but has been declared to be an integer, specfun will return a series representation. (The series representation is not used by specfun for any computations.) You may use this feature to find symbolic values for special values orthogonal polynomials. An example: 
(c1) load("specfun")$
(c2) legendre_p(n,1);
(d2)        legendre_p(n, 1)
/* Declare n to be an integer; now legendre_p(n,1) evaluates to 1. */
(c3) declare(n,integer)$
(c4) legendre_p(n,1);
(d4)        1
(c5) ultraspherical(n,3/2,1);
(d4)         (n+1)*gamma (n+3) / (2*gamma (n+2))
Although the preceding example doesn't show it, two terms of the sum are added outside the summation. Removing these two terms avoids errors associated with 0^0 terms in a sum that should evaluate to 1, but evaluate to 0 in a Maxima summation. Because the sum index runs from 1 to n - 1, the lower sum index will exceed the upper sum index when n = 0; setting sumhack to true provides a fix. For example: 
(c1) load("specfun.o")$
(c2) declare(n,integer)$
(c3) e : legendre_p(n,x)$
(c4) 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);)
(c5) ev(e,sum,n=0),sumhack : true;
(d5)         1
Most functions in specfun have a gradef property; derivatives with respect to the order or other function parameters aren't unevaluated. 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 Maxima and specfun version information. The specfun version may be found using get: 
 (c2) get('specfun,'version);
 (d2)          110
 @end example




Definitions for Orthogonal Polynomials







Function: ASSOC_LEGENDRE_P (n, m, x)

[specfun package] return the associated Legendre function
of the first kind for  integers  n > -1  and 
 m > -1.  When  | m | > n  and  n >= 0,
we have   assoc_legendre_p (n, m, x) = 0.
Reference: A&S 22.5.37 page 779,  A&S 8.6.6 (second 
equation) page 334, and A&S 8.2.5 page 333.
To access this function, load("specfun").
See  ASSOC_LEGENDRE_Q,  LEGENDRE_P, and  LEGENDRE_Q.










Function: ASSOC_LEGENDRE_Q (n, m, x)






[specfun package] return the associated Legendre function
of the second kind for  integers  n > -1  and 
 m > -1. 





Reference: Gradshteyn and Ryzhik  8.706 page 1000.





To access this function, load("specfun").





See also ASSOC_LEGENDRE_P, LEGENDRE_P, and LEGENDRE_Q.










Function: CHEBYSHEV_T (n, x)






[specfun package] return 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.





To access this function, load("specfun").





See also CHEBYSHEV_U.











Function: CHEBYSHEV_U (n, x)






[specfun package] return 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. 





To access this function, load("specfun").





See also CHEBYSHEV_T.











Function: GEN_LAGUERRE (n, a, x)






[specfun package] return the generalized Laguerre polynomial
for  integers  n > -1.





To access this function, load("specfun").





Reference: table on page 789 in A&S.










Function: HERMITE (n,x)






[specfun package] return the Hermite polynomial for
integers  n > -1.





To access this function, load("specfun").





Reference:  A&S 22.5.40 and 22.5.41, page 779.











Function: JACOBI_P (n, a, b, x)






[specfun package] return 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  (1-x)^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.





To access this function, load("specfun").





Reference:  table on page 789 in A&S.











Function: LAGUERRE (n, x)






[specfun package] return the Laguerre polynomial for 
integers  n > -1.  





Reference: A&S 22.5.16, page 778 and A&S page 789.





To access this function, load("specfun").





See also GEN_LAGUERRE.











Function: LEGENDRE_P (n, x)







[specfun package] return the Legendre polynomial of the 
first kind for  integers  n > -1.  





Reference: A&S 22.5.35 page 779.





To access this function, load("specfun").





See  LEGENDRE_Q.











Function: LEGENDRE_Q (n, x)






[specfun package] return the Legendre polynomial of the 
first kind for  integers  n > -1.  





Reference: A&S 8.6.19 page 334.





To access this function, load("specfun").





See also LEGENDRE_P.











Function: SPHERICAL_BESSEL_J (n, x)






[specfun package] return 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.





To access this function, load("specfun").





See also SPHERICAL_HANKEL1, SPHERICAL_HANKEL2, 
and SPHERICAL_BESSEL_Y.











Function: SPHERICAL_BESSEL_Y (n, x)






[specfun package] return 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.





To access this function, load("specfun").





See also SPHERICAL_HANKEL1, SPHERICAL_HANKEL2, 
and SPHERICAL_BESSEL_Y.











Function: SPHERICAL_HANKEL1 (n,x)






[specfun package] return the spherical hankel function  of the
first kind for  integers  n > -1.  





Reference: A&S 10.1.36 page 439.





To access this function, load("specfun").





See also SPHERICAL_HANKEL2, SPHERICAL_BESSEL_J, and
SPHERICAL_BESSEL_Y.











Function: SPHERICAL_HANKEL2 (n,x)






[specfun package] return the spherical hankel function  of the
second kind for  integers  n > -1.  





Reference:  A&S 10.1.17 page 439.





To access this function, load("specfun").





See also SPHERICAL_HANKEL1, SPHERICAL_BESSEL_J, and
SPHERICAL_BESSEL_Y.











Function: SPHERICAL_HARMONIC (n, m, x, y)






[specfun package] return the spherical harmonic function
for  integers  n > -1 and  | m | <= n .





Reference:  Merzbacher 9.64.





To access this function, load("specfun").





See also ASSOC_LEGENDRE_P











Function: ULTRASPHERICAL (n,a,x)






[specfun package] return the ultraspherical polynomials for  
integers  n > -1.  The ultraspherical polynomials are
also known as Gegenbauer polynomials.





Reference: A&S 22.5.27





To access this function, load("specfun").





See also JACOBI_P.









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