15.8 Hypergeometric Functions

The Hypergeometric Functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, A&S Chapters 13 and A&S 15.

Maxima has very limited knowledge of these functions. They can be returned from function hgfred.

Function: %m [k,u] (z) ¶

Whittaker M function (A&S eqn 13.1.32):

\[M_{\kappa,\mu}(z) = e^{-{1\over 2}z} z^{{1\over 2} + \mu} M\left({1\over 2} + \mu - \kappa, 1 + 2\mu, z\right) \]

where \(M(a,b,z)\) is Kummer’s solution of the confluent hypergeometric equation.

This can also be expressed by the series (DLMF 13.14.E6):

\[M_{\kappa,\mu}(z) = e^{-{1\over 2} z} z^{{1\over 2} + \mu} \sum_{s=0}^{\infty} {\left({1\over 2} + \mu - \kappa\right)_s \over (1 + 2\mu)_s s!} z^s \]

Categories: Special functions ·

Function: %w [k,u] (z) ¶

Whittaker W function (A&S eqn 13.1.33):

\[W_{\kappa,\mu}(z) = e^{-{1\over 2}z} z^{{1\over 2} + \mu} U\left({1\over 2} + \mu - \kappa, 1+2\mu,z\right) \]

where \(U(a,b,z)\) is Kummer’s second solution of the confluent hypergeometric equation.

Categories: Special functions ·

Function: %f [p,q] ([a],[b],z) ¶

The \(_{p}F_{q}(a_1,a_2,...,a_p;b_1,b_2,...,b_q;z)\) hypergeometric function, where a a list of length p and b a list of length q.

Categories: Bessel functions · Special functions ·

Function: hypergeometric ([a1, ..., ap],[b1, ... ,bq], x) ¶

The hypergeometric function. Unlike Maxima’s %f hypergeometric function, the function hypergeometric is a simplifying function; also, hypergeometric supports complex double and big floating point evaluation. For the Gauss hypergeometric function, that is \(p = 2\) and \(q = 1\), floating point evaluation outside the unit circle is supported, but in general, it is not supported.

When the option variable expand_hypergeometric is true (default is false) and one of the arguments a1 through ap is a negative integer (a polynomial case), hypergeometric returns an expanded polynomial.

Examples:

(%i1)  hypergeometric([],[],x);
(%o1) %e^x

Polynomial cases automatically expand when expand_hypergeometric is true:

(%i2) hypergeometric([-3],[7],x);
(%o2) hypergeometric([-3],[7],x)

(%i3) hypergeometric([-3],[7],x), expand_hypergeometric : true;
(%o3) -x^3/504+3*x^2/56-3*x/7+1

Both double float and big float evaluation is supported:

(%i4) hypergeometric([5.1],[7.1 + %i],0.42);
(%o4)       1.346250786375334 - 0.0559061414208204 %i
(%i5) hypergeometric([5,6],[8], 5.7 - %i);
(%o5)     .007375824009774946 - .001049813688578674 %i
(%i6) hypergeometric([5,6],[8], 5.7b0 - %i), fpprec : 30;
(%o6) 7.37582400977494674506442010824b-3
                          - 1.04981368857867315858055393376b-3 %i

Function: hypergeometric_simp (e) ¶

hypergeometric_simp simplifies hypergeometric functions by applying hgfred to the arguments of any hypergeometric functions in the expression e.

Only instances of hypergeometric are affected; any %f, %w, and %m in the expression e are not affected. Any unsimplified hypergeometric functions are returned unchanged (instead of changing to %f as hgfred would).

load("hypergeometric"); loads this function.

See also hgfred.

Examples:

(%i1) load ("hypergeometric") $
(%i2) foo : [hypergeometric([1,1], [2], z), hypergeometric([1/2], [1], z)];
(%o2) [hypergeometric([1, 1], [2], z), 
                                                     1
                                     hypergeometric([-], [1], z)]
                                                     2
(%i3) hypergeometric_simp (foo);
                 log(1 - z)              z    z/2
(%o3)         [- ----------, bessel_i(0, -) %e   ]
                     z                   2
(%i4) bar : hypergeometric([n], [m], z + 1);
(%o4)            hypergeometric([n], [m], z + 1)
(%i5) hypergeometric_simp (bar);
(%o5)            hypergeometric([n], [m], z + 1)

Categories: Hypergeometric functions · Simplification functions · Special functions ·

Function: hgfred (a, b, t) ¶

Simplify the generalized hypergeometric function in terms of other, simpler, forms. a is a list of numerator parameters and b is a list of the denominator parameters.

If hgfred cannot simplify the hypergeometric function, it returns an expression of the form %f[p,q]([a], [b], x) where p is the number of elements in a, and q is the number of elements in b. This is the usual \(_pF_q\) generalized hypergeometric function.

(%i1) assume(not(equal(z,0)));
(%o1)                          [notequal(z, 0)]
(%i2) hgfred([v+1/2],[2*v+1],2*%i*z);

                     v/2                               %i z
                    4    bessel_j(v, z) gamma(v + 1) %e
(%o2)               ---------------------------------------
                                       v
                                      z
(%i3) hgfred([1,1],[2],z);

                                   log(1 - z)
(%o3)                            - ----------
                                       z
(%i4) hgfred([a,a+1/2],[3/2],z^2);

                               1 - 2 a          1 - 2 a
                        (z + 1)        - (1 - z)
(%o4)                   -------------------------------
                                 2 (1 - 2 a) z

It can be beneficial to load orthopoly too as the following example shows. Note that L is the generalized Laguerre polynomial.

(%i5) load("orthopoly")$
(%i6) hgfred([-2],[a],z);

                                    (a - 1)
                                 2 L       (z)
                                    2
(%o6)                            -------------
                                   a (a + 1)

(%i7) ev(%);

                                  2
                                 z        2 z
(%o7)                         --------- - --- + 1
                              a (a + 1)    a