15.8 Hypergeometric Functions

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

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

Function: %m [k,u] (z) ¶

Whittaker M function M[k,u](z) = exp(-z/2)*z^(1/2+u)*M(1/2+u-k,1+2*u,z). (A&S 13.1.32)

Categories: Special functions ·

Function: %w [k,u] (z) ¶

Whittaker W function. (A&S 13.1.33)

Categories: Special functions ·

Function: %f [p,q] ([a],[b],z) ¶

The pFq(a1,a2,..ap;b1,b2,..bq;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