(maxima.info)Hypergeometric Functions

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15.8 Hypergeometric Functions
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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)

 -- Function: %w [<k>,<u>] (<z>)
     Whittaker W function.  (A&S 13.1.33)

 -- 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'.

 -- 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