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<a name="Hypergeometric-Functions"></a>
<div class="header">
<p>
Next: <a href="maxima_92.html#Parabolic-Cylinder-Functions" accesskey="n" rel="next">Parabolic Cylinder Functions</a>, Previous: <a href="maxima_90.html#Struve-Functions" accesskey="p" rel="previous">Struve Functions</a>, Up: <a href="maxima_83.html#Special-Functions" accesskey="u" rel="up">Special Functions</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Hypergeometric-Functions-1"></a>
<h3 class="section">15.8 Hypergeometric Functions</h3>
<p>The Hypergeometric Functions are defined in Abramowitz and Stegun,
<i>Handbook of Mathematical Functions</i>, <a href="https://personal.math.ubc.ca/~cbm/aands/page_504.htm">A&S Chapters 13</a> and
<a href="https://personal.math.ubc.ca/~cbm/aands/page_555.htm">A&S 15</a>.
</p>
<p>Maxima has very limited knowledge of these functions. They
can be returned from function <code>hgfred</code>.
</p>
<a name="Item_003a-Special_002fdeffn_002f_0025m"></a><dl>
<dt><a name="index-_0025m"></a>Function: <strong>%m</strong> <em>[<var>k</var>,<var>u</var>] (<var>z</var>) </em></dt>
<dd><p>Whittaker M function (<a href="https://personal.math.ubc.ca/~cbm/aands/page_505.htm">A&S eqn 13.1.32</a>):
</p>
$$
M_{\kappa,\mu}(z) = e^{-{1\over 2}z} z^{{1\over 2} + \mu} M\left({1\over 2} + \mu - \kappa, 1 + 2\mu, z\right)
$$
<p>where <em>M(a,b,z)</em> is Kummer’s solution of the confluent hypergeometric equation.
</p>
<p>This can also be expressed by the series (<a href="https://dlmf.nist.gov/13.14.E6">DLMF 13.14.E6</a>):
$$
M_{\kappa,\mu}(z) = e^{-{1\over 2} z} z^{{1\over 2} + \mu}
\sum_{s=0}^{\infty} {\left({1\over 2} + \mu - \kappa\right)_s \over (1 + 2\mu)_s s!} z^s
$$</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></dd></dl>
<a name="Item_003a-Special_002fdeffn_002f_0025w"></a><dl>
<dt><a name="index-_0025w"></a>Function: <strong>%w</strong> <em>[<var>k</var>,<var>u</var>] (<var>z</var>) </em></dt>
<dd><p>Whittaker W function (<a href="https://personal.math.ubc.ca/~cbm/aands/page_505.htm">A&S eqn 13.1.33</a>):
$$
W_{\kappa,\mu}(z) = e^{-{1\over 2}z} z^{{1\over 2} + \mu} U\left({1\over 2} + \mu - \kappa, 1+2\mu,z\right)
$$</p>
<p>where <em>U(a,b,z)</em> is Kummer’s second solution of the confluent hypergeometric equation.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></dd></dl>
<a name="Item_003a-Special_002fdeffn_002f_0025f"></a><dl>
<dt><a name="index-_0025f"></a>Function: <strong>%f</strong> <em>[<var>p</var>,<var>q</var>] (<var>[a],[b],z</var>) </em></dt>
<dd><p>The
\(_{p}F_{q}(a_1,a_2,...,a_p;b_1,b_2,...,b_q;z)\) hypergeometric function,
where <var>a</var> a list of length <var>p</var> and
<var>b</var> a list of length <var>q</var>.
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Bessel-functions">Bessel functions</a>
·<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></p></dd></dl>
<a name="Item_003a-Special_002fdeffn_002fhypergeometric"></a><dl>
<dt><a name="index-hypergeometric"></a>Function: <strong>hypergeometric</strong> <em>([<var>a1</var>, ..., <var>ap</var>],[<var>b1</var>, ... ,<var>bq</var>], x)</em></dt>
<dd><p>The hypergeometric function. Unlike Maxima’s <code>%f</code> hypergeometric
function, the function <code>hypergeometric</code> is a simplifying
function; also, <code>hypergeometric</code> supports complex double and
big floating point evaluation. For the Gauss hypergeometric function,
that is <em>p = 2</em> and <em>q = 1</em>, floating point evaluation
outside the unit circle is supported, but in general, it is not
supported.
</p>
<p>When the option variable <code>expand_hypergeometric</code> is true (default
is false) and one of the arguments <code>a1</code> through <code>ap</code> is a
negative integer (a polynomial case), <code>hypergeometric</code> returns an
expanded polynomial.
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) hypergeometric([],[],x);
(%o1) %e^x
</pre></div>
<p>Polynomial cases automatically expand when <code>expand_hypergeometric</code> is true:
</p>
<div class="example">
<pre class="example">(%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
</pre></div>
<p>Both double float and big float evaluation is supported:
</p>
<div class="example">
<pre class="example">(%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
</pre></div>
</dd></dl>
<a name="Item_003a-Special_002fdeffn_002fhypergeometric_005fsimp"></a><dl>
<dt><a name="index-hypergeometric_005fsimp"></a>Function: <strong>hypergeometric_simp</strong> <em>(<var>e</var>)</em></dt>
<dd>
<p><code>hypergeometric_simp</code> simplifies hypergeometric functions
by applying <code>hgfred</code>
to the arguments of any hypergeometric functions in the expression <var>e</var>.
</p>
<p>Only instances of <code>hypergeometric</code> are affected;
any <code>%f</code>, <code>%w</code>, and <code>%m</code> in the expression <var>e</var> are not affected.
Any unsimplified hypergeometric functions are returned unchanged
(instead of changing to <code>%f</code> as <code>hgfred</code> would).
</p>
<p><code>load("hypergeometric");</code> loads this function.
</p>
<p>See also <code><a href="#hgfred">hgfred</a></code>.
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%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)
</pre></div>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Hypergeometric-functions">Hypergeometric functions</a>
·<a href="maxima_424.html#Category_003a-Simplification-functions">Simplification functions</a>
·<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
·</div></dd></dl>
<a name="hgfred"></a><a name="Item_003a-Special_002fdeffn_002fhgfred"></a><dl>
<dt><a name="index-hgfred"></a>Function: <strong>hgfred</strong> <em>(<var>a</var>, <var>b</var>, <var>t</var>)</em></dt>
<dd>
<p>Simplify the generalized hypergeometric function in terms of other,
simpler, forms. <var>a</var> is a list of numerator parameters and <var>b</var>
is a list of the denominator parameters.
</p>
<p>If <code>hgfred</code> cannot simplify the hypergeometric function, it returns
an expression of the form <code>%f[p,q]([a], [b], x)</code> where <var>p</var> is
the number of elements in <var>a</var>, and <var>q</var> is the number of elements
in <var>b</var>. This is the usual
\(_pF_q\) generalized hypergeometric
function.
</p>
<div class="example">
<pre class="example">(%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
</pre></div>
<p>It can be beneficial to load orthopoly too as the following example
shows. Note that <var>L</var> is the generalized Laguerre polynomial.
</p>
<div class="example">
<pre class="example">(%i5) load("orthopoly")$
(%i6) hgfred([-2],[a],z);
</pre><pre class="example">
(a - 1)
2 L (z)
2
(%o6) -------------
a (a + 1)
</pre><pre class="example">(%i7) ev(%);
2
z 2 z
(%o7) --------- - --- + 1
a (a + 1) a
</pre></div>
</dd></dl>
<a name="Item_003a-Special_002fnode_002fParabolic-Cylinder-Functions"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_92.html#Parabolic-Cylinder-Functions" accesskey="n" rel="next">Parabolic Cylinder Functions</a>, Previous: <a href="maxima_90.html#Struve-Functions" accesskey="p" rel="previous">Struve Functions</a>, Up: <a href="maxima_83.html#Special-Functions" accesskey="u" rel="up">Special Functions</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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