<title>Rising Factorial</title>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">rising_factorial</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">i</span><span class="special">);</span>
-<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
-<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">rising_factorial</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">i</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
+<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
+<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">rising_factorial</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">i</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
Returns the rising factorial of <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>i</em></span>:
</p>
-<p>
- rising_factorial(x, i) = Γ(x + i) / Γ(x);
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic"><span class="emphasis"><em>rising_factorial(x, i) = Γ(x + i)
+ / Γ(x)</em></span></span>
+ </p></blockquote></div>
<p>
or
</p>
-<p>
- rising_factorial(x, i) = x(x+1)(x+2)(x+3)...(x+i-1)
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic"><span class="emphasis"><em>rising_factorial(x, i) = x(x+1)(x+2)(x+3)...(x+i-1)</em></span></span>
+ </p></blockquote></div>
<p>
Note that both <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>i</em></span> can be negative
as well as positive.
</p>
<p>
- The final <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
+ The final <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
- what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">policy
+ what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
<span class="phrase"><a name="math_toolkit.factorials.sf_rising_factorial.testing"></a></span><a class="link" href="sf_rising_factorial.html#math_toolkit.factorials.sf_rising_factorial.testing">Testing</a>
</h5>
<p>
- The spot tests for the rising factorials use data generated by functions.wolfram.com.
+ The spot tests for the rising factorials use data generated by <a href="https://functions.wolfram.com" target="_top">functions.wolfram.com</a>.
</p>
<h5>
<a name="math_toolkit.factorials.sf_rising_factorial.h2"></a>
<span class="phrase"><a name="math_toolkit.factorials.sf_rising_factorial.implementation"></a></span><a class="link" href="sf_rising_factorial.html#math_toolkit.factorials.sf_rising_factorial.implementation">Implementation</a>
</h5>
<p>
- Rising and falling factorials are implemented as ratios of gamma functions
- using <a class="link" href="../sf_gamma/gamma_ratios.html" title="Ratios of Gamma Functions">tgamma_delta_ratio</a>.
+ Rising and factorials are implemented as ratios of gamma functions using
+ <a class="link" href="../sf_gamma/gamma_ratios.html" title="Ratios of Gamma Functions">tgamma_delta_ratio</a>.
Optimisations for small integer arguments are handled internally by that
function.
</p>