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<div class="section" id="q-functions">
<h1>q-functions<a class="headerlink" href="#q-functions" title="Permalink to this headline">¶</a></h1>
<div class="section" id="q-pochhammer-symbol">
<h2>q-Pochhammer symbol<a class="headerlink" href="#q-pochhammer-symbol" title="Permalink to this headline">¶</a></h2>
<div class="section" id="qp">
<h3><tt class="xref docutils literal"><span class="pre">qp()</span></tt><a class="headerlink" href="#qp" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.qp">
<tt class="descclassname">mpmath.</tt><tt class="descname">qp</tt><big>(</big><em>a</em>, <em>q=None</em>, <em>n=None</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.qp" title="Permalink to this definition">¶</a></dt>
<dd><p>Evaluates the q-Pochhammer symbol (or q-rising factorial)</p>
<div class="math">
<p><img src="../_images/math/fc763511f4703ac6549f7e4aeffb74dc5f461c46.png" alt="(a; q)_n = \prod_{k=0}^{n-1} (1-a q^k)" /></p>
</div><p>where <img class="math" src="../_images/math/f24dff9e8b573b740dfe94ec0f515ec397febca2.png" alt="n = \infty"/> is permitted if <img class="math" src="../_images/math/1ac3bec0deb1b3d8164b2938edd7bb85d132b005.png" alt="|q| < 1"/>. Called with two arguments,
<tt class="docutils literal"><span class="pre">qp(a,q)</span></tt> computes <img class="math" src="../_images/math/07744e6ccc7038c02383ab1cc180593bc7eca5d6.png" alt="(a;q)_{\infty}"/>; with a single argument, <tt class="docutils literal"><span class="pre">qp(q)</span></tt>
computes <img class="math" src="../_images/math/7d3edb4bff19034d5d9c9ba55849a1e5334602b7.png" alt="(q;q)_{\infty}"/>. The special case</p>
<div class="math">
<p><img src="../_images/math/aaad2fdb58b63f2db45dc3d4296709293da07f91.png" alt="\phi(q) = (q; q)_{\infty} = \prod_{k=1}^{\infty} (1-q^k) =
\sum_{k=-\infty}^{\infty} (-1)^k q^{(3k^2-k)/2}" /></p>
</div><p>is also known as the Euler function, or (up to a factor <img class="math" src="../_images/math/5a4e8132b6c466816240ade9a33ef2b4453f441e.png" alt="q^{-1/24}"/>)
the Dirichlet eta function.</p>
<p><strong>Examples</strong></p>
<p>If <img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/> is a positive integer, the function amounts to a finite product:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">qp</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
<span class="go">-725305.0</span>
<span class="gp">>>> </span><span class="n">fprod</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="mi">3</span><span class="o">**</span><span class="n">k</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
<span class="go">-725305.0</span>
<span class="gp">>>> </span><span class="n">qp</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
<span class="go">1.0</span>
</pre></div>
</div>
<p>Complex arguments are allowed:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">qp</span><span class="p">(</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="n">j</span><span class="p">,</span> <span class="mf">0.75</span><span class="n">j</span><span class="p">)</span>
<span class="go">(0.4628842231660149089976379 + 4.481821753552703090628793j)</span>
</pre></div>
</div>
<p>The regular Pochhammer symbol <img class="math" src="../_images/math/671243179eef0081e2811f59a14a8e99d0542ffb.png" alt="(a)_n"/> is obtained in the
following limit as <img class="math" src="../_images/math/9db120b46ade50f2e63c13d506a8ec951e8c6f92.png" alt="q \to 1"/>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">a</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">7</span>
<span class="gp">>>> </span><span class="n">limit</span><span class="p">(</span><span class="k">lambda</span> <span class="n">q</span><span class="p">:</span> <span class="n">qp</span><span class="p">(</span><span class="n">q</span><span class="o">**</span><span class="n">a</span><span class="p">,</span><span class="n">q</span><span class="p">,</span><span class="n">n</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">q</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="go">604800.0</span>
<span class="gp">>>> </span><span class="n">rf</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">n</span><span class="p">)</span>
<span class="go">604800.0</span>
</pre></div>
</div>
<p>The Taylor series of the reciprocal Euler function gives
the partition function <img class="math" src="../_images/math/7dfd17dc4d3100462df726e8eb7f9262783429eb.png" alt="P(n)"/>, i.e. the number of ways of writing
<img class="math" src="../_images/math/174fadd07fd54c9afe288e96558c92e0c1da733a.png" alt="n"/> as a sum of positive integers:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">q</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">qp</span><span class="p">(</span><span class="n">q</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="go">[1.0, 1.0, 2.0, 3.0, 5.0, 7.0, 11.0, 15.0, 22.0, 30.0, 42.0]</span>
</pre></div>
</div>
<p>Special values include:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">qp</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="go">1.0</span>
<span class="gp">>>> </span><span class="n">findroot</span><span class="p">(</span><span class="n">diffun</span><span class="p">(</span><span class="n">qp</span><span class="p">),</span> <span class="o">-</span><span class="mf">0.4</span><span class="p">)</span> <span class="c"># location of maximum</span>
<span class="go">-0.4112484791779547734440257</span>
<span class="gp">>>> </span><span class="n">qp</span><span class="p">(</span><span class="n">_</span><span class="p">)</span>
<span class="go">1.228348867038575112586878</span>
</pre></div>
</div>
<p>The q-Pochhammer symbol is related to the Jacobi theta functions.
For example, the following identity holds:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span> <span class="c"># arbitrary</span>
<span class="gp">>>> </span><span class="n">qp</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
<span class="go">0.2887880950866024212788997</span>
<span class="gp">>>> </span><span class="n">root</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="n">q</span><span class="p">,</span><span class="o">-</span><span class="mi">24</span><span class="p">)</span><span class="o">*</span><span class="n">jtheta</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">pi</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span><span class="n">root</span><span class="p">(</span><span class="n">q</span><span class="p">,</span><span class="mi">6</span><span class="p">))</span>
<span class="go">0.2887880950866024212788997</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="q-gamma-and-factorial">
<h2>q-gamma and factorial<a class="headerlink" href="#q-gamma-and-factorial" title="Permalink to this headline">¶</a></h2>
<div class="section" id="qgamma">
<h3><tt class="xref docutils literal"><span class="pre">qgamma()</span></tt><a class="headerlink" href="#qgamma" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.qgamma">
<tt class="descclassname">mpmath.</tt><tt class="descname">qgamma</tt><big>(</big><em>z</em>, <em>q</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.qgamma" title="Permalink to this definition">¶</a></dt>
<dd><p>Evaluates the q-gamma function</p>
<div class="math">
<p><img src="../_images/math/2105c28decf1bb8f0d13c5d68a29cb44997dd0a1.png" alt="\Gamma_q(z) = \frac{(q; q)_{\infty}}{(q^z; q)_{\infty}} (1-q)^{1-z}." /></p>
</div><p><strong>Examples</strong></p>
<p>Evaluation for real and complex arguments:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">qgamma</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mf">0.75</span><span class="p">)</span>
<span class="go">4.046875</span>
<span class="gp">>>> </span><span class="n">qgamma</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="mi">6</span><span class="p">)</span>
<span class="go">121226245.0</span>
<span class="gp">>>> </span><span class="n">qgamma</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4</span><span class="n">j</span><span class="p">,</span> <span class="mf">0.5</span><span class="n">j</span><span class="p">)</span>
<span class="go">(0.1663082382255199834630088 + 0.01952474576025952984418217j)</span>
</pre></div>
</div>
<p>The q-gamma function satisfies a functional equation similar
to that of the ordinary gamma function:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">0.25</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">z</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">2.5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">qgamma</span><span class="p">(</span><span class="n">z</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">q</span><span class="p">)</span>
<span class="go">1.428277424823760954685912</span>
<span class="gp">>>> </span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">q</span><span class="o">**</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">q</span><span class="p">)</span><span class="o">*</span><span class="n">qgamma</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="n">q</span><span class="p">)</span>
<span class="go">1.428277424823760954685912</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="qfac">
<h3><tt class="xref docutils literal"><span class="pre">qfac()</span></tt><a class="headerlink" href="#qfac" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.qfac">
<tt class="descclassname">mpmath.</tt><tt class="descname">qfac</tt><big>(</big><em>z</em>, <em>q</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.qfac" title="Permalink to this definition">¶</a></dt>
<dd><p>Evaluates the q-factorial,</p>
<div class="math">
<p><img src="../_images/math/3e50edea0eb21ef537c4f26a2fc230aafd973ecb.png" alt="[n]_q! = (1+q)(1+q+q^2)\cdots(1+q+\cdots+q^{n-1})" /></p>
</div><p>or more generally</p>
<div class="math">
<p><img src="../_images/math/41d27f52860f33422bbc768f64238f0d9d3167ed.png" alt="[z]_q! = \frac{(q;q)_z}{(1-q)^z}." /></p>
</div><p><strong>Examples</strong></p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">qfac</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
<span class="go">1.0</span>
<span class="gp">>>> </span><span class="n">qfac</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="go">2080.0</span>
<span class="gp">>>> </span><span class="n">qfac</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">)</span>
<span class="go">121226245.0</span>
<span class="gp">>>> </span><span class="n">qfac</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">1</span><span class="n">j</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">1</span><span class="n">j</span><span class="p">)</span>
<span class="go">(0.4370556551322672478613695 + 0.2609739839216039203708921j)</span>
</pre></div>
</div>
</dd></dl>
</div>
</div>
<div class="section" id="hypergeometric-q-series">
<h2>Hypergeometric q-series<a class="headerlink" href="#hypergeometric-q-series" title="Permalink to this headline">¶</a></h2>
<div class="section" id="qhyper">
<h3><tt class="xref docutils literal"><span class="pre">qhyper()</span></tt><a class="headerlink" href="#qhyper" title="Permalink to this headline">¶</a></h3>
<dl class="function">
<dt id="mpmath.qhyper">
<tt class="descclassname">mpmath.</tt><tt class="descname">qhyper</tt><big>(</big><em>a_s</em>, <em>b_s</em>, <em>q</em>, <em>z</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.qhyper" title="Permalink to this definition">¶</a></dt>
<dd><p>Evaluates the basic hypergeometric series or hypergeometric q-series</p>
<div class="math">
<p><img src="../_images/math/a8efaffc0e5fe799f97388cdc4eef302ba675f3f.png" alt="\,_r\phi_s \left[\begin{matrix}
a_1 & a_2 & \ldots & a_r \\
b_1 & b_2 & \ldots & b_s
\end{matrix} ; q,z \right] =
\sum_{n=0}^\infty
\frac{(a_1;q)_n, \ldots, (a_r;q)_n}
{(b_1;q)_n, \ldots, (b_s;q)_n}
\left((-1)^n q^{n\choose 2}\right)^{1+s-r}
\frac{z^n}{(q;q)_n}" /></p>
</div><p>where <img class="math" src="../_images/math/b785fe47a412e3791d236225cb0cc77774ca869a.png" alt="(a;q)_n"/> denotes the q-Pochhammer symbol (see <a title="mpmath.qp" class="reference internal" href="#mpmath.qp"><tt class="xref docutils literal"><span class="pre">qp()</span></tt></a>).</p>
<p><strong>Examples</strong></p>
<p>Evaluation works for real and complex arguments:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
<span class="gp">>>> </span><span class="n">qhyper</span><span class="p">([</span><span class="mf">0.5</span><span class="p">],</span> <span class="p">[</span><span class="mf">2.25</span><span class="p">],</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="go">-0.1975849091263356009534385</span>
<span class="gp">>>> </span><span class="n">qhyper</span><span class="p">([</span><span class="mf">0.5</span><span class="p">],</span> <span class="p">[</span><span class="mf">2.25</span><span class="p">],</span> <span class="mf">0.25</span><span class="o">-</span><span class="mf">0.25</span><span class="n">j</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="go">(2.806330244925716649839237 + 3.568997623337943121769938j)</span>
<span class="gp">>>> </span><span class="n">qhyper</span><span class="p">([</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="o">+</span><span class="mf">0.5</span><span class="n">j</span><span class="p">],</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mi">3</span><span class="o">+</span><span class="mi">4</span><span class="n">j</span><span class="p">)</span>
<span class="go">(9.112885171773400017270226 - 1.272756997166375050700388j)</span>
</pre></div>
</div>
<p>Comparing with a summation of the defining series, using
<a title="mpmath.nsum" class="reference external" href="../calculus/sums_limits.html#mpmath.nsum"><tt class="xref docutils literal"><span class="pre">nsum()</span></tt></a>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">b</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">qhyper</span><span class="p">([],</span> <span class="p">[</span><span class="n">b</span><span class="p">],</span> <span class="n">q</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
<span class="go">0.6221136748254495583228324</span>
<span class="gp">>>> </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">z</span><span class="o">**</span><span class="n">n</span> <span class="o">/</span> <span class="n">qp</span><span class="p">(</span><span class="n">q</span><span class="p">,</span><span class="n">q</span><span class="p">,</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">qp</span><span class="p">(</span><span class="n">b</span><span class="p">,</span><span class="n">q</span><span class="p">,</span><span class="n">n</span><span class="p">)</span> <span class="o">*</span> <span class="n">q</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
<span class="go">0.6221136748254495583228324</span>
</pre></div>
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<li><a class="reference external" href="#">q-functions</a><ul>
<li><a class="reference external" href="#q-pochhammer-symbol">q-Pochhammer symbol</a><ul>
<li><a class="reference external" href="#qp"><tt class="docutils literal"><span class="pre">qp()</span></tt></a></li>
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<li><a class="reference external" href="#q-gamma-and-factorial">q-gamma and factorial</a><ul>
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