Q Functions

Q Functions#

q-Pochhammer symbol#

ctx.q_pochhammer(a, q=None, n=None)#

where ctx is dec, mpm, fpm, or gmp.

Returns the q-Pochhammer symbol. See also Wikipedia [1516], MathWorld [1113], Mpmath [765].

Evaluates the q-Pochhammer symbol (or q-rising factorial)

\[(a; q)_n = \prod_{k=0}^{n-1} (1-a q^k)\]

where \(n = \infty\) is permitted if \(|q| < 1\). Called with two arguments, qp(a,q) computes \((a;q)_{\infty}\); with a single argument, qp(q) computes \((q;q)_{\infty}\). The special case

\[\phi(q) = (q; q)_{\infty} = \prod_{k=1}^{\infty} (1-q^k) = \sum_{k=-\infty}^{\infty} (-1)^k q^{(3k^2-k)/2}\]

is also known as the Euler function, or (up to a factor \(q^{-1/24}\)) the Dedekind eta function.

If \(n\) is a positive integer, the function amounts to a finite product:

>>> from mpfunlab import *
>>> mp.dps = 25; mp.pretty = True
>>> qp(2,3,5)
-725305.0
>>> fprod(1-2*3**k for k in range(5))
-725305.0
>>> qp(2,3,0)
1.0

Complex arguments are allowed:

>>> qp(2-1j, 0.75j)
(0.4628842231660149089976379 + 4.481821753552703090628793j)

q-gamma function#

ctx.q_gamma(z, q)#

where ctx is dec, mpm, fpm, or gmp.

Returns the q-gamma function. See also Wikipedia [1517], MathWorld [1111], NIST [851], Mpmath [767].

Evaluates the q-gamma function

\[\Gamma_q(z) = \frac{(q; q)_{\infty}}{(q^z; q)_{\infty}} (1-q)^{1-z}.\]

Evaluation for real and complex arguments:

>>> from mpfunlab import *
>>> mp.dps = 25; mp.pretty = True
>>> qgamma(4,0.75)
4.046875
>>> qgamma(6,6)
121226245.0
>>> qgamma(3+4j, 0.5j)
(0.1663082382255199834630088 + 0.01952474576025952984418217j)

The q-gamma function satisfies a functional equation similar to that of the ordinary gamma method:

>>> q = mpf(0.25)
>>> z = mpf(2.5)
>>> qgamma(z+1,q)
1.428277424823760954685912
>>> (1-q**z)/(1-q)*qgamma(z,q)
1.428277424823760954685912

q-factorial#

ctx.q_factorial(z, q)#

where ctx is dec, mpm, fpm, or gmp.

Returns the q-factorial. See also Wikipedia [1508], MathWorld [1110], NIST [850], Mpmath [766].

Evaluates the q-factorial,

\[[n`q! = (1+q)(1+q+q^2)\cdots(1+q+\cdots+q^{n-1})\]

or more generally

\[[z`q! = \frac{(q;q)_z}{(1-q)^z}.\]

Examples

>>> from mpfunlab import *
>>> mp.dps = 25; mp.pretty = True
>>> qfac(0,0)
1.0
>>> qfac(4,3)
2080.0
>>> qfac(5,6)
121226245.0
>>> qfac(1+1j, 2+1j)
(0.4370556551322672478613695 + 0.2609739839216039203708921j)

Hypergeometric q-series#

ctx.q_hyperg(a_s, b_s, q, z)#

where ctx is dec, mpm, fpm, or gmp.

Returns the hypergeometric q-series. See also MathWorld [1112], NIST [842], Mpmath [754].

Evaluates the basic hypergeometric series or hypergeometric q-series

\[\begin{split}\,_r\phi_s \left[\begin{matrix} a_1 & a_2 & \ldots & a_r \\ b_1 & b_2 & \ldots & b_s \end{matrix} ; q,z \right] = \sum_{n=0}^\infty \frac{(a_1;q)_n, \ldots, (a_r;q)_n} {(b_1;q)_n, \ldots, (b_s;q)_n} \left((-1)^n q^{n\choose 2}\right)^{1+s-r} \frac{z^n}{(q;q)_n}\end{split}\]

where \((a;q)_n\) denotes the q-Pochhammer symbol (see qp()).

Examples

Evaluation works for real and complex arguments:

>>> from mpfunlab import *
>>> mp.dps = 25; mp.pretty = True
>>> qhyper([0.5], [2.25], 0.25, 4)
-0.1975849091263356009534385
>>> qhyper([0.5], [2.25], 0.25-0.25j, 4)
(2.806330244925716649839237 + 3.568997623337943121769938j)
>>> qhyper([1+j], [2,3+0.5j], 0.25, 3+4j)
(9.112885171773400017270226 - 1.272756997166375050700388j)
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