q-functions

q-Pochhammer symbol

qp()

mpmath.qp(a, q=None, n=None, **kwargs)

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

$$(a;q{)}_n=\prod _{k=0}^{n-1}(1-a{q}^k)$$

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

$$\varphi (q)=(q;q{)}_{\mathrm{\infty }}=\prod _{k=1}^{\mathrm{\infty }}(1-q^k)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}(-1{)}^k{q}^{(3k^2-k)/2}$$

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

Examples

If $n$ is a positive integer, the function amounts to a finite product:

>>> from mpmath 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)

The regular Pochhammer symbol $(a{)}_n$ is obtained in the following limit as $q\to 1$:

>>> a, n = 4, 7
>>> limit(lambda q: qp(q**a,q,n) / (1-q)**n, 1)
604800.0
>>> rf(a,n)
604800.0

The Taylor series of the reciprocal Euler function gives the partition function $P(n)$, i.e. the number of ways of writing $n$ as a sum of positive integers:

>>> taylor(lambda q: 1/qp(q), 0, 10)
[1.0, 1.0, 2.0, 3.0, 5.0, 7.0, 11.0, 15.0, 22.0, 30.0, 42.0]

Special values include:

>>> qp(0)
1.0
>>> findroot(diffun(qp), -0.4)   # location of maximum
-0.4112484791779547734440257
>>> qp(_)
1.228348867038575112586878

The q-Pochhammer symbol is related to the Jacobi theta functions. For example, the following identity holds:

>>> q = mpf(0.5)    # arbitrary
>>> qp(q)
0.2887880950866024212788997
>>> root(3,-2)*root(q,-24)*jtheta(2,pi/6,root(q,6))
0.2887880950866024212788997

q-gamma and factorial

qgamma()

mpmath.qgamma(z, q, **kwargs)

Evaluates the q-gamma function

$${\mathrm{\Gamma }}_q(z)=\frac{(q;q{)}_{\mathrm{\infty }}}{(q^z;q{)}_{\mathrm{\infty }}}(1-q{)}^{1-z}.$$

Examples

Evaluation for real and complex arguments:

>>> from mpmath 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 function:

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

qfac()

mpmath.qfac(z, q, **kwargs)

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 mpmath 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

qhyper()

mpmath.qhyper(a_s, b_s, q, z, **kwargs)

Evaluates the basic hypergeometric series or hypergeometric q-series

$$\begin{array}{r}{}_r{\varphi }_s[\begin{array}{cccc}{a}_1& a_2& \dots & a_r\\ b_1& b_2& \dots & b_s\end{array};q,z]=\sum _{n=0}^{\mathrm{\infty }}\frac{(a_1;q{)}_n,\dots ,(a_r;q{)}_n}{(b_1;q{)}_n,\dots ,(b_s;q{)}_n}{((-1{)}^n{q}^{(\genfrac{}{}{0ex}{}{n}{2})})}^{1+s-r}\frac{z^n}{(q;q{)}_n}\end{array}$$

where $(a;q{)}_n$ denotes the q-Pochhammer symbol (see qp()).

Examples

Evaluation works for real and complex arguments:

>>> from mpmath 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)

Comparing with a summation of the defining series, using nsum():

>>> b, q, z = 3, 0.25, 0.5
>>> qhyper([], [b], q, z)
0.6221136748254495583228324
>>> nsum(lambda n: z**n / qp(q,q,n)/qp(b,q,n) * q**(n*(n-1)), [0,inf])
0.6221136748254495583228324