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 \(n = \infty\) 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 Dirichlet 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

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

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

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

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

Examples

Evaluation works for real and complex arguments:

>>> from 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

Table Of Contents

Previous topic

Number-theoretical, combinatorial and integer functions

Next topic

Numerical calculus

This Page