QPochhammer

QPochhammer[a,q,n]

gives the -Pochhammer symbol .

QPochhammer[a,q]

gives the -Pochhammer symbol .

gives the -Pochhammer symbol .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
$TemplateBox[{a, q}, QPochhammer2]=product_(k=0)^infty(1-a q^k)$ .
$TemplateBox[{a, q, n}, QPochhammer]=TemplateBox[{a, q}, QPochhammer2]/TemplateBox[{{a, , {q, ^, n}}, q}, QPochhammer2]$ .
QPochhammer automatically threads over lists.

Examples

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Basic Examples (4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Scope (20)

Numerical Evaluation (4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

Specific Values (4)

Values of QPochhammer at fixed points:

QPochhammer for symbolic parameters:

Finite products evaluate for all Gaussian rational numbers:

Find the maximum of QPochhammer[x]:

Visualization (2)

Plot the QPochhammer function:

Plot the real part of :

Plot the imaginary part of :

Function Properties (9)

The real domain of :

Approximate function range of :

is not an analytic function:

Has both singularities and discontinuities for x≤-1 or for x≥1:

is neither nonincreasing nor nondecreasing:

QPochhammer is not injective:

QPochhammer is not surjective:

QPochhammer is neither non-negative nor non-positive:

QPochhammer is neither convex nor concave:

TraditionalForm formatting:

Series Expansions (1)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Applications (5)

-series are building blocks of other -factorial functions:

The -binomial theorem:

Build -analogs of sine and cosine:

Verify some analogs of the usual trigonometric identities:

Plot the functions:

-analog of :

-analog of :

Find RamanujanTau from its generating function, the modular discriminant:

Neat Examples (2)

Hirschhorn's modular identity $(TemplateBox[{q, q}, QPochhammer2])^5=TemplateBox[{{q, ^, 5}, {q, ^, 5}}, QPochhammer2] mod 5$ :

The boundary of the unit circle contains a dense subset of essential singularities of :

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