QPochhammer

QPochhammer[a,q,n]

gives the -Pochhammer symbol .

QPochhammer[a,q]

gives the -Pochhammer symbol .

QPochhammer[q]

gives the -Pochhammer symbol .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • .
  • .
  • QPochhammer automatically threads over lists.

Examples

open allclose all

Basic Examples  (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Generate partition numbers:

Scope  (14)

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 TemplateBox[{{x, +, {i,  , y}}, {1, /, 2}}, QPochhammer2]:

Plot the imaginary part of TemplateBox[{{x, +, {i,  , y}}, {1, /, 2}}, QPochhammer2]:

Function Properties  (3)

The real domain of QPochhammer:

Approximate function range of QPochhammer:

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:

Build -analogs of sine and cosine:

-analog of :

-analog of :

Triple product identity:

Find RamanujanTau from its generating function:

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 TemplateBox[{q}, QPochhammer1]:

Introduced in 2008
 (7.0)