# 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(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 x1: 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:

### Series Expansions(1)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

## Applications(6) -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:

The -binomial theorem:

## Neat Examples(2)

Hirschhorn's modular identity :

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