# 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

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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 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:

TraditionalForm formatting:

### Series Expansions(1)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

## Applications(10)

-series are building blocks of other -factorial functions:

The -binomial theorem:

RogersRamanujan identities:

Build -analogs of sine and cosine:

Verify some analogs of the usual trigonometric identities:

Plot the functions:

-analog of :

-analog of :

Demonstrate the pentagonal number theorem:

An alternative formulation in terms of a Dirichlet character modulo 12:

Generate partition numbers:

Verify Jacobi's triple product identity through series expansion:

Find RamanujanTau from its generating function, the modular discriminant:

The probability that the determinant of a random uniform matrix in a finite field of characteristic is zero:

Compute the probability for a matrix in a field of characteristic 2:

Compare with a simulation:

## Neat Examples(4)

Hirschhorn's modular identity :

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

Expand the RogersRamanujan continued fraction into a series:

Compare with the closed form in terms of QPochhammer:

Visualize the RogersRamanujan continued fraction over the unit disk:

Visualize a partial sum of the "strange function" of Kontsevich and Zagier in the complex plane:

Wolfram Research (2008), QPochhammer, Wolfram Language function, https://reference.wolfram.com/language/ref/QPochhammer.html.

#### Text

Wolfram Research (2008), QPochhammer, Wolfram Language function, https://reference.wolfram.com/language/ref/QPochhammer.html.

#### CMS

Wolfram Language. 2008. "QPochhammer." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/QPochhammer.html.

#### APA

Wolfram Language. (2008). QPochhammer. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/QPochhammer.html

#### BibTeX

@misc{reference.wolfram_2024_qpochhammer, author="Wolfram Research", title="{QPochhammer}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/QPochhammer.html}", note=[Accessed: 26-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_qpochhammer, organization={Wolfram Research}, title={QPochhammer}, year={2008}, url={https://reference.wolfram.com/language/ref/QPochhammer.html}, note=[Accessed: 26-May-2024 ]}