QPochhammer

QPochhammer[a,q,n]

gives the -Pochhammer symbol TemplateBox[{a, q, n}, QPochhammer].

QPochhammer[a,q]

gives the -Pochhammer symbol TemplateBox[{a, q}, QPochhammer2].

QPochhammer[q]

gives the -Pochhammer symbol TemplateBox[{q}, QPochhammer1].

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 TemplateBox[{z, {1, /, 2}}, QPochhammer2]:

Plot the imaginary part of TemplateBox[{z, {1, /, 2}}, QPochhammer2]:

Function Properties  (9)

The real domain of TemplateBox[{x}, QPochhammer1]:

Approximate function range of TemplateBox[{x}, QPochhammer1]:

TemplateBox[{x}, QPochhammer1] is not an analytic function:

Has both singularities and discontinuities for x-1 or for x1:

TemplateBox[{x}, QPochhammer1] 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 (TemplateBox[{q, q}, QPochhammer2])^5=TemplateBox[{{q, ^, 5}, {q, ^, 5}}, QPochhammer2] mod 5:

The boundary of the unit disk contains a dense subset of essential singularities of TemplateBox[{q}, QPochhammer1]:

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: 27-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: 27-May-2024 ]}