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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 (22)

Numerical Evaluation (6)

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:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix QPochhammer function using MatrixFunction:

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 (11)

-series are building blocks of other -factorial functions:

The -binomial theorem:

Rogers–Ramanujan 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:

Define a function for computing the conjugate of an integer partition:

Count the number of integer partitions of that are self-conjugate:

Compute the same result from the generating function:

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 :

Expand the Rogers–Ramanujan continued fraction into a series:

Compare with the closed form in terms of QPochhammer:

Visualize the Rogers–Ramanujan continued fraction over the unit disk:

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

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