QPochhammer
QPochhammer[a,q,n]
gives the -Pochhammer symbol
.
QPochhammer[a,q]
gives the -Pochhammer symbol
.
QPochhammer[q]
gives the -Pochhammer symbol
.
Details
![](Files/QPochhammer.en/details_1.png)
- Mathematical function, suitable for both symbolic and numerical manipulation.
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- QPochhammer automatically threads over lists.
Examples
open allclose allBasic Examples (4)
Scope (22)
Numerical Evaluation (6)
The precision of the output tracks the precision of the input:
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)
Function Properties (9)
Approximate function range of :
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:
Applications (11)
-series are building blocks of other
-factorial functions:
Build -analogs of sine and cosine:
Verify some analogs of the usual trigonometric identities:
Demonstrate the pentagonal number theorem:
An alternative formulation in terms of a Dirichlet character modulo 12:
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:
Neat Examples (4)
Hirschhorn's modular identity :
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:
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