QPochhammer

The q-Pochhammer symbol $(a; q)_n = \prod_{k=0}^{n-1} (1 - a q^k)$, for a nonnegative integer $n$ or $n = \infty$. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageQPochhammer!↗

QPochhammer(a, q, n)the q-Pochhammer symbol $(a; q)_n$.

SageMathenumeratio_q_pochhammer(a, q, n)

Typed signatures

Type	Package	Applies
(complex, complex, infinity | integer) -> number	analytic	always

Domain: Special functions

Details

• Finite $n \ge 0$ builds the product from exact boxed arithmetic — an exact rational $a$ or $q$ stays exact, and $n = 0$ gives the empty product, 1, even for symbolic $a$ and $q$.
• $n = \infty$ is Euler's infinite product, numeric-only and only where it converges ($|q| < 1$); it stays symbolic for exact operands (use N()) and for $|q| \ge 1$.
• Negative or symbolic $n$ is not implemented and stays symbolic — Wolfram's extension to negative $n$ via $(a;q)_{-n} = 1/\prod_{k=1}^n(1 - a q^{-k})$ is not carried here.

Plot

Examples

6 more kept as data, checked against the oracles

Basic #

$(1-2)(1-6)(1-18) = -85$

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In$\mathrm{QPochhammer}(2, 3, 3)$Out$-85$

Exact rational arithmetic, no N() needed

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In$\mathrm{QPochhammer}(\frac{1}{2}, \frac{1}{2}, 3)$Out$\frac{21}{64}$

Scope #

The infinite product, Euler's function at $q = \tfrac12$

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In$\mathrm{QPochhammer}(0.5, 0.5, ∞)$Out$0.288\,788\,095\,086\,602\,4$

$(3;5)_9$ as an exact integer, a product of nine factors $1-3\cdot5^k$

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In$\mathrm{QPochhammer}(3, 5, 9)$Out$-175\,258\,163\,137\,268\,414\,822\,846\,915\,072$

With one factor, $(a;2)_1 = 1-a$

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In$\mathrm{FunctionExpand}(\mathrm{QPochhammer}(a, 2, 1))$Out$1-a$

Complex base and ratio give an exact complex rational

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In$\mathrm{QPochhammer}(\frac{1}{2}+\frac{1}{2}i, \frac{1}{2}-\frac{1}{2}i, 10)$Out$\frac{95\,264\,271}{268\,435\,456}-\frac{25\,488\,603}{268\,435\,456}i$

A matrix function with exact entries at eigenvalues $-1$ and 5

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In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\mathrm{QPochhammer}(3, \operatorname{\_1}, 5), \bigl\lbrack\bigl\lbrack-1, 0\bigr\rbrack, \bigl\lbrack0, 5\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack-128, 0\bigr\rbrack, \bigl\lbrack0, -1\,452\,215\,072\bigr\rbrack\bigr\rbrack$

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In$\mathrm{FunctionExpand}(\mathrm{QPochhammer}(2, q, 5))$Out$(1-2q^4)(1-2q^3)(1-2q^2)(2q-1)$

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In$\operatorname{Normal}(\operatorname{Series}(\mathrm{QPochhammer}(n, q, x), q, 0, 3))$Out$\frac{1}{6}\mathrm{QPochhammer}^{(0, 3, 0)}(n, 0, x)q^3+\frac{1}{2}\mathrm{QPochhammer}^{(0, 2, 0)}(n, 0, x)q^2+q\mathrm{QPochhammer}^{(0, 1, 0)}(n, 0, x)+\mathrm{QPochhammer}(n, 0, x)$

Properties #

The empty product

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In$\mathrm{QPochhammer}(a, q, 0)$Out$1$

Zero factors give the empty product 1

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In$\mathrm{QPochhammer}(0, 0, 0)$Out$1$

Known value · Wolfram MathematicalFunctionData, QPochhammer

With $a=q=0$ and one factor the value is still 1

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In$\mathrm{QPochhammer}(0, 0, 1)$Out$1$

Known value · Wolfram MathematicalFunctionData, QPochhammer

About

Examples	18
Checked against	Wolfram Language 17 agree1 differSageMath 11 agree1 unevaluated3 error
Crosswalk	Wolfram LanguageSageMath

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/q-series.ts

finite n: exact boxed Multiply/Subtract/Power, evaluated in place. Infinite n: a capped product until |q^k| underflows.

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

QPochhammer[a, q, n]; mpmath.qp(a, q, n).

mappedsagepackages/oracle/src/mappings.ts

See also: QFactorial, QBinomial