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QPochhammer

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

QPochhammer[a, q, n]

gives the q-Pochhammer symbol prod_{k=0}^{n-1} (1 - a q^k).

QPochhammer[a, q]

gives the infinite q-Pochhammer (a;q)_Inf for |q|\<1. The finite form is exact/symbolic for a non-negative integer n; the infinite form evaluates for machine-real a, q. Listable, NumericFunction.

Examples (5)

Every input below was run against the current Mathilda build and its output recorded.

Basic examples (1)

In[1]:= QPochhammer[a, q, 3]
Out[1]= (1 - a) (1 - a q) (1 - a q^2)

Applications (4)

The finite q-shifted factorial

In[2]:= QPochhammer[a, q, 3]
Out[2]= (1 - a) (1 - a q) (1 - a q^2)

In[3]:= QPochhammer[a, q, 0]
Out[3]= 1

Exact rational arguments collapse to a rational

In[4]:= QPochhammer[2, 1/3, 4]
Out[4]= -175/729

The infinite form at machine precision

In[5]:= QPochhammer[0.5, 0.5]
Out[5]= 0.288788

Algorithm

Mathilda -- QPochhammer, the q-Pochhammer symbol (q-shifted factorial).

  QPochhammer[a, q, n] = prod_{k=0}^{n-1} (1 - a q^k)
  QPochhammer[a, q]     = prod_{k=0}^{Infinity} (1 - a q^k)   ((a;q)_inf)

Finite form (3 args): for a non-negative integer n the product is built and handed to the evaluator, which reduces it exactly for exact a, q and at

machine / MPFR precision for inexact a, q (so N works through it).  A

symbolic / non-integer n is left unevaluated -- which is exactly what Product relies on to emit QPochhammer[a, q, n] as a closed form.

Infinite form (2 args): evaluated for machine-real a, q with |q| < 1 by

accumulating factors until they fall below machine epsilon.  Symbolic or
|q| >= 1 inputs stay unevaluated.

Memory: honours the builtin ownership contract (never frees res).

Implementation notes

Algorithm. builtin_qpochhammer evaluates the q-Pochhammer symbol. The three-argument finite form QPochhammer[a, q, n] = prod_{k=0}^{n-1} (1 - a q^k): for a non-negative integer n the product is built as an Expr and handed to the evaluator, which reduces it exactly for exact a, q and at machine / MPFR precision for inexact a, q; n = 0 -> 1; a symbolic or non-integer n is left unevaluated (which is exactly what Product relies on to emit the closed form). The two-argument infinite form QPochhammer[a, q] = (a; q)_inf is evaluated only for machine-real a, q with |q| < 1 and at least one inexact operand, by accumulating (1 - a q^k) in double until the factor falls below machine epsilon; an all-exact or |q| >= 1 input stays symbolic.

Data structures. Expr; the finite form goes through the ordinary evaluator (GMP for exact, libm/MPFR for inexact); the infinite form is a double accumulation only. ND: binary kernel NDK_BIN2(QPochhammer, sf_machine_qpochhammer), registered REG_B, so packed_aware. Attributes: Listable, NumericFunction, Protected.

Complexity / limits. The finite form is O(n) factors; the infinite form is machine double only (no MPFR kernel — so N[QPochhammer[a, q], p] returns a machine-precision value regardless of p) and requires |q| < 1, summing up to 100000 factors. Compile[] lowers at both scalar and rank-1 array shapes (Compiled -> True).

Attributes: Listable, NumericFunction, Protected.

References

See also: Product

Notes & additional examples

Notes

QPochhammer[a, q, n] is the finite q-Pochhammer symbol (q-shifted factorial) prod_{k=0}^{n-1} (1 - a q^k), with QPochhammer[a, q, 0] = 1. For a non-negative integer n the product is formed and evaluated — exactly for exact a, q, numerically when they are inexact. A symbolic or non-integer n is left unevaluated, which is what Product relies on to return the symbol as a closed form.

The two-argument QPochhammer[a, q] is the infinite product (a; q)_inf, evaluated for machine-real a, q with |q| < 1. That infinite form is a machine-precision kernel only, so a request for extra digits via N[..., p] still returns a machine-precision value.