q-Factorial

Author

Eric W. Weisstein
September 23, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/q-Factorial.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/q-Factorial.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Notation

QFactorial[k,q]//TraditionalForm

Numerics

Do[Print[{n,QFactorial[-.5,1.-10^-n]//Timing}],{n,0,7}]

Plots

<<Utilities`Plot`

Plot[Table[QFactorial[n,q],{q,.2,.8,.2}],{n,0,10},Evaluated->True,AxesLabel->{n,QFactorial[n,q]}]

Enumeration

V6

QFactorial[#,q]&/@Range[4]//TraditionalForm

Expand[%]//TraditionalForm

V7

QFactorial[#,q]&/@Range[4]//TraditionalForm

QFactorial[#,q]&/@Range[4]//FunctionExpand

As Ratio

Table[QPochhammer[q,q,k]/(1-q)^k,{k,4}]//FunctionExpand//Cancel//TraditionalForm

Limits

Limit[QFactorial[#,q]&/@Range[10],q->1]

(Subscript[qFaq, 2])[k_,q_]:=qSeries[q,q,k]/(1-q)^k;

{(Subscript[qFaq, 2])[5,q],Simplify[(Subscript[qFaq, 2])[5,q]]}

Identity

(QFactorial[k, q])==((QPochhammer[q, q, k])/(((1-q))^k))

FullSimplify[QFactorial[k,q]-((QPochhammer[q,q,k])/(((1-q))^k))]

FullSimplify[%]

Table[QFactorial[k,q]-(((1-q))^(-k)) QPochhammer[q,q,k]/.{q->RandomReal[WorkingPrecision->20],k->RandomInteger[{1,10}]},{1000}]//Chop//Union

Derivative and Integral

Derivatives

D[QFactorial[n,q],q]

Integral

Integrate[QFactorial[n,q],q]