q-Binomial Coefficient

Author

Eric W. Weisstein
March 23, 2009

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

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Notation

QBinomial[n,k,q]//TraditionalForm

Definition

(QBinomial[n, k, q])==((QPochhammer[q, q, n])/((QPochhammer[q, q, k]) (QPochhammer[q, q, n-k])))

FullSimplify[((QPochhammer[q,q,n])/(QPochhammer[q,q,k]QPochhammer[q,q,n-k]))]

FullSimplify[((QPochhammer[q,q,n])/(QPochhammer[q,q,k]QPochhammer[q,q,n-k]))-QBinomial[n,k,q]]

Table[{((QPochhammer[q,q,n])/(QPochhammer[q,q,k]QPochhammer[q,q,n-k])),QBinomial[n,k,q]}/.{k->RandomInteger[{1,10}],n->RandomInteger[{10,10}],q->RandomComplex[]},{1}]

Product

(∏_(i=0)^(m-1))((1-(q^(n-i)))/(1-(q^(i+1))))==(QBinomial[n, m, q])

Product[((1-(q^(n-i)))/(1-(q^(i+1)))),{i,0,m-1}]

FullSimplify[%]

Table[{(((q^(-(1/2) (-1+m) m)) (((-(q^n)))^m) QPochhammer[(q^(-n)),q,m])/(QPochhammer[q,q,m])),QBinomial[n,m,q]}/.{n->RandomInteger[{1,10}],m->RandomInteger[{1,10}],q->RandomComplex[]},{1}]

Sum

(∑_(i=0)^(n-1))(q^i)==(((q^n)-1)/(q-1))==(QBinomial[n, 1, q])

Sum[(q^i),{i,0,n-1}]

FullSimplify[%]

QBinomial[n,1,q]//FunctionExpand

FullSimplify[%]

FullSimplify[((-1+(q^n))/(-1+q))-((QPochhammer[(q^n),q])/((1-q) QPochhammer[(q^(1+n)),q]))]

Identities

(QBinomial[n, m, q])==(((q^(-(1/2) (m-1) m)) (((-(q^n)))^m) (QPochhammer[(q^(-n)), q, m]))/(QPochhammer[q, q, m]))

Product[((1-(q^(n-i)))/(1-(q^(i+1)))),{i,0,m-1}]

FullSimplify[%,(m|n)∈Integers&&m>0&&n>0]

Table[{(((q^(-(1/2) (-1+m) m)) (((-(q^n)))^m) QPochhammer[(q^(-n)),q,m])/(QPochhammer[q,q,m])),QBinomial[n,m,q]}/.{n->RandomInteger[{1,10}],m->RandomInteger[{1,10}],q->RandomComplex[]},{1}]

Recurrence

(QBinomial[n+1, k, q])==(q^k) (QBinomial[n, k, q])+(QBinomial[n, k-1, q])

RSolve[{f[n+1,k]==f[n,k]+f[n,k-1],f[0,0]==1},f[n,k],{n,k}]

Subtract@@@Flatten[Table[{Binomial[n+1,k],Binomial[n,k]+Binomial[n,k-1]},{n,10},{k,n}]//FunctionExpand,1]//Expand//Union

RSolve[f[n+1,k]==(q^k)f[n,k]+f[n,k-1],f,{n,k}]

Subtract@@@Flatten[Table[{QBinomial[n+1,k,q],(q^k)QBinomial[n,k,q]+QBinomial[n,k-1,q]},{n,10},{k,n}]//FunctionExpand,1]//Expand//Union

Tables

Old

{Simplify[qBinomial[{4,1},q]],Simplify[qBinomial[{4,3},q]]}

Simplify[qBinomial[{4,2},q]]

Simplify[qBinomial2[{4,2},q]]

Expand[%]

Table[qBinomial[{i,j},q],{i,1,5},{j,1,i}]

Simplify[Table[qBinomial[{i,j},q],{i,1,5},{j,1,i}]]

Limit[Table[qBinomial[q,i,j],{i,1,4},{j,1,i}],q->1]

Table[Binomial[i,j],{i,1,4},{j,1,i}]

Table[{i,j},{i,1,4},{j,1,i}]

Table[Simplify[Expand[qBinomial[{n+1,k+1},q]/qBinomial[{n,k+1},q]-(1-q^(n+1))/(1-q^(n-k)) ] /. {n->Random[Integer,{6,10}],k->Random[Integer,{2,5}]}],{5}]

Table[Simplify[Expand[qBinomial[{n+1,k+1},q]/qBinomial[{n+1,k},q]-(1-q^(n-k+1))/(1-q^(k+1)) ] /. {n->Random[Integer,{6,10}],k->Random[Integer,{2,5}]}],{5}]

V7

TableForm[Flatten[Table[HoldForm[QBinomial[n,k,q]]==QBinomial[n,k,q]//FunctionExpand,{n,4},{k,n-1}],1],TableDepth->1]//TraditionalForm

Combinatorial

Identity

MyQBinomial[n_,m_,q_]:=Total[q^((Total/@Subsets[Range[n],{m}])-m (m+1)/2)]

MyQBinomial[10,3,q]//Factor

QBinomial[10,3,q]//FunctionExpand//Factor

Diagram

Partitions[1]

Partitions[2]

Partitions[3]

Partitions[4]

Partitions[5]

Show[GraphicsArray[Show[CellGraphics[#],DisplayFunction->Identity,PlotRange->{{0,3},{-3,0}},AspectRatio->Automatic]&/@{{{1}},{{1,1}},{{1},{1}},{{1,1},{1,0}},{{1,1},{1,1}}}]]

Derivative and Integral

Derivatives

D[QBinomial[n,k,q],q]

Integral

Integrate[QBinomial[n,k,q],q]