q-Product

Author

Eric W. Weisstein
August 17, 2008

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

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

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

Products

(Subscript[Q, 0])==(∏_n^∞)(1-(q^(2 n)))==(QPochhammer2[(q^2), (q^2)])

Product[1-(q^(2n)),{n,∞}]

Q0[q_]:=QPochhammer[(q^2)]

(Subscript[Q, 1])==(∏_n^∞)((q^(2 n))+1)==((QPochhammer2[-1, (q^2)])/2)

Product[1+(q^(2n)),{n,∞}]

Q1[q_]:=(1/2) QPochhammer[-1,(q^2)]

(Subscript[Q, 2])==(∏_n^∞)((q^(2 n))+1-1)==((q (QPochhammer2[-(1/q), (q^2)]))/(q+1))

Product[1+(q^(2n-1)),{n,∞}]//FullSimplify

Q2[q_]:=((q QPochhammer[-(1/q),(q^2)])/(1+q))

(Subscript[Q, 3])==(∏_n^∞)(1-(q^(2 n-1)))==((q (QPochhammer2[(1/q), (q^2)]))/(q-1))

Product[1-(q^(2n-1)),{n,∞}]

Q3[q_]:=((q QPochhammer[(1/q),(q^2)])/(-1+q))

Identities

Q0(q) Q1(q)==Q0((q^2))

FullSimplify[Q0[q] Q1[q]-Q0[(q^2)]]

Series[%,{q,0,50}]

Q0(q)Q3(q)=Q0((q^(1/2)))

FullSimplify[Q0[q]Q3[q]-Q0[(q^(1/2))]]

Series[%,{q,0,50}]

Q2(q)Q3(q)==Q3((q^2))

FullSimplify[Q2[q]Q3[q]-Q3[(q^2)]]

Series[%,{q,0,50}]

Q1(q) Q2(q)==Q1((q^(1/2)))

FullSimplify[Q1[q]Q2[q]-Q1[(q^(1/2))]]

Series[%,{q,0,50}]