q-Exponential Function

Author

Eric W. Weisstein
September 23, 2008

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

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

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

Code

Sums

(∑_(n=0)^∞)(((z^n) (q^(Binomial[n, 2])))/(QPochhammer[q, q, n]))==(Subscript[e, q])(z)==(QPochhammer2[-z, q])

Sum[((q^(Binomial[n,2]))/(QPochhammer[q,q,n]))(z^n),{n,0,∞}]//InputForm

Sum[((q^(Binomial[n,2]))/(QPochhammer[q,q,n]))(z^n),{n,0,20}]/.{q->.2,z->.3}

QExpE[z,q]

QExpE[.3,.2]

(∑_(n=0)^∞)((z^n)/(QPochhammer[q, q, n]))==(Subscript[e, q])(z)==QHypergeometricPFQ({0},{},z,q)

Sum[((z^n)/(QPochhammer[q,q,n])),{n,0,∞}]//InputForm

Sum[((z^n)/(QPochhammer[q,q,n])),{n,0,20}]/.{q->.2,z->.3}

QExpe[z,q]

QExpe[.2,.3]

Identities

(Subscript[e, q])(z)==QHypergeometricPFQ({0},{},z,q)==(1/(QPochhammer2[z, q]))/;(Abs[z])<1

{QHypergeometricPFQ[{0},{},z,q],(1/(QPochhammer[z,q]))}/.{q->.2,z->.2}

FunctionExpand[QHypergeometricPFQ[{0},{},z,q],Abs[z]<1]

QHypergeometricPFQ[{0},{},z,q]//TraditionalForm