q-Hypergeometric Function

Author

Eric W. Weisstein
March 12, 2009

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

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

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

Sums

(∑_(n=0)^∞)(((z^n) (QPochhammer[a, q, n]) (QPochhammer[b, q, n]))/((QPochhammer[q, q, n]) (QPochhammer[c, q, n])))==(QHypergeometricPFQSeq[a,b, c, q, z, 2, 1])

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

(∑_(n=0)^∞)(((z^n) (QPochhammer[a, q, n]) (QPochhammer[b, q, n]))/((QPochhammer[q, q, n]) (QPochhammer[c, q, n]) (QPochhammer[d, q, n])))==(QHypergeometricPFQSeq[a,b, c,d, q, z, 2, 2])

Sum[((QPochhammer[a,q,n]QPochhammer[b,q,n])/(QPochhammer[q,q,n]QPochhammer[c,q,n]QPochhammer[d,q,n]))((((-1))^(n ))(q^(n(n-1)/2)))(z^n),{n,0,∞}]

Special Case

2(Subscript[ϕ, 1])

((((((-1))^n)(q^(Binomial[n,2]))))^(1+s-r))/.{r->2,s->1}

Identities

(QHypergeometricPFQSeq[a,b, c, q, (c/(a b)), 2, 1])==(((QPochhammer2[(c/a), q]) (QPochhammer2[(c/b), q]))/((QPochhammer2[c, q]) (QPochhammer2[(c/(a b)), q])))

Last/@Select[Table[With[{a=RandomComplexDisk[],b=RandomComplexDisk[],c=RandomComplexDisk[],z=RandomComplexDisk[],q=RandomComplexDisk[]},{{a,b,c,q,z},QHypergeometricPFQ[{a,b},{c},q,(c/(a b))]==((QPochhammer[(c/a),q]QPochhammer[(c/b),q])/(QPochhammer[c,q]QPochhammer[(c/(a b)),q]))}],{50}],Abs[((#[[1,3]])/(#[[1,1]]#[[1,2]]))]<1&]

(QHypergeometricPFQSeq[a,b, c, q, z, 2, 1])==((((QPochhammer2[b, q]) (QPochhammer2[a z, q])) (QHypergeometricPFQSeq[(c/b),z, a z, q, b, 2, 1]))/((QPochhammer2[c, q]) (QPochhammer2[z, q])))

Table[With[{a=RandomComplexDisk[],b=RandomComplexDisk[],c=RandomComplexDisk[],z=RandomComplexDisk[],q=RandomComplexDisk[]},QHypergeometricPFQ[{a,b},{c},q,z]==((QPochhammer[b,q]*QPochhammer[a*z,q])/(QPochhammer[c,q]*QPochhammer[z,q]))*QHypergeometricPFQ[{c/b,z},{a*z},q,b]],{10}]//Union

(QHypergeometricPFQSeq[a,b, c, q, z, 2, 1])==(((QPochhammer2[(c/b), q]) (QPochhammer2[b z, q]))/((QPochhammer2[c, q]) (QPochhammer2[z, q])))(QHypergeometricPFQSeq[b,((a b z)/c), b z, q, (c/b), 2, 1])

Table[With[{a=RandomComplexDisk[],b=RandomComplexDisk[],c=RandomComplexDisk[],z=RandomComplexDisk[],q=RandomComplexDisk[]},QHypergeometricPFQ[{a,b},{c},q,z]==((QPochhammer[(c/b),q]QPochhammer[b z,q])/(QPochhammer[z,q]QPochhammer[c,q]))QHypergeometricPFQ[{b,((a b z)/c)},{b z},q,(c/b)]],{10}]

(QHypergeometricPFQSeq[a,b, c, q, z, 2, 1])==(((QPochhammer2[((a b z)/c), q]) (QHypergeometricPFQSeq[(c/a),(c/b), c, q, ((a b z)/c), 2, 1]))/(QPochhammer2[z, q]))

Table[With[{a=RandomComplexDisk[],b=RandomComplexDisk[],c=RandomComplexDisk[],z=RandomComplexDisk[],q=RandomComplexDisk[]},QHypergeometricPFQ[{a,b},{c},q,z]==(QPochhammer[(a*b*z)/c,q]/QPochhammer[z,q])*QHypergeometricPFQ[{c/a,c/b},{c},q,(a*b*z)/c]],{10}]//Union