Digital Library of Mathematical Functions
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17 q-Hypergeometric and Related FunctionsProperties

§17.8 Special Cases of ψrr Functions

Jacobi’s Triple Product

17.8.1 n=-(-z)nqn(n-1)/2=(q,z,q/z;q);

compare (20.5.9).

Ramanujan’s ψ11 Summation

Quintuple Product Identity

17.8.3 n=-(-1)nqn(3n-1)/2z3n(1+zqn)=(q,-z,-q/z;q)(qz2,q/z2;q2).

Bailey’s Bilateral Summations

17.8.4 ψ22(b,c;aq/b,aq/c;q,-aq/(bc)) =(aq/(bc);q)(aq2/b2,aq2/c2,q2,aq,q/a;q2)(aq/b,aq/c,q/b,q/c,-aq/(bc);q),
17.8.5 ψ33(b,c,dq/b,q/c,q/d;q,qbcd) =(q,q/(bc),q/(bd),q/(cd);q)(q/b,q/c,q/d,q/(bcd);q),
17.8.6 ψ44(-qa12,b,c,d-a12,aq/b,aq/c,aq/d;q,qa32bcd)=(aq,aq/(bc),aq/(bd),aq/(cd),qa12/b,qa12/c,qa12/d,q,q/a;q)(aq/b,aq/c,aq/d,q/b,q/c,q/d,qa12,qa-12,qa32/(bcd);q),
17.8.7 ψ66(qa12,-qa12,b,c,d,ea12,-a12,aq/b,aq/c,aq/d,aq/e;q,qa2bcde)=(aq,aq/(bc),aq/(bd),aq/(be),aq/(cd),aq/(ce),aq/(de),q,q/a;q)(aq/b,aq/c,aq/d,aq/e,q/b,q/c,q/d,q/e,qa2/(bcde);q).