What's New
About the Project
NIST
17 q-Hypergeometric and Related FunctionsProperties

§17.10 Transformations of ψrr Functions

Bailey’s ψ22 Transformations

17.10.1 ψ22(a,bc,d;q,z)=(az,d/a,c/b,dq/(abz);q)(z,d,q/b,cd/(abz);q)ψ22(a,abz/daz,c;q,da),
17.10.2 ψ22(a,bc,d;q,z)=(az,bz,cq/(abz),dq/(abz);q)(q/a,q/b,c,d;q)×ψ22(abz/c,abz/daz,bz;q,cdabz).

Other Transformations

17.10.3 ψ88(qa12,-qa12,c,d,e,f,aq-n,q-na12,-a12,aq/c,aq/d,aq/e,aq/f,qn+1,aqn+1;q,a2q2n+2cdef)=(aq,q/a,aq/(cd),aq/(ef);q)n(q/c,q/d,aq/e,aq/f;q)nψ44(e,f,aqn+1/(cd),q-naq/c,aq/d,qn+1,ef/(aqn);q,q),
17.10.4 ψ22(e,faq/c,aq/d;q,aqef)=(q/c,q/d,aq/e,aq/f;q)(aq,q/a,aq/(cd),aq/(ef);q)×n=-(1-aq2n)(c,d,e,f;q)n(1-a)(aq/c,aq/d,aq/e,aq/f;q)n(qa3cdef)nqn2.
17.10.5 (aq/b,aq/c,aq/d,aq/e,q/(ab),q/(ac),q/(ad),q/(ae);q)(fa,ga,f/a,g/a,qa2,q/a2;q)×ψ88(qa,-qa,ba,ca,da,ea,fa,gaa,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g;q,q2bcdefg)=(q,q/(bf),q/(cf),q/(df),q/(ef),qf/b,qf/c,qf/d,qf/e;q)(fa,q/(fa),aq/f,f/a,g/f,fg,qf2;q)×ϕ78(f2,qf,-qf,fb,fc,fd,fe,fgf,-f,fq/b,fq/c,fq/d,fq/e,fq/g;q,q2bcdefg)+idem(f;g).
17.10.6 (aq/b,aq/c,aq/d,aq/e,aq/f,q/(ab),q/(ac),q/(ad),q/(ae),q/(af);q)(ag,ah,ak,g/a,h/a,k/a,qa2,q/a2;q)×ψ1010(qa,-qa,ba,ca,da,ea,fa,ga,ha,kaa,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g,aq/h,aq/k;q,q2bcdefghk)=(q,q/(bg),q/(cg),q/(dg),q/(eg),q/(fg),qg/b,qg/c,qg/d,qg/e,qg/f;q)(gh,gk,h/g,ag,q/(ag),g/a,aq/g,qg2;q)×ϕ910(g2,qg,-qg,gb,gc,gd,ge,gf,gh,gkg,-g,qg/b,qg/c,qg/d,qg/e,qg/f,qg/h,qg/k;q,q2bcdefghk)+idem(g;h,k).