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17 q-Hypergeometric and Related FunctionsProperties

§17.9 Transformations of Higher ϕrr Functions

Contents

§17.9(i) ϕ12ϕ22, ϕ13, or ϕ23

F. H. Jackson’s Transformations

17.9.1 ϕ12(a,bc;q,z) =(za;q)(z;q)ϕ22(a,c/bc,az;q,bz),
17.9.2 ϕ12(q-n,bc;q,z) =(c/b;q)n(c;q)nbnϕ13(q-n,b,q/cbq1-n/c;q,z/c),
17.9.3 ϕ12(a,bc;q,z) =(abz/c;q)(bz/c;q)ϕ23(a,c/b,0c,cq/bz;q,q),
17.9.4 ϕ12(q-n,bc;q,z) =(c/b;q)n(c;q)n(bzq)nϕ23(q-n,q/z,q1-n/cbq1-n/c,0;q,q),
17.9.5 ϕ12(q-n,bc;q,z) =(c/b;q)n(c;q)nϕ23(q-n,b,bzq-n/cbq1-n/c,0;q,q).

§17.9(ii) ϕ23ϕ23

Transformations of ϕ23-Series

17.9.6 ϕ23(a,b,cd,e;q,de/(abc)) =(e/a,de/(bc);q)(e,de/(abc);q)ϕ23(a,d/b,d/cd,de/(bc);q,e/a),
17.9.7 ϕ23(a,b,cd,e;q,de/(abc)) =(b,de/(ab),de/(bc);q)(d,e,de/(abc);q)ϕ23(d/b,e/b,de/(abc)de/(ab),de/(bc);q,b),
17.9.8 ϕ23(q-n,b,cd,e;q,q) =(de/(bc);q)n(e;q)n(bcd)nϕ23(q-n,d/b,d/cd,de/(bc);q,q),
17.9.9 ϕ23(q-n,b,cd,e;q,q) =(e/c;q)n(e;q)ncnϕ23(q-n,c,d/bd,cq1-n/e;q,bqe),
17.9.10 ϕ23(q-n,b,cd,e;q,deqnbc) =(e/c;q)n(e;q)nϕ23(q-n,c,d/bd,cq1-n/e;q,q).

q-Sheppard Identity

17.9.11 ϕ23(q-n,b,cd,e;q,q)=(e/c,d/c;q)n(e,d;q)ncnϕ23(q-n,c,cbq1-n/(de)cq1-n/e,cq1-n/d;q,q),
17.9.12 ϕ23(a,b,cd,e;q,deabc)=(e/b,e/c,cq/a,q/d;q)(e,cq/d,q/a,e/(bc);q)ϕ23(c,d/a,cq/ecq/a,bcq/e;q,bqd)-(q/d,eq/d,b,c,d/a,de/(bcq),bcq2/(de);q)(d/q,e,bq/d,cq/d,q/a,e/(bc),bcq/e;q)×ϕ23(aq/d,bq/d,cq/dq2/d,eq/d;q,deabc),
17.9.13 ϕ23(a,b,cd,e;q,deabc)=(e/b,e/c;q)(e,e/(bc);q)ϕ23(d/a,b,cd,bcq/e;q,q)+(d/a,b,c,de/(bc);q)(d,e,bc/e,de/(abc);q)ϕ23(e/b,e/c,de/(abc)de/(bc),eq/(bc);q,q).

§17.9(iii) Further ϕsr Functions

Sears’ Balanced ϕ34 Transformations

With def=abcq1-n

17.9.14 ϕ34(q-n,a,b,cd,e,f;q,q)=(e/a,f/a;q)n(e,f;q)nanϕ34(q-n,a,d/b,d/cd,aq1-n/e,aq1-n/f;q,q)=(a,ef/(ab),ef/(ac);q)n(e,f,ef/(abc);q)nϕ34(q-n,e/a,f/a,ef/(abc)ef/(ab),ef/(ac),q1-n/a;q,q).

Watson’s q-Analog of Whipple’s Theorem

With n a nonnegative integer

17.9.15 (aq,aq/(de);q)n(aq/d,aq/e;q)nϕ34(aq/(bc),d,e,q-naq/b,aq/c,deq-n/a;q,q)=ϕ78(a,qa12,-qa12,b,c,d,e,q-na12,-a12,aq/b,aq/c,aq/d,aq/e,aqn+1;q,a2q2+nbcde).

Bailey’s Transformation of Very-Well-Poised ϕ78

17.9.16 ϕ78(a,qa12,-qa12,b,c,d,e,fa12,-a12,aq/b,aq/c,aq/d,aq/e,aq/f;q,a2q2bcdef)=(aq,aq/(de),aq/(df),aq/(ef);q)(aq/d,aq/e,aq/f,aq/(def);q)ϕ34(aq/(bc),d,e,faq/b,aq/c,def/a;q,q)+(aq,aq/(bc),d,e,f,a2q2/(bdef),a2q2/(cdef);q)(aq/b,aq/c,aq/d,aq/e,aq/f,a2q2/(bcdef),def/(aq);q)×ϕ34(aq/(de),aq/(df),aq/(ef),a2q2/(bcdef)a2q2/(bdef),a2q2/(cdef),aq2/(def);q,q).

Sears–Carlitz Transformation

With a=q-n and n a nonnegative integer,

17.9.17 ϕ23(a,b,caq/b,aq/c;q,aqzbc)=(az;q)(z;q)ϕ45(a12,-a12,(aq)12,-(aq)12,aq/(bc)aq/b,aq/c,az,q/z;q,q).

Gasper’s q-Analog of Clausen’s Formula

17.9.18 (ϕ34(a,b,abz,ab/zabq12,-abq12,-ab;q,q))2=ϕ45(a2,b2,ab,abz,ab/zabq12,-abq12,-ab,a2b2;q,q),

provided that the series expansions of both ϕ’s terminate.

§17.9(iv) Bibasic Series

Mixed-Base Heine-Type Transformations

17.9.19 n=0(a;q2)n(b;q)n(q2;q2)n(c;q)nzn=(b;q)(az;q2)(c;q)(z;q2)n=0(c/b;q)2n(z;q2)nb2n(q;q)2n(az;q2)n+(b;q)(azq;q2)(c;q)(zq;q2)n=0(c/b;q)2n+1(zq;q2)nb2n+1(q;q)2n+1(azq;q2)n.
17.9.20 n=0(a;qk)n(b;q)knzn(qk;qk)n(c;q)kn=(b;q)(az;qk)(c;q)(z;qk)n=0(c/b;q)n(z;qk)nbn(q;q)n(az;qk)n,
k=1,2,3,.