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17 q-Hypergeometric and Related FunctionsProperties

§17.7 Special Cases of Higher ϕsr Functions

Contents

§17.7(i) ϕ22 Functions

q-Analog of Bailey’s F12(-1) Sum

q-Analog of Gauss’s F12(-1) Sum

17.7.2 ϕ22(a2,b2abq12,-abq12;q,-q)=(a2q,b2q;q2)(q,a2b2q;q2).

Sum Related to (17.6.4)

17.7.3 ϕ22(c2/b2,b2c,cq;q2,q)=12(b2,q;q2)(c,cq;q2)((c/b;q)(b;q)+(-c/b;q)(-b;q)).

§17.7(ii) ϕ23 Functions

q-Pfaff–Saalschütz Sum

Nonterminating Form of the q-Saalschütz Sum

17.7.5 ϕ23(a,b,ce,f;q,q)+(q/e,a,b,c,qf/e;q)(e/q,aq/e,bq/e,cq/e,f;q)ϕ23(aq/e,bq/e,cq/eq2/e,qf/e;q,q)=(q/e,f/a,f/b,f/c;q)(aq/e,bq/e,cq/e,f;q),

where ef=abcq.

F. H. Jackson’s Terminating q-Analog of Dixon’s Sum

17.7.6 ϕ23(q-2n,b,cq1-2n/b,q1-2n/c;q,q2-nbc)=(b,c;q)n(q,bc;q)2n(q,bc;q)n(b,c;q)2n.

Continued Fractions

For continued-fraction representations of a ratio of ϕ23 functions, see Cuyt et al. (2008, pp. 399–400).

§17.7(iii) Other ϕsr Functions

q-Analog of Dixon’s F23(1) Sum

17.7.7 ϕ34(a,-qa12,b,c-a12,aq/b,aq/c;q,qa12bc)=(aq,qa12/b,qa12/c,aq/(bc);q)(aq/b,aq/c,qa12,qa12/(bc);q).

Gasper–Rahman q-Analog of Watson’s F23 Sum

17.7.8 ϕ78(λ,qλ12,-qλ12,a,b,c,-c,λq/c2λ12,-λ12,λq/a,λq/b,λq/c,-λq/c,c2;q,-λqab)=(λq,c2/λ;q)(aq,bq,c2q/a,c2q/b;q2)(λq/a,λq/b;q)(q,abq,c2q,c2q/(ab);q2),

where λ=-c(ab/q)12.

Andrews’ Terminating q-Analog of (17.7.8)

17.7.9 ϕ34(q-n,aqn,c,-c(aq)12,-(aq)12,c2;q,q)={0,n odd,cn(q,aq/c2;q2)n/2(aq,c2q;q2)n/2,n even.

Gasper–Rahman q-Analog of Whipple’s F23 Sum

17.7.10 ϕ78(-c,q(-c)12,-q(-c)12,a,q/a,c,-d,-q/d(-c)12,-(-c)12,-cq/a,-ac,-q,cq/d,cd;q,c)=(-c,-cq;q)(acd,acq/d,cdq/a,cq2/(ad);q2)(cd,cq/d,-ac,-cq/a;q).

Andrews’ Terminating q-Analog

17.7.11 ϕ34(q-n,qn+1,c,-ce,c2q/e,-q;q,q)=(eq-n,eqn+1,c2q1-n/e,c2qn+2/e;q2)(e,c2q/e;q).

First q-Analog of Bailey’s F34(1) Sum

17.7.12 ϕ34(a,aq,b2q2n,q-2nb,bq,a2q2;q2,q2)=an(-q,b/a;q)n(-aq,b;q)n.

Second q-Analog of Bailey’s F34(1) Sum

17.7.13 ϕ34(a,aq,b2q2n-2,q-2nb,bq,a2;q2,q2)=an(-q,b/a;q)n(1-bqn-1)(-a,b;q)n(1-bq2n-1).

F. H. Jackson’s q-Analog of Dougall’s F67(1) Sum

17.7.14 ϕ78(a,qa12,-qa12,b,c,d,e,q-na12,-a12,aq/b,aq/c,aq/d,aq/e,aqn+1;q,q)=(aq,aq/(bc),aq/(bd),aq/(cd);q)n(aq/b,aq/c,aq/d,aq/(bcd);q)n,

where a2q=bcdeq-n.

Limiting Cases of (17.7.14)

17.7.15 ϕ56(a,qa12,-qa12,b,c,da12,-a12,aq/b,aq/c,aq/d;q,aqbcd)=(aq,aq/(bc),aq/(bd),aq/(cd);q)(aq/b,aq/c,aq/d,aq/(bcd);q),

and when d=q-n,

17.7.16 ϕ56(a,qa12,-qa12,b,c,q-na12,-a12,aq/b,aq/c,aqn+1;q,aqn+1bc)=(aq,aq/(bc);q)n(aq/b,aq/c;q)n.

Bailey’s Nonterminating Extension of Jackson’s ϕ78 Sum

17.7.17 ϕ78(a,qa12,-qa12,b,c,d,e,fa12,-a12,aq/b,aq/c,aq/d,aq/e,aq/f;q,q)-ba(aq,c,d,e,f,bq/a,bq/c,bq/d,bq/e,bq/f;q)(aq/b,aq/c,aq/d,aq/e,aq/f,bc/a,bd/a,be/a,bf/a,b2q/a;q)×ϕ78(b2/a,qba-12,-qba-12,b,bc/a,bd/a,be/a,bf/aba-12,-ba-12,bq/a,bq/c,bq/d,bq/e,bq/f;q,q)=(aq,b/a,aq/(cd),aq/(ce),aq/(cf),aq/(de),aq/(df),aq/(ef);q)(aq/c,aq/d,aq/e,aq/f,bc/a,bd/a,be/a,bf/a;q),

where qa2=bcdef.

Gasper–Rahman q-Analogs of the Karlsson–Minton Sums

17.7.18 ϕr+1r+2(a,b,b1qm1,,brqmrbq,b1,,br;q,a-1q1-(m1++mr))=(q,bq/a;q)(b1/b;q)m1(br/b;q)mr(bq,q/a;q)(b1;q)m1(br;q)mrbm1++mr,

and

17.7.19 ϕrr+1(a,b1qm1,,brqmrb1,,br;q,a-1q1-(m1++mr))=0,

where m1,m2,,mr are arbitrary nonnegative integers.

Gosper’s Bibasic Sum

17.7.20 k=0n1-apkqk1-a(a;p)k(c;q)k(q;q)k(ap/c;p)kc-k=(ap;p)n(cq;q)n(q;q)n(ap/c;p)nc-n.

Gasper’s Extensions of Gosper’s Bibasic Sum

17.7.21 k=0n(1-apkqk)(1-bpkq-k)(1-a)(1-b)(a,b;p)k(c,a/(bc);q)k(q,aq/b;q)k(ap/c,bcp;p)kqk=(ap,bp;p)n(cq,aq/(bc);q)n(q,aq/b;q)n(ap/c,bcp;p)n,
17.7.22 k=-mn(1-adpkqk)(1-bpk/(dqk))(1-ad)(1-(b/d))×(a,b;p)k(c,ad2/(bc);q)k(dq,adq/b;q)k(adp/c,bcp/d;p)kqk=(1-a)(1-b)(1-c)(1-(ad2/(bc)))d(1-ad)(1-(b/d))(1-(c/d))(1-(ad/(bc)))×((ap,bp;p)n(cq,ad2q/(bc);q)n(dq,adq/b;q)n(adp/c,bcp/d;p)n-(c/(ad),d/(bc);p)m+1(1/d,b/(ad);q)m+1(1/c,bc/(ad2);q)m+1(1/a,1/b;p)m+1,)

and n-th difference generalization:

17.7.23 (1-aq)(1-bq)k=0n(apk,bp-k;q)n-1(1-(ap2k/b))(p;p)n(p;p)n-k(apk/b;q)n+1(-1)kp(k2)=δn,0.