Digital Library of Mathematical Functions
About the Project
NIST
17 q-Hypergeometric and Related FunctionsProperties

§17.6 ϕ12 Function

Contents

§17.6(i) Special Values

q-Gauss Sum

First q-Chu–Vandermonde Sum

Second q-Chu–Vandermonde Sum

Andrews–Askey Sum

17.6.4 ϕ12(b2,b2/cc;q2,cq/b2)=12(b2,q;q2)(c,cq/b2;q2)((c/b;q)(b;q)+(-c/b;q)(-b;q)),
|cq|<|b2|.

Bailey–Daum q-Kummer Sum

17.6.5 ϕ12(a,baq/b;q,-q/b)=(-q;q)(aq,aq2/b2;q2)(-q/b,aq/b;q),
|b|>|q|.

§17.6(ii) ϕ12 Transformations

Heine’s First Transformation

Heine’s Second Tranformation

17.6.7 ϕ12(a,bc;q,z)=(c/b,bz;q)(c,z;q)ϕ12(abz/c,bbz;q,c/b),
|z|<1,|c|<|b|.

Heine’s Third Transformation

17.6.8 ϕ12(a,bc;q,z)=(abz/c;q)(z;q)ϕ12(c/a,c/bc;q,abz/c),
|z|<1,|abz|<|c|.

Fine’s First Transformation

17.6.9 ϕ12(q,aqbq;q,z)=-(1-b)(aq/b)(1-(aq/b))n=0(aq,azq/b;q)nqn(azq2/b;q)n+(aq,azq/b;q)(aq/b;q)ϕ12(q,0bq;q,z),
|z|<1.

Fine’s Second Transformation

17.6.10 (1-z)ϕ12(q,aqbq;q,z)=n=0(b/a;q)n(-az)nq(n2+n)/2(bq,zq;q)n,
|z|<1.

Fine’s Third Transformation

17.6.11 1-z1-bϕ12(q,aqbq;q,z)=n=0(aq;q)n(azq/b;q)2nbn(zq,aq/b;q)n-aqn=0(aq;q)n(azq/b;q)2n+1(bq)n(zq;q)n(aq/b;q)n+1,
|z|<1,|b|<1.

Rogers–Fine Identity

17.6.12 (1-z)ϕ12(q,aqbq;q,z)=n=0(aq,azq/b;q)n(bq,zq;q)n(1-azq2n+1)(bz)nqn2,
|z|<1.

Nonterminating Form of the q-Vandermonde Sum

17.6.13 ϕ12(a,b;c;q,q)+(q/c,a,b;q)(c/q,aq/c,bq/c;q)ϕ12(aq/c,bq/c;q2/c;q,q)=(q/c,abq/c;q)(aq/c,bq/c;q),
17.6.14 n=0(a;q)n(b;q2)nzn(q;q)n(azb;q2)n=(az,bz;q2)(z,azb;q2)ϕ12(a,bbz;q2,zq).

Three-Term ϕ12 Transformations

17.6.15 ϕ12(a,bc;q,z)=(abz/c,q/c;q)(az/c,q/a;q)ϕ12(c/a,cq/(abz)cq/(az);q,bq/c)-(b,q/c,c/a,az/q,q2/(az);q)(c/q,bq/c,q/a,az/c,cq/(az);q)ϕ12(aq/c,bq/cq2/c;q,z),
|z|<1,|bq|<|c|.
17.6.16 ϕ12(a,bc;q,z)=(b,c/a,az,q/(az);q)(c,b/a,z,q/z;q)ϕ12(a,aq/caq/b;q,cq/(abz))+(a,c/b,bz,q/(bz);q)(c,a/b,z,q/z;q)ϕ12(b,bq/cbq/a;q,cq/(abz)),
|z|<1, |abz|<|cq|.

§17.6(iii) Contiguous Relations

Heine’s Contiguous Relations

17.6.17 ϕ12(a,bc/q;q,z)-ϕ12(a,bc;q,z) =cz(1-a)(1-b)(q-c)(1-c)ϕ12(aq,bqcq;q,z),
17.6.18 ϕ12(aq,bc;q,z)-ϕ12(a,bc;q,z) =az1-b1-cϕ12(aq,bqcq;q,z),
17.6.19 ϕ12(aq,bcq;q,z)-ϕ12(a,bc;q,z) =az(1-b)(1-(c/a))(1-c)(1-cq)ϕ12(aq,bqcq2;q,z),
17.6.20 ϕ12(aq,b/qc;q,z)-ϕ12(a,bc;q,z) =az(1-b/(aq))1-cϕ12(aq,bcq;q,z),
17.6.21 b(1-a)ϕ12(aq,bc;q,z)-a(1-b)ϕ12(a,bqc;q,z) =(b-a)ϕ12(a,bc;q,z),
17.6.22 a(1-bc)ϕ12(a,b/qc;q,z)-b(1-ac)ϕ12(a/q,bc;q,z) =(a-b)(1-abzcq)ϕ12(a,bc;q,z),
17.6.23 q(1-ac)ϕ12(a/q,bc;q,z)+(1-a)(1-abzc)ϕ12(aq,bc;q,z)=(1+q-a-aqc+a2zc-abzc)ϕ12(a,bc;q,z),
17.6.24 (1-c)(q-c)(abz-c)ϕ12(a,bc/q;q,z)+z(c-a)(c-b)ϕ12(a,bcq;q,z)=(c-1)(c(q-c)+z(ca+cb-ab-abq))ϕ12(a,bc;q,z).

§17.6(iv) Differential Equations

Iterations of 𝒟

17.6.25 𝒟qnϕ12(a,bc;q,zd) =(a,b;q)ndn(c;q)n(1-q)nϕ12(aqn,bqncqn;q,dz),
17.6.26 𝒟qn((z;q)(abz/c;q)ϕ12(a,bc;q,z)) =(c/a,c/b;q)n(c;q)n(1-q)n(abc)n(zqn;q)(abz/c;q)ϕ12(a,bcqn;q,zqn).

q-Differential Equation

17.6.27 z(c-abqz)𝒟q2ϕ12(a,bc;q,z)+(1-c1-q+(1-a)(1-b)-(1-abq)1-qz)×𝒟qϕ12(a,bc;q,z)-(1-a)(1-b)(1-q)2ϕ12(a,bc;q,z)=0.

(17.6.27) reduces to the hypergeometric equation (15.10.1) with the substitutions aqa, bqb, cqc, followed by limq1-.

§17.6(v) Integral Representations

17.6.28 ϕ12(qα,qβqγ;q,z) =Γq(γ)Γq(β)Γq(γ-β)01tβ-1(tq;q)γ-β-1(xt;q)αqt.
17.6.29 ϕ12(a,bc;q,z) =(-12π)(a,b;q)(q,c;q)-(q1+ζ,cqζ;q)(aqζ,bqζ;q)π(-z)ζsin(πζ)ζ,

where |z|<1, |ph(-z)|<π, and the contour of integration separates the poles of (q1+ζ,cqζ;q)/sin(πζ) from those of 1/(aqζ,bqζ;q), and the infimum of the distances of the poles from the contour is positive.

§17.6(vi) Continued Fractions

For continued-fraction representations of the ϕ12 function, see Cuyt et al. (2008, pp. 395–399).