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17 q-Hypergeometric and Related FunctionsProperties

§17.6 ϕ12 Function

Contents
  1. §17.6(i) Special Values
  2. §17.6(ii) ϕ12 Transformations
  3. §17.6(iii) Contiguous Relations
  4. §17.6(iv) Differential Equations
  5. §17.6(v) Integral Representations
  6. §17.6(vi) Continued Fractions

§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|.

Related formulas are (17.7.3), (17.8.8) and

17.6.4_5 ϕ12(b2,b2/ccq2;q2,cq3/b2)=12b(b2,q;q2)(cq2,cq/b2;q2)((cq/b;q)(b;q)(cq/b;q)(b;q)),
|cq3|<|b2|.

For similar formulas see Verma and Jain (1983).

Bailey–Daum q-Kummer Sum

§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)=(1b)(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

Fine’s Third Transformation

17.6.11 1z1bϕ12(q,aqbq;q,z)=n=0(aq;q)n(azq/b;q)2nbn(zq,aq/b;q)naqn=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 (1z)ϕ12(q,aqbq;q,z)=n=0(aq,azq/b;q)n(bq,zq;q)n(1azq2n+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|.

For a similar result for q-confluent hypergeometric functions see Morita (2013).

§17.6(iii) Contiguous Relations

Heine’s Contiguous Relations

17.6.17 ϕ12(a,bc/q;q,z)ϕ12(a,bc;q,z) =cz(1a)(1b)(qc)(1c)ϕ12(aq,bqcq;q,z),
17.6.18 ϕ12(aq,bc;q,z)ϕ12(a,bc;q,z) =az1b1cϕ12(aq,bqcq;q,z),
17.6.19 ϕ12(aq,bcq;q,z)ϕ12(a,bc;q,z) =az(1b)(1(c/a))(1c)(1cq)ϕ12(aq,bqcq2;q,z),
17.6.20 ϕ12(aq,b/qc;q,z)ϕ12(a,bc;q,z) =az(1b/(aq))1cϕ12(aq,bcq;q,z),
17.6.21 b(1a)ϕ12(aq,bc;q,z)a(1b)ϕ12(a,bqc;q,z) =(ba)ϕ12(a,bc;q,z),
17.6.22 a(1bc)ϕ12(a,b/qc;q,z)b(1ac)ϕ12(a/q,bc;q,z) =(ab)(1abzcq)ϕ12(a,bc;q,z),
17.6.23 q(1ac)ϕ12(a/q,bc;q,z)+(1a)(1abzc)ϕ12(aq,bc;q,z)=(1+qaaqc+a2zcabzc)ϕ12(a,bc;q,z),
17.6.24 (1c)(qc)(abzc)ϕ12(a,bc/q;q,z)+z(ca)(cb)ϕ12(a,bcq;q,z)=(c1)(c(qc)+z(ca+cbababq))ϕ12(a,bc;q,z).

§17.6(iv) Differential Equations

Iterations of 𝒟

q-Differential Equation

17.6.27 z(cabqz)𝒟q2ϕ12(a,bc;q,z)+(1c1q+(1a)(1b)(1abq)1qz)×𝒟qϕ12(a,bc;q,z)(1a)(1b)(1q)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(γβ)01tβ1(tq;q)γβ1(xt;q)αdqt.
17.6.29 ϕ12(a,bc;q,z) =(12πi)(a,b;q)(q,c;q)ii(q1+ζ,cqζ;q)(aqζ,bqζ;q)π(z)ζsin(πζ)dζ,

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).