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17 q-Hypergeometric and Related FunctionsProperties

Β§17.4 Basic Hypergeometric Functions

  1. Β§17.4(i) Ο•sr Functions
  2. §17.4(ii) ψsr Functions
  3. Β§17.4(iii) q-Appell Functions
  4. Β§17.4(iv) Classification

Β§17.4(i) Ο•sr Functions

17.4.1 Ο•sr+1⁑(a0,a1,a2,…,arb1,b2,…,bs;q,z)=Ο•sr+1⁑(a0,a1,…,ar;b1,b2,…,bs;q,z)=βˆ‘n=0∞(a0;q)n⁒(a1;q)n⁒⋯⁒(ar;q)n(q;q)n⁒(b1;q)n⁒⋯⁒(bs;q)n⁒((βˆ’1)n⁒q(n2))sβˆ’r⁒zn.

Here and elsewhere it is assumed that the bj do not take any of the values qβˆ’n. The infinite series converges for all z when s>r, and for |z|<1 when s=r.

17.4.2 limqβ†’1βˆ’β‘Ο•rr+1⁑(qa0,qa1,…,qarqb1,…,qbr;q,z)=Frr+1⁑(a0,a1,…,arb1,…,br;z).

For the function on the right-hand side see Β§16.2(i).

This notation is from Gasper and Rahman (2004). It is slightly at variance with the notation in Bailey (1964) and Slater (1966). In these references the factor ((βˆ’1)n⁒q(n2))sβˆ’r is not included in the sum. In practice this discrepancy does not usually cause serious problems because the case most often considered is r=s.

§17.4(ii) ψsr Functions

17.4.3 ψsr⁑(a1,a2,…,arb1,b2,…,bs;q,z)=ψsr⁑(a1,a2,…,ar;b1,b2,…,bs;q,z)=βˆ‘n=βˆ’βˆžβˆž(a1,a2,…,ar;q)n⁒(βˆ’1)(sβˆ’r)⁒n⁒q(sβˆ’r)⁒(n2)⁒zn(b1,b2,…,bs;q)n=βˆ‘n=0∞(a1,a2,…,ar;q)n⁒(βˆ’1)(sβˆ’r)⁒n⁒q(sβˆ’r)⁒(n2)⁒zn(b1,b2,…,bs;q)n+βˆ‘n=1∞(q/b1,q/b2,…,q/bs;q)n(q/a1,q/a2,…,q/ar;q)n⁒(b1⁒b2⁒⋯⁒bsa1⁒a2⁒⋯⁒ar⁒z)n.

Here and elsewhere the bj must not take any of the values qβˆ’n, and the aj must not take any of the values qn+1. The infinite series converge when sβ‰₯r provided that |(b1⁒⋯⁒bs)/(a1⁒⋯⁒ar⁒z)|<1 and also, in the case s=r, |z|<1.

17.4.4 limqβ†’1βˆ’β‘Οˆrr⁑(qa1,qa2,…,qarqb1,qb2,…,qbr;q,z)=Hrr⁑(a1,a2,…,arb1,b2,…,br;z).

For the function Hrr see Β§16.4(v).

Β§17.4(iii) q-Appell Functions

The following definitions apply when |x|<1 and |y|<1:

17.4.5 Ξ¦(1)⁑(a;b,bβ€²;c;q;x,y) =βˆ‘m,nβ‰₯0(a;q)m+n⁒(b;q)m⁒(bβ€²;q)n⁒xm⁒yn(q;q)m⁒(q;q)n⁒(c;q)m+n,
17.4.6 Ξ¦(2)⁑(a;b,bβ€²;c,cβ€²;q;x,y) =βˆ‘m,nβ‰₯0(a;q)m+n⁒(b;q)m⁒(bβ€²;q)n⁒xm⁒yn(q,c;q)m⁒(q,cβ€²;q)n,
17.4.7 Ξ¦(3)⁑(a,aβ€²;b,bβ€²;c;q;x,y) =βˆ‘m,nβ‰₯0(a,b;q)m⁒(aβ€²,bβ€²;q)n⁒xm⁒yn(q;q)m⁒(q;q)n⁒(c;q)m+n,
17.4.8 Ξ¦(4)⁑(a,b;c,cβ€²;q;x,y) =βˆ‘m,nβ‰₯0(a,b;q)m+n⁒xm⁒yn(q,c;q)m⁒(q,cβ€²;q)n.

Β§17.4(iv) Classification

The series (17.4.1) is said to be balanced or SaalschΓΌtzian when it terminates, r=s, z=q, and

17.4.9 q⁒a0⁒a1⁒⋯⁒as=b1⁒b2⁒⋯⁒bs.

The series (17.4.1) is said to be k-balanced when r=s and

17.4.10 qk⁒a0⁒a1⁒⋯⁒as=b1⁒b2⁒⋯⁒bs.

The series (17.4.1) is said to be well-poised when r=s and

17.4.11 a0⁒q=a1⁒b1=a2⁒b2=β‹―=as⁒bs.

The series (17.4.1) is said to be very-well-poised when r=s, (17.4.11) is satisfied, and

17.4.12 b1=βˆ’b2=a0.

The series (17.4.1) is said to be nearly-poised when r=s and

17.4.13 a0⁒q=a1⁒b1=a2⁒b2=β‹―=asβˆ’1⁒bsβˆ’1.