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17 q-Hypergeometric and Related FunctionsProperties

§17.4 Basic Hypergeometric Functions

Contents

§17.4(i) ϕsr Functions

17.4.1 ϕsr+1(a0,a1,a2,,arb1,b2,,bs;q,z)=ϕsr+1(a0,a1,,ar;b1,b2,,bs;q,z)=n=0(a0;q)n(a1;q)n(ar;q)n(q;q)n(b1;q)n(bs;q)n((-1)nq(n2))s-rzn.

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

17.4.2 limq1-ϕrr+1(qa0,qa1,,qarqb1,,qbr;q,z)=Frr+1(a0,a1,,arb1,,br;z).

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

This notation is from Gasper and Rahman (2004). It is slightly at variance with the notation in Bailey (1935) and Slater (1966). In these references the factor ((-1)nq(n2))s-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)=n=-(a1,a2,,ar;q)n(-1)(s-r)nq(s-r)(n2)zn(b1,b2,,bs;q)n=n=0(a1,a2,,ar;q)n(-1)(s-r)nq(s-r)(n2)zn(b1,b2,,bs;q)n+n=1(q/b1,q/b2,,q/bs;q)n(q/a1,q/a2,,q/ar;q)n(b1b2bsa1a2arz)n.

Here and elsewhere the bj must not take any of the values q-n, and the aj must not take any of the values qn+1. The infinite series converge when sr provided that |(b1bs)/(a1arz)|<1 and also, in the case s=r, |z|<1.

§17.4(iii) Appell Functions

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

17.4.5 Φ(1)(a;b,b;c;x,y) =m,n0(a;q)m+n(b;q)m(b;q)nxmyn(q;q)m(q;q)n(c;q)m+n,
17.4.6 Φ(2)(a;b,b;c,c;x,y) =m,n0(a;q)m+n(b;q)m(b;q)nxmyn(q;q)m(q;q)n(c;q)m(c;q)n,
17.4.7 Φ(3)(a,a;b,b;c;x,y) =m,n0(a,b;q)m(a,b;q)nxmyn(q;q)m(q;q)n(c;q)m+n,
17.4.8 Φ(4)(a;b;c,c;x,y) =m,n0(a,b;q)m+nxmyn(q,c;q)m(q,c;q)n.

§17.4(iv) Classification

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

17.4.9 qa0a1as=b1b2bs.

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

17.4.10 qka0a1as=b1b2bs.

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

17.4.11 a0q=a1b1=a2b2==asbs.

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=-b2=a0.

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

17.4.13 a0q=a1b1=a2b2==as-1bs-1.