What's New
About the Project
NIST
17 q-Hypergeometric and Related FunctionsNotation

§17.1 Special Notation

(For other notation see Notation for the Special Functions.)

k,j,m,n,r,s nonnegative integers.
z complex variable.
x real variable.
q() base: unless stated otherwise |q|<1.
(a;q)n q-shifted factorial: (1-a)(1-aq)(1-aqn-1).

The main functions treated in this chapter are the basic hypergeometric (or q-hypergeometric) function ϕsr(a1,a2,,ar;b1,b2,,bs;q,z), the bilateral basic hypergeometric (or bilateral q-hypergeometric) function ψsr(a1,a2,,ar;b1,b2,,bs;q,z), and the q-analogs of the Appell functions Φ(1)(a;b,b;c;x,y), Φ(2)(a;b,b;c,c;x,y), Φ(3)(a,a;b,b;c;x,y), and Φ(4)(a;b;c,c;x,y).

Another function notation used is the “idem” function:

f(χ1;χ2,,χn)+idem(χ1;χ2,,χn)=j=1nf(χj;χ1,χ2,,χj-1,χj+1,,χn).

These notations agree with Gasper and Rahman (2004) (except for the q-Appell functions which are not considered in this reference). A slightly different notation is that in Bailey (1935) and Slater (1966); see §17.4(i). Fine (1988) uses F(a,b;t:q) for a particular specialization of a ϕ12 function.