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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;q;x,y), Φ(2)(a;b,b;c,c;q;x,y), Φ(3)(a,a;b,b;c;q;x,y), and Φ(4)(a,b;c,c;q;x,y).

Another function notation used is the “idem” function:


These notations agree with Gasper and Rahman (2004). A slightly different notation is that in Bailey (1964) and Slater (1966); see §17.4(i). Fine (1988) uses F(a,b;t:q) for a particular specialization of a ϕ12 function.