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17 q-Hypergeometric and Related FunctionsNotation

Β§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 (βˆˆβ„‚)

base: unless stated otherwise |q|<1.

(a;q)n

q-shifted factorial: (1βˆ’a)⁒(1βˆ’a⁒q)⁒⋯⁒(1βˆ’a⁒qnβˆ’1).

The main functions treated in this chapter are the basic hypergeometric (or q-hypergeometric) function Ο•sr⁑(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 Ξ¦(1)⁑(a;b,bβ€²;c;q;x,y), Ξ¦(2)⁑(a;b,bβ€²;c,cβ€²;q;x,y), Ξ¦(3)⁑(a,aβ€²;b,bβ€²;c;q;x,y), and Ξ¦(4)⁑(a,b;c,cβ€²;q;x,y).

Another function notation used is the β€œidem” function:

f⁑(Ο‡1;Ο‡2,…,Ο‡n)+idem⁑(Ο‡1;Ο‡2,…,Ο‡n)=βˆ‘j=1nf⁑(Ο‡j;Ο‡1,Ο‡2,…,Ο‡jβˆ’1,Ο‡j+1,…,Ο‡n).

These notations agree with Gasper and Rahman (2004). A slightly different notation is that in Bailey (1964) and Slater (1966); see Β§17.4(i). Fine (1988) uses F⁑(a,b;t:q) for a particular specialization of a Ο•12 function.