17 q-Hypergeometric and Related Functions Notation 17 q-Hypergeometric and Related Functions 17.2 Calculus

§17.1 Special Notation

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Defines:: $\mathrm{idem}\left(\NVar{\chi_{1}};\NVar{\chi_{2},\dots,\chi_{n}}\right)$ : $\mathrm{idem}$ function
Keywords:: bilateral basic hypergeometric function, bilateral $q$ -hypergeometric function, idem function, notation, $q$ -hypergeometric function
Referenced by:: Erratum (V1.0.10) for Section 17.1
Permalink:: http://dlmf.nist.gov/17.1
Correction (effective with 1.0.10):: The notation for $q$ -Appell functions has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.
See also:: Annotations for Ch.17

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

$k, j, m, n, r, s$	nonnegative integers.
$z$	complex variable.
$x$	real variable.
$q$ $(\in\mathbb{C})$	base: unless stated otherwise $\|q\|<1$ .
$\left(a;q\right)_{n}$	$q$ -shifted factorial: $(1-a)(1-aq)\cdots\left(1-aq^{n-1}\right)$ .

The main functions treated in this chapter are the basic hypergeometric (or $q$ -hypergeometric) function ${{}_{r}\phi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)$ , the bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function ${{}_{r}\psi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)$ , and the $q$ -analogs of the Appell functions $\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)$ , $\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)$ , $\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)$ , and $\Phi^{(4)}\left(a,b;c,c^{\prime};q;x,y\right)$ .

Another function notation used is the ‘‘idem’’ function:

f(\chi_{1};\chi_{2},\dots,\chi_{n})+\mathrm{idem}\left(\chi_{1};\chi_{2},\dots% ,\chi_{n}\right)=\sum_{j=1}^{n}f(\chi_{j};\chi_{1},\chi_{2},\dots,\chi_{j-1},% \chi_{j+1},\dots,\chi_{n}).

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 ${{}_{2}\phi_{1}}$ function.