§17.1 Special Notation

Defines:: $\mathop{\mathrm{idem}\/}\nolimits\!\left(\chi _{1};\chi _{2},\dots,\chi _{n}\right)$ : $\mathop{\mathrm{idem}\/}\nolimits$ function
Keywords:: bilateral basic hypergeometric function, bilateral $q$ -hypergeometric function, $q$ -hypergeometric function
Permalink:: http://dlmf.nist.gov/17.1

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

$k,j,m,n,r,s$	nonnegative integers.
$z$	complex variable.
$x$	real variable.
$q$ $(\in\Complex)$	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 $\mathop{{{}_{{r}}\phi _{{s}}}\/}\nolimits\!\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)$ , the bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function $\mathop{{{}_{{r}}\psi _{{s}}}\/}\nolimits\!\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)$ , and the $q$ -analogs of the Appell functions $\mathop{\Phi^{{(1)}}\/}\nolimits\!\left(a;b,b^{{\prime}};c;x,y\right)$ , $\mathop{\Phi^{{(2)}}\/}\nolimits\!\left(a;b,b^{{\prime}};c,c^{{\prime}};x,y\right)$ , $\mathop{\Phi^{{(3)}}\/}\nolimits\!\left(a,a^{{\prime}};b,b^{{\prime}};c;x,y\right)$ , and $\mathop{\Phi^{{(4)}}\/}\nolimits\!\left(a;b;c,c^{{\prime}};x,y\right)$ .

Another function notation used is the “idem” function:

$f(\chi _{1};\chi _{2},\dots,\chi _{n})+\mathop{\mathrm{idem}\/}\nolimits\!\left(\chi _{1};\chi _{2},\dots,\chi _{n}\right)=\sum _{{j=1}}^{n}f(\chi _{j};\chi _{1},\chi _{2},\dots,\chi _{{j-1}},\chi _{{j+1}},\dots,\chi _{n}).$

These notations agree with Gasper and Rahman (2004) (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 $\mathop{{{}_{{2}}\phi _{{1}}}\/}\nolimits$ function.