§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 -hypergeometric function, -hypergeometric function
Permalink:: http://dlmf.nist.gov/17.1

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

	nonnegative integers.
	complex variable.
	real variable.
$(\in\Complex)$	base: unless stated otherwise .
$\left(a;q\right)_{{n}}$	-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 -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 -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 -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 -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 for a particular specialization of a $\mathop{{{}_{{2}}\phi_{{1}}}\/}\nolimits$ function.

-Hypergeometric and Related Functions 17.2 Calculus