# §17.1 Special Notation

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

$k,j,m,n,r,s$ nonnegative integers. complex variable. real variable. base: unless stated otherwise $|q|<1$. $q$-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 $\mathop{\Phi^{(1)}\/}\nolimits\!\left(a;b,b^{\prime};c;q;x,y\right)$, $\mathop{\Phi^{(2)}\/}\nolimits\!\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)$, $\mathop{\Phi^{(3)}\/}\nolimits\!\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)$, and $\mathop{\Phi^{(4)}\/}\nolimits\!\left(a,b;c,c^{\prime};q;x,y\right)$.

Another function notation used is the “idem” function:

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

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