# q-hypergeometric function

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##### 1: 17.1 Special Notation
###### §17.1 Special Notation
 $k,j,m,n,r,s$ nonnegative integers. …
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: …
##### 2: 17.17 Physical Applications
###### §17.17 Physical Applications
See Kassel (1995). …
##### 6: 17.5 ${{}_{0}\phi_{0}},{{}_{1}\phi_{0}},{{}_{1}\phi_{1}}$ Functions
###### Euler’s Second Sum
17.5.2 ${{}_{1}\phi_{0}}\left(a;-;q,z\right)=\frac{\left(az;q\right)_{\infty}}{\left(z% ;q\right)_{\infty}},$ $|z|<1$;
17.5.3 ${{}_{1}\phi_{0}}\left(q^{-n};-;q,z\right)=\left(zq^{-n};q\right)_{n}.$
##### 8: 17.4 Basic Hypergeometric Functions
###### §17.4(ii) ${{}_{r}\psi_{s}}$Functions
The series (17.4.1) is said to be balanced or Saalschützian when it terminates, $r=s$, $z=q$, and … The series (17.4.1) is said to be k-balanced when $r=s$ and … The series (17.4.1) is said to be well-poised when $r=s$ and …
##### 10: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions
###### Bailey’s Bilateral Summations
For similar formulas see Verma and Jain (1983).