Basic hypergeometric phi
The basic hypergeometric series ${}_j\phi{}_{\ell}$ is defined by $${\small{}_j\phi_{\ell}(a_1,\ldots,a_j;b_1,\ldots,b_{\ell};q,z)={}_j \phi_{\ell} \left[ \begin{array}{llllll} a_1 & a_2 & \ldots & a_j \\
& & & & ; q,z \\
b_1 & b_2 & \ldots & b_{\ell} \end{array}\right]=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1;q)_k \ldots (a_j;q)_k}{(b_1;q)_k \ldots (b_{\ell};q)_k} \left((-1)^kq^{k \choose 2} \right)^{1+\ell-j}z^k=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1;q)_k \ldots (a_j;q)_k}{(b_1;q)_k \ldots (b_{\ell};q)_k} \left(-q^{\frac{k-1}{2}} \right)^{k(1+\ell-j)}z^k.}$$
Properties
Theorem: The following formula holds: $$\displaystyle\lim_{q \rightarrow 1^-} {}_j \phi_{\ell} \left[ \begin{array}{l|l} q^{a_1}, \ldots, q^{a_j} \\ q^{b_1}, \ldots, q^{b_{\ell}} \end{array} \Bigg| q,z(1-q)^{1+\ell-j} \right]={}_j F_{\ell}\left(a_1,\ldots,a_j;b_1,\ldots,b_{\ell};(-1)^{1+\ell-j}z \right)$$
Proof: █
Theorem: ($q$-Pfaff-Saalschütz) The following formula holds: $${}_3\phi_2(q^{-n},a,b;c,d;q,q) = \dfrac{\left(\frac{d}{a};q \right)_n \left( \frac{d}{b};q \right)_n}{\left(d;q \right)_n \left(\frac{d}{ab};q \right)_n},$$ with $cd=abq^{1-n}$.
Proof: █
Theorem: ($q$-Chu-Vandermonde sum) The following formula holds: $${}_2\phi_1(q^{-n},a;c;q,q) = \dfrac{\left( \frac{c}{a};q \right)_n}{(c;q)_n} a^n.$$
Proof: █
Theorem: ($q$-Gauss theorem) The following formula holds: $${}_2\phi_1 \left( a,b;c;q,\dfrac{c}{ab} \right) = \dfrac{\left(\frac{c}{a};q \right)_{\infty} \left( \frac{c}{b};q \right)_{\infty}}{\left(c;q\right)_{\infty} \left(\frac{c}{ab};q \right)_{\infty}}; \left| \dfrac{c}{ab} \right|<1.$$
Proof: █
Theorem: If $|z|<1$ or $a=q^{-n}$ then $${}_1\phi_0(a;-;q,z) = \dfrac{(az;q)_{\infty}}{(z;q)_{\infty}}.$$
Proof: █
Theorem: The following formula holds: $${}_3\phi_2 (q^{-n},a,b;c,d;q,q) = \dfrac{b^n (\frac{d}{b};q)_n}{(d;q)_n} {}_3 \phi_2 \left(q^{-n},b,\frac{c}{a};c,q^{1-n}\frac{b}{d};q,\frac{aq}{d} \right).$$
Proof: █
Theorem: (Heine Transformation) The following formula holds for $|z|<1$ and $|b|<1$: $${}_2\phi_1(a,b;c;q,z)=\dfrac{(b;q)_{\infty} (az;q)_{\infty}}{(c;q)_{\infty} (z;q)_{\infty}} {}_2\phi_1 \left( \frac{c}{b},z; az; q,b \right).$$
Proof: █
Theorem: The following formula holds for $|z|<1$ and $|abz|<|c|$: $${}_2\phi_1(a,b;c;q,z) = \dfrac{(\frac{abz}{c};q)_{\infty})}{(z;q)_{\infty}} {}_2\phi_1 \left(\frac{c}{a},\frac{c}{b};c;q,\frac{abz}{c}\right).$$
Proof: █
Theorem: (Bailey-Daum sum) The following formula holds for $|q|<|b|$: $${}_2\phi_1\left( a,b;\frac{aq}{b}; q, -\frac{q}{b} \right) = \dfrac{(-q;q)_{\infty} (aq;q^2)_{\infty} (\frac{aq^2}{b^2};q^2)_{\infty}}{(-\frac{q}{b};q)_{\infty} (\frac{aq}{b};q)_{\infty}}$$
Proof: █
Theorem
The following formula holds: $$e_q(z) = {}_1\phi_0(0;-;q;z),$$ where $e_q$ is the $q$-exponential $e$ and ${}_1\phi_0$ denotes the basic hypergeometric phi.
