The q-binomial theorem and other summation formulas

<< Basic hypergeometric functions

The q-binomial theorem and other summation formulas

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The q-binomial theorem and other summation formulas

${\displaystyle{\displaystyle{\displaystyle{}{}{{}_{1}\phi_{0}}\!\left({a\atop-};q,z\right)=\sum_{n=0}^{\infty}\frac{\qPochhammer{a}{q}{n}}{\qPochhammer{q}{q}{n}}z^{n}=\frac{\qPochhammer{az}{q}{\infty}}{\qPochhammer{z}{q}{\infty}},\quad 0<|q|<1}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle|z|<1}}}$

${\displaystyle{\displaystyle{\displaystyle{{}_{1}\phi_{0}}\!\left({q^{-n}\atop-};q,z\right)=\qPochhammer{zq^{-n}}{q}{n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots}}}$

${\displaystyle{\displaystyle{\displaystyle{}{}{{}_{2}\phi_{1}}\!\left({a,b\atop c};q,\frac{c}{ab}\right)=\frac{\qPochhammer{a^{-1}c,b^{-1}c}{q}{\infty}}{\qPochhammer{c,a^{-1}b^{-1}c}{q}{\infty}},\quad 0<|q|<1}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\left|\frac{c}{ab}\right|<1}}}$

${\displaystyle{\displaystyle{\displaystyle{}{}{}{}{{}_{2}\phi_{1}}\!\left({q^{-n},b\atop c};q,\frac{cq^{n}}{b}\right)=\frac{\qPochhammer{b^{-1}c}{q}{n}}{\qPochhammer{c}{q}{n}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots}}}$

${\displaystyle{\displaystyle{\displaystyle{{}_{2}\phi_{1}}\!\left({q^{-n},b\atop c};q,q\right)=\frac{\qPochhammer{b^{-1}c}{q}{n}}{\qPochhammer{c}{q}{n}}b^{n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots}}}$

${\displaystyle{\displaystyle{\displaystyle{}{{}_{1}\phi_{1}}\!\left({a\atop c};q,\frac{c}{a}\right)=\frac{\qPochhammer{a^{-1}c}{q}{\infty}}{\qPochhammer{c}{q}{\infty}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}}$

${\displaystyle{\displaystyle{\displaystyle{}{{}_{2}\phi_{0}}\!\left({q^{-n},b\atop-};q,\frac{q^{n}}{b}\right)=\frac{1}{b^{n}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots}}}$

${\displaystyle{\displaystyle{\displaystyle{}{{}_{0}\phi_{1}}\!\left({-\atop c};q,c\right)=\frac{1}{\qPochhammer{c}{q}{\infty}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}}$

${\displaystyle{\displaystyle{\displaystyle{}{}{}{}{{}_{3}\phi_{2}}\!\left({q^{-n},a,b\atop c,abc^{-1}q^{1-n}};q,q\right)=\frac{\qPochhammer{a^{-1}c,b^{-1}c}{q}{n}}{\qPochhammer{c,a^{-1}b^{-1}c}{q}{n}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots}}}$

${\displaystyle{\displaystyle{\displaystyle{}{}{}{{}_{6}\phi_{5}}\!\left({q\sqrt{a},-q\sqrt{a},a,b,c,d\atop\sqrt{a},-\sqrt{a},ab^{-1}q,ac^{-1}q,ad^{-1}q};q,\frac{aq}{bcd}\right){}=\frac{\qPochhammer{aq,ab^{-1}c^{-1}q,ab^{-1}d^{-1}q,ac^{-1}d^{-1}q}{q}{\infty}}{\qPochhammer{ab^{-1}q,ac^{-1}q,ad^{-1}q,ab^{-1}c^{-1}d^{-1}q}{q}{\infty}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\left|\frac{aq}{bcd}\right|<1}}}$

${\displaystyle{\displaystyle{\displaystyle{{}_{6}\phi_{5}}\!\left({q\sqrt{a},-q\sqrt{a},a,b,c,q^{-n}\atop\sqrt{a},-\sqrt{a},ab^{-1}q,ac^{-1}q,aq^{n+1}};q,\frac{aq^{n+1}}{bc}\right){}=\frac{\qPochhammer{aq,ab^{-1}c^{-1}q}{q}{n}}{\qPochhammer{ab^{-1}q,ac^{-1}q}{q}{n}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots}}}$

${\displaystyle{\displaystyle{\displaystyle{{}_{6}\phi_{4}}\!\left({q\sqrt{a},-q\sqrt{a},a,b,0,q^{-n}\atop\sqrt{a},-\sqrt{a},ab^{-1}q,aq^{n+1}};q,\frac{q^{n}}{b}\right){}=\frac{\qPochhammer{aq}{q}{n}}{\qPochhammer{ab^{-1}q}{q}{n}}b^{-n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots}}}$

${\displaystyle{\displaystyle{\displaystyle{{}_{6}\phi_{3}}\!\left({q\sqrt{a},-q\sqrt{a},a,0,0,q^{-n}\atop\sqrt{a},-\sqrt{a},aq^{n+1}};q,\frac{q^{n-1}}{a}\right){}=(-1)^{n}a^{-n}q^{-\binomial{n+1}{2}}\qPochhammer{aq}{q}{n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots}}}$

<< Basic hypergeometric functions

The q-binomial theorem and other summation formulas

More integrals >>