Transformation formulas

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Transformation formulas

The q-shifted factorial >>

Transformation formulas

${\displaystyle{\displaystyle{\displaystyle{}{}\HyperpFq{2}{1}@@{a,b}{c}{z}=(1-z)^{c-a-b}\,\HyperpFq{2}{1}@@{c-a,c-b}{c}{z}}}}$
${\displaystyle{\displaystyle{\displaystyle{}{}\HyperpFq{2}{1}@@{a,b}{c}{z}=(1-z)^{-a}\,\HyperpFq{2}{1}@@{a,c-b}{c}{\frac{z}{z-1}}}}}$
${\displaystyle{\displaystyle{\displaystyle{}{}\HyperpFq{1}{1}@@{a}{c}{z}={\mathrm{e}^{z}}\,\HyperpFq{1}{1}@@{c-a}{c}{-z}}}}$
${\displaystyle{\displaystyle{\displaystyle\HyperpFq{1}{1}@@{-n}{a}{x}=\frac{(-x)^{n}}{{\left(a\right)_{n}}}\,\HyperpFq{2}{0}@@{-n,-a-n+1}{-}{-\frac{1}{x}}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\HyperpFq{2}{1}@@{-n,b}{c}{x}=\frac{{\left(b\right)_{n}}}{{\left(c\right)_{n}}}(-x)^{n}\,\HyperpFq{2}{1}@@{-n,-c-n+1}{-b-n+1}{\frac{1}{x}}}}}$

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

${\displaystyle{\displaystyle{\displaystyle{}{}\HyperpFq{4}{3}@@{-n,a,b,c}{d,e,f}{1}{}=\frac{{\left(e-a\right)_{n}}{\left(f-a\right)_{n}}}{{\left(e\right)_{n}}{\left(f\right)_{n}}}\HyperpFq{4}{3}@@{-n,a,d-b,d-c}{d,a-e-n+1,a-f-n+1}{1}}}}$

<< Some integrals

Transformation formulas

The q-shifted factorial >>