Transformation formulas

<< Some integrals

Transformation formulas

The q-shifted factorial >>

Transformation formulas

${\displaystyle{\displaystyle{\displaystyle{}{}{{}_{2}F_{1}}\!\left({a,b\atop c};z\right)=(1-z)^{c-a-b}\,{{}_{2}F_{1}}\!\left({c-a,c-b\atop c};z\right)}}}$
${\displaystyle{\displaystyle{\displaystyle{}{}{{}_{2}F_{1}}\!\left({a,b\atop c};z\right)=(1-z)^{-a}\,{{}_{2}F_{1}}\!\left({a,c-b\atop c};\frac{z}{z-1}\right)}}}$
${\displaystyle{\displaystyle{\displaystyle{}{}{{}_{1}F_{1}}\!\left({a\atop c};z\right)=\expe^{z}\,{{}_{1}F_{1}}\!\left({c-a\atop c};-z\right)}}}$
${\displaystyle{\displaystyle{\displaystyle{{}_{1}F_{1}}\!\left({-n\atop a};x\right)=\frac{(-x)^{n}}{\pochhammer{a}{n}}\,{{}_{2}F_{0}}\!\left({-n,-a-n+1\atop-};-\frac{1}{x}\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle{{}_{2}F_{1}}\!\left({-n,b\atop c};x\right)=\frac{\pochhammer{b}{n}}{\pochhammer{c}{n}}(-x)^{n}\,{{}_{2}F_{1}}\!\left({-n,-c-n+1\atop-b-n+1};\frac{1}{x}\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle{}{}{{}_{4}F_{3}}\!\left({-n,a,b,c\atop d,e,f};1\right){}=\frac{\pochhammer{e-a}{n}\pochhammer{f-a}{n}}{\pochhammer{e}{n}\pochhammer{f}{n}}{{}_{4}F_{3}}\!\left({-n,a,d-b,d-c\atop d,a-e-n+1,a-f-n+1};1\right)}}}$

<< Some integrals

Transformation formulas

The q-shifted factorial >>