Formula:6645

Digital Library of Mathematical Functions ID 18.26.E19

${\displaystyle{\displaystyle(1-z)^{-c+iy}{{}_{2}F_{1}}\left({a+iy,b+iy\atop a+b};z\right)=\sum_{n=0}^{\infty}\frac{S_{n}\left(y^{2};a,b,c\right)}{{\left(a+b\right)_{n}}n!}z^{n},}}$

Constraint(s)

${\displaystyle{\displaystyle|z|<1}}$

Symbols List

• ${\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}$: = F ( a , b ; c ; z ) notation for Gauss’ hypergeometric function
• ${\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}$:
• ${\displaystyle{\displaystyle S_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)}}$: continuous dual Hahn polynomial
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle y}}$: real variable
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
 

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