Formula:6647

Digital Library of Mathematical Functions ID 18.26.E21

${\displaystyle{\displaystyle(1-z)^{y}{{}_{2}F_{1}}\left({y-N,y+\gamma+1\atop-\delta-N};z\right)=\sum_{n=0}^{N}\frac{{\left(\gamma+1\right)_{n}}{\left(-N\right)_{n}}}{{\left(-\delta-N\right)_{n}}n!}\*R_{n}\left(y(y+\gamma+\delta+1);\gamma,\delta,N\right)z^{n}.}}$

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
• ${\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}$:
• ${\displaystyle{\displaystyle R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)}}$: dual Hahn polynomial
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle y}}$: real variable
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle N}}$: positive integer
• ${\displaystyle{\displaystyle\delta}}$: arbitary small positive constant
 

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