Formula:10504

Digital Library of Mathematical Functions ID 35.7.E1

${\displaystyle{\displaystyle{{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}{k!}\sum_{|\kappa|=k}\frac{{\left[a\right]_{\kappa}}{\left[b\right]_{\kappa}}}{{\left[c\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right)},}}$

Constraint(s)

${\displaystyle{\displaystyle-c+\frac{1}{2}(j+1)\notin\mathbb{N}}}$
${\displaystyle{\displaystyle 1\leq j\leq m}}$
${\displaystyle{\displaystyle||\mathbf{T}||<1}}$

Symbols List

• ${\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{\mathbf{T}}\right)}}$: generalized hypergeometric function of matrix argument
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\notin}}$:
• ${\displaystyle{\displaystyle\mathbb{N}}}$:
• ${\displaystyle{\displaystyle{\left[\NVar{a}\right]_{\NVar{\kappa}}}}}$: partitional shifted factorial
• ${\displaystyle{\displaystyle Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)}}$:
• ${\displaystyle{\displaystyle a}}$: complex variable
• ${\displaystyle{\displaystyle\mathbf{T}}}$: space of real symmetric matrices
• ${\displaystyle{\displaystyle b}}$: complex variable
• ${\displaystyle{\displaystyle c}}$: complex variable
• ${\displaystyle{\displaystyle j}}$: nonnegative integer
• ${\displaystyle{\displaystyle k}}$: nonnegative integer
• ${\displaystyle{\displaystyle m}}$: positive integer
 

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