Pagina:Werkecarlf03gausrich.djvu/144

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SECTIO PRIMA.
Relationes inter functiones contiguas.

7.

Functionem ipsi $F^{\prime }(\alpha ,\beta ,\gamma ,x)$ contiguam vocamus, quae ex illa oritur, dum elementum primum, secundum, vel tertium unitate vel augetur vel diminuitur, manentibus tribus reliquis elementis. Functio itaque primaria $F(\alpha .\beta ,\gamma ,x)$ sex contiguas suppeditat, inter quarum binas ipsamque primariam aequatio persimplex linearis datur. Has aequationes, numero quindecim, hic in conspectum producimus, brevitatis gratia elementum quartum quod semper subintelligitur $=x$ omittentes, functionemque primariam simpliciter per $F$ denotantes.

[1]	${\textstyle 0=(\gamma -2\alpha -(\beta -\alpha )x)F+\alpha (1-x)F(\alpha +1,\beta ,\gamma )-(\gamma -\alpha )F(\alpha -1,\beta ,\gamma )}$
[2]	${\textstyle 0=(\beta -\alpha )F+\alpha F(\alpha +1,\beta ,\gamma )-\beta F(\alpha ,\beta +1,\gamma )}$
[3]	${\textstyle 0=(\gamma -\alpha -\beta )F+\alpha (1-x)F(\alpha +1,\beta ,\gamma )-(\gamma -\beta )F(\alpha ,\beta -1,\gamma )}$
[4]	${\textstyle 0=\gamma (\alpha -(\gamma -\beta )x)F-\alpha \gamma (1-x)F(\alpha +1,\beta ,\gamma )+(\gamma -\alpha )(\gamma -\beta )xF(\alpha ,\beta ,\gamma +1)}$
[5]	${\textstyle 0=(\gamma -\alpha -1)F+\alpha F(\alpha +1,\beta ,\gamma )-(\gamma -1)F(\alpha ,\beta ,\gamma -1)}$
[6]	${\textstyle 0=(\gamma -\alpha -\beta )F-(\gamma -\alpha )F(\alpha -1,\beta ,\gamma )+\beta (1-x)F(\alpha ,\beta +1,\gamma )}$
[7]	${\textstyle 0=(\beta -\alpha )(1-x)F-(\gamma -\alpha )F(\alpha -1,\beta ,\gamma )+(\gamma -\beta )F(\alpha ,\beta -1,\gamma )}$
[8]	${\textstyle 0=\gamma (1-x)F-\gamma F(\alpha -1,\beta ,\gamma )+(\gamma -\beta )xF(\alpha ,\beta ,\gamma +1)}$
[9]	${\textstyle 0=(\alpha -1-(\gamma -\beta -1)x)F+(\gamma -\alpha )F(\alpha -1,\beta ,\gamma )-(\gamma -1)(1-x)F(\alpha ,\beta ,\gamma -1)}$
[10]	${\textstyle 0=(\gamma -2\beta +(\beta -\alpha )x)F+\beta (1-x)F(\alpha ,\beta +1,\gamma )-(\gamma -\beta )F(\alpha ,\beta -1,\gamma )}$
[11]	${\textstyle 0=\gamma (\beta -(\gamma -\alpha )x)F-\beta \gamma (1-x)F(\alpha ,\beta +1,\gamma )-(\gamma -\alpha )(\gamma -\beta )F(\alpha ,\beta ,\gamma +1)}$
[12]	${\textstyle 0=(\gamma -\beta -1)F+\beta F(\alpha ,\beta +1,\gamma )-(\gamma -1)F(\alpha ,\beta ,\gamma -1)}$
[13]	${\textstyle 0=\gamma (1-x)F-\gamma F(\alpha ,\beta -1,\gamma )+(\gamma -\alpha )xF(\alpha ,\beta ,\gamma +1)}$
[14]	${\textstyle 0=(\beta -1-(\gamma -\alpha -1)x)F+(\gamma -\beta )F(\alpha ,\beta -1,\gamma )-(\gamma -1)(1-x)F(\alpha ,\beta ,\gamma -1)}$
[15]	${\textstyle {\begin{aligned}0=\gamma (\gamma -1-(2\gamma -\alpha -\beta -1)x)F+(\gamma -\alpha )(\gamma -\beta )xF(\alpha ,\beta ,\gamma +1)&\\-\gamma (\gamma -1)(1-x){F}(\alpha ,\beta ,\gamma -1)&\end{aligned}}}$