Pagina:Werkecarlf03gausrich.djvu/158

$F(\alpha ,\beta ,\gamma ,1)$ aequalis producto ex ${F}(\alpha ,\beta ,\gamma +k,1)$

in $\quad (\gamma -\alpha )(\gamma +1-\alpha )(\gamma +2-\alpha )\ldots (\gamma +k-1-\alpha )$
in $\quad (\gamma -\beta )(\gamma +1-\beta )(\gamma +2-\beta )\ldots (\gamma +k-1-\beta )$
diviso per productum
ex $\quad \gamma (\gamma +1)(\gamma +2)\ldots (\gamma +k-1)$
in $\quad (\gamma -\alpha -\beta )(\gamma +1-\alpha -\beta )(\gamma +2-\alpha -\beta )\ldots (\gamma +k-1-\alpha -\beta )$

18.

Introducamus abhinc sequentem notationem:

[38]	$\Pi (k,z)={\frac {1\;.\;2\;.\;3\ldots \ldots k}{(z+1)(z+2)(z+3)\ldots (z+k)}}k^{z}$

ubi $k$ natura sua subintelligitur designare integrum positivum, qua restrictione $\Pi (k,z)$ exhibet functionem duarum quantitatum $k,$ $z$ prorsus determinatam. Hoc modo facile intelligetur, theorema in fine art. praec. propositum ita exhiberi posse

[39]	$F(\alpha ,\beta ,\gamma ,1)={\frac {\Pi (k,\gamma -1).\Pi (k,\gamma -\alpha -\beta -1)}{\Pi (k,\gamma -\alpha -1).\Pi (k,\gamma -\beta -1)}}.F(\alpha ,\beta ,\gamma +k,1)$

19.

Operae pretium erit, indolem functionis $\Pi (k,z)$ accuratius perpendere. Quoties $z$ est integer negativus, functio manifesto valorem infinite magnum obtinet, simulac ipsi $k$ tribuitur valor satis magnus. Pro valoribus integris ipsius ${z}$ non negativis autem habemus

${\begin{aligned}&\Pi (k,0)=1\\&\Pi (k,1)={\frac {1}{1+{\frac {1}{k}}}}\\&\Pi (k,2)={\frac {1\;.\;2}{(1+{\frac {1}{k}})(1+{\frac {2}{k}})}}\\&\Pi (k,3)={\frac {1\;.\;2\;.\;3}{(1+{\frac {1}{k}})(1+{\frac {2}{k}})(1+{\frac {3}{k}})}}\end{aligned}}$

etc. sive generaliter

[40]	$\Pi (k,z)={\frac {1\;.\;2\;.\;3\;\ldots \ldots \;z}{(1+{\frac {1}{k}})(1+{\frac {2}{k}})(1+{\frac {3}{k}})\ldots (1+{\frac {z}{k}})}}$