Pagina:Werkecarlf03gausrich.djvu/145

erit coëfficiens potestatis $x^{m}$

in $F\;\;.\quad .\quad .\quad .\quad$	$.\quad .\quad \alpha (\beta +m-1)M$
in $F(\alpha ,\beta -1,\gamma )\quad$	$.\quad .\quad \alpha (\beta -1)M$
in $F(\alpha +1,\beta ,\gamma )\quad$	$.\quad .\quad (\alpha +m)(\beta +m-1)M$
in $F(\alpha ,\beta ,\gamma -1)\quad$	$.\quad .\quad {\frac {\alpha (\beta +m-1)(\gamma +m-1)M}{\gamma -1}}$

coëfficiens autem potestatis $x^{m-1}$ in $F(\alpha +1,\beta ,\gamma ),$ seu coëfficiens potestatis $x^{m}$ in $xF(\alpha +1,\beta ,\gamma )$

$=m(\gamma +m-1)M$

Hinc statim demanat veritas formularum 5 et 3; permutando $\alpha$ cum $\beta ,$ oritur ex 5 formula 12, atque ex his duabus per eliminationem 2. Perinde per eandem permutationem ex 3 oritur 6; ex 6 et 12 combinatis oritur 9, hinc per permutationem 14, quibus combinatis habetur 7; denique ex 2 et 6 eruitur 1, atque hinc permutando 10. Formula 8 simili modo ut supra formulae 5 et 3, e consideratione coëfficientium derivari potest (eodemque modo, si placeret, omnes 15 formulae erui possent), vel elegantius ex iam notis sequenti modo. Mutando in formula 5 elementum $\alpha$ in $\alpha -1,$ atque $\gamma$ in $\gamma +1,$ prodit

$0=(\gamma -\alpha +1)F(\alpha -1,\beta ,\gamma +1)+(\alpha -1)F(\alpha ,\beta ,\gamma +1)-\gamma F(\alpha -1,\beta ,\gamma )$

Mutando vero in formula 9 tantummodo $\gamma$ in $\gamma +1,$ fit

$0=(\alpha -1-(\gamma -\beta )x)F(\alpha ,\beta ,\gamma +1)+(\gamma -\alpha +1)F(\alpha -1,\beta ,\gamma +1)-\gamma (1-x)F(\alpha ,\beta ,\gamma )$

E subtractione harum formularum statim oritur 8, atque hinc per permutationem 13. Ex 1 et 8 prodit 4, hincque permutando 11. Denique ex 8 et 9 deducitur 15.

9.

Si $\alpha ^{\prime }-\alpha ,$ $\beta ^{\prime }-\beta ,$ $\gamma ^{\prime }-\gamma ,$ nec non $\alpha ^{\prime \prime }-\alpha ,$ $\beta ^{\prime \prime }-\beta ,$ $\gamma ^{\prime \prime }-\gamma$ sunt numeri integri (positivi seu negativi), a functione $F(\alpha ,\beta ,\gamma )$ ad functionem $F(\alpha ^{\prime },\beta ^{\prime },\gamma ^{\prime }),$ et perinde ab hac usque ad functionem $F(\alpha ^{\prime \prime },\beta ^{\prime \prime },\gamma ^{\prime \prime })$ transire licet per seriem similium functionum, ita ut quaelibet contigua sit antecedenti et consequenti, mutando scilicet primo elementum unum e.g. $\alpha$ continuo unitate, donec a $F(\alpha ,\beta ,\gamma )$ perventum sit ad $F(\alpha ^{\prime },\beta ,\gamma ),$ dein mutando elementum secundum, donec perven-