Pagina:Werkecarlf03gausrich.djvu/147

VI. ${\phantom {\text{II}}}0=\gamma {F}-\gamma (1-x){F}^{\prime \prime }-(\gamma -\alpha -\beta -1)x{F}^{\prime \prime \prime }$

Hinc atque ex III, eliminando ${F}^{\prime \prime }$

VII. ${\phantom {\text{I}}}0=\gamma {F}-(\gamma -\alpha -1-\beta x){F}^{\prime \prime \prime }-(\alpha +1)(1-x){F}^{\prime \prime \prime \prime }$

Porro ex IV atque V, eliminando ${F}^{\prime \prime \prime \prime \prime }$

VIII. $0=(\gamma +1){F}^{\prime \prime \prime }-(\gamma +1){F}^{\prime \prime \prime \prime }+(\beta +1)x{F}^{\prime \prime \prime \prime \prime \prime }$

Hinc atque ex VII, eliminando ${F}^{\prime \prime \prime \prime },$

IX. ${\phantom {\text{II}}}0=\gamma (\gamma +1)F-(\gamma +1)(\gamma -(\alpha +\beta +1)x){F}^{\prime \prime \prime }-(\alpha +1)(\beta +1)x(1-x){F}^{\prime \prime \prime \prime \prime \prime }$

11.

Si omnes relationes inter ternas functiones ${F}(\alpha ,\beta ,\gamma ),$ ${F}(\alpha +\lambda ,\beta +\mu ,\gamma +\nu ),$ $F(\alpha +\lambda ^{\prime },\beta +\mu ^{\prime },\gamma +\nu ^{\prime }),$ in quibus $\lambda ,$ $\mu ,$ $\nu ,$ $\lambda ^{\prime },$ $\mu ^{\prime },$ $\nu ^{\prime }$ vel $=0$ vel $=+1$ vel $=-1,$ exhaurire vellemus, formularum multitudo usque ad 325 ascenderet. Haud inutilis foret talis collectio, saltem simpliciorum ex his formulis: hoc vero loco sufficiat, paucas tantummodo apposuisse, quas vel ex formulis art. 7, vel si magis placet, simili modo ut duae priores ex illis in art. 8 erutae sunt, quivis nullo negotio sibi demonstrare poterit.

[16]	$F(\alpha ,\beta ,\gamma )-F(\alpha ,\beta ,\gamma -1)$	$=-{\frac {\alpha \beta x}{\gamma (\gamma -1)}}F(\alpha +1,\beta +1,\gamma +1)$
[17]	$F(\alpha ,\beta +1,\gamma )-F(\alpha ,\beta ,\gamma )$	$={\frac {\alpha x}{\gamma }}F(\alpha +1,\beta +1,\gamma +1)$
[18]	$F(\alpha +1,\beta ,\gamma )-F(\alpha ,\beta ,\gamma )$	$={\frac {\beta x}{\gamma }}F(\alpha +1,\beta +1,\gamma +1)$
[19]	$F(\alpha ,\beta +1,\gamma +1)-F(\alpha ,\beta ,\gamma )$	$={\frac {\alpha (\gamma -\beta )x}{\gamma (\gamma +1)}}F(\alpha +1,\beta +1,\gamma +2)$
[20]	$F(\alpha +1,\beta ,\gamma +1)-F(\alpha ,\beta ,\gamma )$	$={\frac {\beta (\gamma -\alpha )x}{\gamma (\gamma +1)}}F(\alpha +1,\beta +1,\gamma +2)$
[21]	$F(\alpha -1,\beta +1,\gamma )-F(\alpha ,\beta ,\gamma )$	$={\frac {(\alpha -\beta -1)x}{\gamma }}F(\alpha ,\beta +1,\gamma +1)$
[22]	$F(\alpha +1,\beta -1,\gamma )-F(\alpha ,\beta ,\gamma )$	$={\frac {(\beta -\alpha -1)x}{\gamma }}F(\alpha +1,\beta ,\gamma +1)$
[23]	$F(\alpha -1,\beta +1,\gamma )-F(\alpha +1,\beta -1,\gamma )$	$={\frac {(\alpha -\beta )x}{\gamma }}F(\alpha +1,\beta +1,\gamma +1)$

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