Pagina:Werkecarlf03gausrich.djvu/149

[26]	${\frac {F(\alpha ,\beta +1,\gamma ,x)}{F(\alpha ,\beta ,\gamma ,x)}}={\frac {1}{1-{\frac {\alpha x}{\gamma }}G(\beta +1,\alpha ,\gamma ,x)}}$
[27]	${\frac {F(\alpha +1,\beta ,\gamma ,x)}{F(\alpha ,\beta ,\gamma ,x)}}={\frac {1}{1-{\frac {\beta x}{\gamma }}G(\alpha +1,\beta ,\gamma ,x)}}$
[28]	${\frac {F(\alpha -1,\beta +1,\gamma ,x)}{F(\alpha ,\beta ,\gamma ,x)}}={\frac {1}{1-{\frac {(\alpha -\beta -1)x}{\gamma }}G(\beta +1,\alpha -1,\gamma ,x)}}$
[29]	${\frac {F(\alpha +1,\beta -1,\gamma ,x)}{F(\alpha ,\beta ,\gamma ,x)}}={\frac {1}{1-{\frac {(\beta -\alpha -1)x}{\gamma }}G(\alpha +1,\beta -1,\gamma ,x)}}$

unde, substitutis pro functione $G$ eius valoribus in fractionibus continuis, totidem fractiones continuae novae prodeunt.

Ceterum sponte patet, fractionem continuam in formula 25 abrumpi, si e numeris $\alpha ,$ $\beta ,$ $\gamma -\alpha ,$ $\gamma -\beta$ aliquis fuerit integer negativus, alioquin vero in infinitum excurrere.

13.

Fractiones continuae in art. praec. erutae maximi sunt momenti, asserique potest, vix ullas fractiones continuas secundum legem obviam progredientes ab analystis hactenus erutas esse, quae sub nostris tamquam casus speciales non sint contentae. Imprimis memorabilis est casus is, ubi in formula 25 statuitur $\beta =0,$ unde $F(\alpha ,\beta ,\gamma ,x)=1,$ adeoque, scribendo $\gamma -1$ pro $\gamma$

[30]	$F(\alpha ,1,\gamma )=1+{\frac {\alpha }{\gamma }}x+{\frac {\alpha (\alpha +1)}{\gamma (\gamma +1)}}xx+{\frac {\alpha (\alpha +1)(\alpha +2)}{\gamma (\gamma +1)(\gamma +2)}}x^{3}+{\text{etc.}}$
	$={\cfrac {1}{1-{\cfrac {ax}{1-{\cfrac {bx}{1-{\cfrac {cx}{1-{\cfrac {dx}{1-{\text{etc.}}}}}}}}}}}}$

ubi

${\begin{alignedat}{2}&a={\frac {\alpha }{\gamma }}&&b={\frac {\gamma -\alpha }{\gamma (\gamma +1)}}\\&c={\frac {(\alpha +1)\gamma }{(\gamma +1)(\gamma +2)}}\qquad &&d={\frac {2(\gamma +1-a)}{(\gamma +2)(\gamma +3)}}\\&e={\frac {(\alpha +2)(\gamma +1)}{(\gamma +3)(\gamma +4)}}&&f={\frac {3(\gamma +2-\alpha )}{(\gamma +4)(\gamma +5)}}\\\end{alignedat}}$

etc.