Pagina:Werkecarlf03gausrich.djvu/157

summa duorum terminorum	$={\tfrac {n-1}{h}}-{\tfrac {(n-h-1)(n-h)}{hn}}$
summa trium terminorum	$={\tfrac {n-1}{h}}-{\tfrac {(n-h-1)(n-h)(n-h+1)}{hn(n+1)}}$ etc.

et quum pars altera (per II) formet seriem ultra omnes limites decrescentem, summa illa in infinitum continuata statui debet $={\tfrac {n-1}{h}}.$ Hinc $N+N^{\prime }+N^{\prime \prime }$ etc. in infinitum continuata semper manebit minor quam ${\tfrac {N(n-1)}{h}},$ et proin $M+M^{\prime }+M^{\prime \prime }$ etc. certo ad summam finitam converget. Q. E. D.

VI. Ut ea, quae generaliter de serie $M,$ $M^{\prime },$ $M^{\prime \prime }$ etc. demonstravimus, ad coëfficientes potestatum $x^{m},$ $x^{m+1},$ $x^{m+2}$ etc. in serie ${F}(\alpha ,\beta ,\gamma ,x),$ applicentur, statuere oportebit $\lambda =2,$ $A=\alpha +\beta ,$ $B=\alpha \beta ,$ $a=\gamma +1,$ $b=\gamma ,$ unde quinque assertiones in art. praec. propositae sponte demanant.

17.

Disquisitio itaque de indole summae seriei $F(\alpha ,\beta ,\gamma ,1)$ natura sua restringitur ad casum, quo $\gamma -\alpha -\beta$ est quantitas positiva, ubi illa summa semper exhibet quantitatem finitam. Praemittimus autem observationem sequentem. Si coëfficientes seriei $1+ax+bxx+cx^{3}+$ etc. $=S$ inde a certo termino ultra omnes limites decrescunt, productum

$(1-x)S=1+(a-1)x+(b-a)xx+(c-b)x^{3}+$ etc.

pro $x=1$ statuere oportet $=0,$ etiamsi summa ipsius seriei $S$ infinite magna evadat. Quoniam enim collectis duobus terminis summa fit $=a,$ collectis tribus $=b,$ collectis quatuor $=c$ etc., limes summae in infinitum continuatae est $=0.$ Quoties itaque $\gamma -\alpha -\beta$ est quantitas positiva, statuere oportet $(1-x)F(\alpha ,\beta ,\gamma -1,x)=0$ pro $x=1,$ unde per aequationem 15 art. 7 habebimus

	$0=\gamma (\alpha +\beta -\gamma )F(\alpha ,\beta ,\gamma ,1)+(\gamma -\alpha )(\gamma -\beta )F(\alpha ,\beta ,\gamma +1,1),$ sive
[37]	$F(\alpha ,\beta ,\gamma ,1)={\frac {(\gamma -\alpha )(\gamma -\beta )}{\gamma (\gamma -\alpha -\beta )}}F(\alpha ,\beta ,\gamma +1,1)$

Quare quum perinde fiat

${\begin{aligned}&F(\alpha ,\beta ,\gamma +1,1)={\frac {(\gamma +1-\alpha )(\gamma +1-\beta )}{(\gamma +1)(\gamma +1-\alpha -\beta )}}F(\alpha ,\beta ,\gamma +2,1)\\&F(\alpha ,\beta ,\gamma +2,1)={\frac {(\gamma +2-\alpha )(\gamma +2-\beta )}{(\gamma +2)(\gamma +2-\alpha -\beta )}}F(\alpha ,\beta ,\gamma +3,1)\end{aligned}}$

et sic porro, erit generaliter, $k$ denotante integrum positivum quemcunque