Pagina:Werkecarlf03gausrich.djvu/202

18.

Disquisitiones art. praec. functiones idoneas ${\textstyle U}$ pro singulis valoribus numeri ${\textstyle n}$ invenire quidem docent, sed successive tantum, dum a valoribus minoribus ad maiores transeundum est. Facile autem animadvertimus, has functiones generaliter exprimi per

$${\begin{aligned}u^{n+1}-{\frac {(n+1)n}{2.(2n+1)}}u^{n-1}&+{\frac {(n+1)n(n-1)(n-2)}{2.4(2n+1)(2n-1)}}u^{n-3}-{\frac {(n+1)n(n-1)(n-2)(n-3)(n-4)}{2.4.6.(2n+1)(2n-1)(2n-3)}}u^{n-5}+{\text{ etc. }}\end{aligned}}$$

sive etiam, si characteristica ${\textstyle F}$ ad normam commentationis supra citatae utimur, per

$$u^{n+1}F(-{\frac {1}{2}}n,-{\frac {1}{2}}(n+1),-(n+{\frac {1}{2}}),u^{-2})$$

Haecce inductio facile in demonstrationem rigorosam convertitur per methodum vulgo notam, aut, si ita videtur, adiumento formulae 19 in comment. cit. Functio ${\textstyle U,}$ si magis placet, etiam ordine terminorum inverso, exprimi potest per

$$\pm {\frac {3.5.7\ldots (n+1)}{(n+3)(n+5)\ldots (2n+1)}}\cdot uF(-{\frac {1}{2}}n,{\frac {1}{2}}(n+3),{\frac {3}{2}},uu)$$

pro ${\textstyle n}$ pari, valente signo superiori vel inferiori, prout ${\textstyle {\frac {1}{2}}n}$ par est vel impar aut per

$$\pm {\frac {1.3.5\ldots n}{(n+2)(n+4)\ldots (2n+1)}}F(-{\frac {1}{2}}(n+1),{\frac {1}{2}}n+1,{\frac {1}{2}},uu)$$

pro ${\textstyle n}$ impari, valente signo superiori vel inferiori, prout ${\textstyle {\frac {1}{2}}(n+1)}$ par est vel impar.

Functio ${\textstyle U^{\prime }}$ expressionem generalem aeque simplicem non admittit: facile tamen ex ipsa genesi quantitatum ${\textstyle V,}$ ${\textstyle V^{\prime },}$ ${\textstyle V^{\prime \prime }}$ etc. colligitur, terminum ultimum ipsius ${\textstyle U^{\prime }}$ pro ${\textstyle n}$ pari fieri

$$=\pm {\frac {2.2.4.4.6.6\ldots \ldots n.n}{3.5.7.9.11.13\ldots \ldots (2n-1)(2n+1)}}$$

signo superiori vel inferiori valente, prout ${\textstyle {\frac {1}{2}}n}$ par est vel impar.

Functio ${\textstyle U^{\prime \prime }=\varphi W^{(n+1)}-V^{(n+1)},}$ cuius terminum primum iam in art. praec. assignare docuimus, etiam per algorithmum recurrentem evolvi potest, quum manifesto generaliter habeatur

$${\begin{aligned}&\varphi W^{\prime \prime }-V^{\prime \prime }=w^{\prime }(\varphi W^{\prime }-V^{\prime })+v^{\prime }(\varphi W-V)\\&\varphi W^{\prime \prime \prime }-V^{\prime \prime \prime }=w^{\prime \prime }(\varphi W^{\prime \prime }-V^{\prime \prime })+v^{\prime \prime }(\varphi W^{\prime }-V^{\prime })\\&\varphi W^{\prime \prime \prime \prime }-V^{\prime \prime \prime \prime }=w^{\prime \prime \prime }(\varphi W^{\prime \prime \prime }-V^{\prime \prime \prime })+v^{\prime \prime \prime }(\varphi W^{\prime \prime }-V^{\prime \prime })\end{aligned}}$$