Pagina:Werkecarlf03gausrich.djvu/203

etc. adeoque eo quem tractamus casu

$$\varphi W^{(m+2)}-V^{(m+2)}=u(\varphi W^{(m+1)}-V^{(m+1)})-{\frac {(m+1)^{2}}{(2m-1)(2m+1)}}(\varphi W^{(m)}-V^{(m)})$$

Ita invenimus

$${\begin{alignedat}{6}\varphi W-V&=&u^{-1}&+&{\frac {1}{3}}u^{-3}&+&{\frac {1}{5}}u^{-5}&+&{\frac {1}{7}}u^{-7}&+&{\text{ etc. }}\\\varphi W^{\prime }-V^{\prime }&=&{\frac {1}{3}}u^{-2}&+&{\frac {1}{5}}u^{-4}&+&{\frac {1}{7}}u^{-6}&+&{\frac {1}{9}}u^{-8}&+&{\text{ etc. }}\\\varphi W^{\prime \prime }-V^{\prime \prime }&=&{\frac {4}{45}}u^{-3}&+&{\frac {8}{105}}u^{-5}&+&{\frac {4}{63}}u^{-7}&+&{\frac {112}{2079}}u^{-9}&+&{\text{ etc. }}\\\varphi W^{\prime \prime \prime }-V^{\prime \prime \prime }&=&{\frac {4}{175}}u^{-4}&+&{\frac {8}{315}}u^{-6}&+&{\frac {4}{165}}u^{-8}&+&{\frac {16}{715}}u^{-10}&+&{\text{ etc. }}\end{alignedat}}$$

etc. quas series ita quoque exhibere licet

$${\begin{alignedat}{7}\varphi W-V&=&u^{-1}&(1+&{\frac {1.2}{2.3}}u^{-2}&+&{\frac {1.2.3.4}{2.4.3.5}}u^{-4}&+&{\frac {1.2.3.4.5.6}{2.4.6.3.5.7}}u^{-6}&+&{\text{ etc.}})\\\varphi W^{\prime }-V^{\prime }&=&{\frac {1}{3}}u^{-2}&(1+&{\frac {2.3}{2.5}}u^{-4}&+&{\frac {2.3.4.5}{2.4.5.7}}u^{-4}&+&{\frac {2.3.4.5.6.7}{2.4.6.5.7.9}}u^{-6}&+&{\text{ etc.}})\\\varphi W^{\prime \prime }-V^{\prime \prime }&=&{\frac {4}{45}}u^{-3}&(1+&{\frac {3.4}{2.7}}u^{-2}&+&{\frac {3.4.5.6}{2.4.7.9}}u^{-4}&+&{\frac {3.4.5.6.7.8}{2.4.6.7.9.11}}u^{-6}&+&{\text{ etc.}})\\\varphi W^{\prime \prime \prime }-V^{\prime \prime \prime }&=&{\frac {4}{175}}u^{-4}&(1+&{\frac {4.5}{2.9}}u^{-2}&+&{\frac {4.5.6.7}{2.4.9.11}}u^{-4}&+&{\frac {4.5.6.7.8.9}{2.4.6.9.11.13}}u^{-6}&+&{\text{ etc.}})\end{alignedat}}$$

etc. Hanc inductionem sequentes habebimus generaliter

$${\begin{aligned}U^{\prime \prime }=&\varphi W^{(n+1)}-V^{(n+1)}{\text{ aequalem producto ex }}\\&{\frac {2.2.3.3.4.4\ldots \ldots (n+1).(n+1)}{3.3.5.5.7.7.9\ldots (2n+1)(2n+3)}}u^{-(n+2)}\end{aligned}}$$

in seriem infinitam

$$1+{\frac {(n+2)(n+3)}{2(2n+5)}}u^{-2}+{\frac {(n+2)(n+3)(n+4)(n+5)}{2.4.(2n+5)(2n+7)}}u^{-4}+{\text{etc. }}$$

aut si magis placet in ${\textstyle F({\frac {1}{2}}n+1,{\frac {1}{2}}n+{\frac {3}{2}},n+{\frac {5}{2}},u^{-2}).}$ Haec quoque inductio facillime ad plenam certitudinem evehitur vel per methodum vulgo notam vel adiumento formulae 19 in commentatione saepius citatae.

19.

Quum sufficiat, functionum ${\textstyle T,}$ ${\textstyle U}$ alterutram nosse, posterioris determinationem tamquam simpliciorem praetulimus. Quae quemadmodum evolutioni seriei ${\textstyle u^{-1}+{\frac {1}{3}}u^{-3}+{\frac {1}{5}}u^{-5}+}$ etc. in fractionem continuam innixa est, per ratiocinia similia ex evolutione seriei ${\textstyle t^{-1}+{\frac {1}{2}}t^{-2}+{\frac {1}{3}}t^{-3}+{\frac {1}{4}}t^{-4}+}$ etc. in fractionem continuam