Pagina:Werkecarlf03gausrich.djvu/204

$${\cfrac {1}{t-{\cfrac {\frac {1}{2}}{1-{\cfrac {\frac {1}{6}}{t-{\cfrac {\frac {2}{6}}{1-{\cfrac {\frac {2}{10}}{t-{\cfrac {\frac {3}{10}}{1-{\text{ etc.}}}}}}}}}}}}}}$$

derivare potuissemus algorithmum ad determinandam functionem ${\textstyle T}$ pro valoribus successivis numeri ${\textstyle n.}$ Ad eandem vero conclusionem pervenimus perpendendo, ${\textstyle T}$ nihil aliud esse quam ${\textstyle {\frac {U}{2^{n+1}}}}$ seu ${\textstyle {\frac {W^{(n+1)}}{2^{n+1}}},}$ si pro ${\textstyle u}$ scribitur ${\textstyle 2t-1,}$ quo pacto functiones successive pro ${\textstyle T}$ adoptandae habebuntur per algorithmum sequentem:

$${\begin{alignedat}{2}W^{\phantom {\prime \prime \prime \prime }}&=1\\{\frac {1}{2}}W^{\prime {\phantom {\prime \prime \prime }}}&=t-{\frac {1}{2}}\\{\frac {1}{4}}W^{\prime \prime {\phantom {\prime \prime }}}&=(t-{\frac {1}{2}})\cdot {\frac {1}{2}}W^{\prime }-{\frac {1.1}{2.6}}W&&=tt-t+{\frac {1}{6}}\\{\frac {1}{8}}W^{\prime \prime \prime {\phantom {\prime }}}&=(t-{\frac {1}{2}})\cdot {\frac {1}{4}}W^{\prime \prime }-{\frac {2.2}{6.10}}\cdot {\frac {1}{2}}W^{\prime }&&=t^{3}-{\frac {3}{2}}tt+{\frac {3}{5}}t-{\frac {1}{20}}\\{\frac {1}{10}}W^{\prime \prime \prime \prime }&=(t-{\frac {1}{2}})\cdot {\frac {1}{8}}W^{\prime \prime \prime }-{\frac {3.3}{10.14}}\cdot {\frac {1}{4}}W^{\prime \prime }&&=t^{4}-2t^{3}+{\frac {9}{7}}tt-{\frac {2}{7}}t+{\frac {1}{70}}\end{alignedat}}$$

etc. Per inductionem hinc resultat generaliter

$$T=t^{n+1}-{\frac {(n+1)^{2}}{1.(2n+2)}}t^{n}+{\frac {(n+1)^{2}.nn}{1.2.(2n+2)(2n+1)}}t^{n-1}-{\frac {(n+1)^{2}.nn.(n-1)^{2}}{1.2.3.(2n+2)(2n+1).2n}}t^{n-2}+{\text{etc.}}$$

sive ${\textstyle T=t^{n+1}F(-(n+1),-(n+1),-2(n+1),t^{-1}),}$ cui inductioni facile est demonstrationis vim conciliare. Si magis arridet, ${\textstyle T}$ ordine terminorum inverso etiam per

$$\pm {\frac {1.2.3.4\ldots (n+1)}{2.6.10.14\ldots .(4n+2)}}F(n+2,-(n+1),1,t)$$

exprimi potest, ubi signum superius valet pro ${\textstyle n}$ impari, inferius pro pari. Simili denique modo generaliter ${\textstyle T^{\prime \prime }}$ aequalis invenitur producto ex

$${\frac {1.1.2.2.3.3\ldots \ldots (n+1).(n+1)}{2.6.6.10.10.14\ldots \ldots (4n+2).(4n+6)}}t^{-(n+2)}$$

in seriem infinitam

$$1+{\frac {(n+2)^{2}}{1.(2n+4)}}t^{-1}+{\frac {(n+2)^{2}(n+3)^{2}}{1.2.(2n+4)(2n+5)}}t^{-2}+{\frac {(n+2)^{2}.(n+3)^{2}(n+4)^{2}}{1.2.3.(2n+4)(2n+5)(2n+6)}}t^{-3}+{\text{etc. }}$$

sive in ${\textstyle F(n+2,n+2,2n+4,t^{-1})}$