Pagina:Werkecarlf02gausrich.djvu/30

$$1+{\frac {1}{x-1}}+{\frac {1}{x-1}}\cdot {\frac {1}{xx-1}}+{\frac {1}{x-1}}\cdot {\frac {1}{xx-1}}\cdot {\frac {1}{x^{3}-1}}+{\text{etc.}}$$

Haec itaque series aequalis est producto infinito

$${\frac {1}{1-{\frac {1}{x}}}}\cdot {\frac {1}{1-{\frac {1}{x^{3}}}}}\cdot {\frac {1}{1-{\frac {1}{x^{5}}}}}\cdot {\frac {1}{1-{\frac {1}{x^{7}}}}}{\text{etc. in infin. }}$$

Porro quum generaliter sit

$$f(x,m)=f(x,m-2\lambda ).(1-x^{m-1})(1-x^{m-3})(1-x^{m-5})\ldots (1-x^{m-2\lambda +1})$$

erit

$${\begin{aligned}f(x,m)&=f(x,-\infty )\cdot (1-x^{m-1})(1-x^{m-3})(1-x^{m-5}){\text{ etc. in infin. }}\\&={\frac {1-x^{m-1}}{1-x^{-1}}}\cdot {\frac {1-x^{m-3}}{1-x^{-3}}}\cdot {\frac {1-x^{m-5}}{1-x^{-5}}}\cdot {\frac {1-x^{m-7}}{1-x^{-7}}}{\text{ etc. in infin. }}\end{aligned}}$$

quos factores tandem continuo magis ad unitatem convergere palam est.

Attentionem peculiarem meretur casus ${\textstyle m=-1}$, ubi fit

$$f(x,-1)=1+x^{-1}+x^{-3}+x^{-6}+x^{-10}+{\text{etc.}}$$

Haec itaque series aequatur producto infinito

$${\frac {1-x^{-2}}{1-x^{-1}}}\cdot {\frac {1-x^{-2}}{1-x^{-3}}}\cdot {\frac {1-x^{-6}}{1-x^{-5}}}{\text{ etc.}}$$

sive scribendo ${\textstyle x}$ pro ${\textstyle x^{-1}}$, erit

$$1+x+x^{3}+x^{6}+{\text{etc.}}={\frac {1-xx}{1-x}}\cdot {\frac {1-x^{4}}{1-x^{3}}}\cdot {\frac {1-x^{6}}{1-x^{5}}}\cdot {\frac {1-x^{8}}{1-x^{7}}}{\text{ etc.}}$$

Haec aequalitas inter duas expressiones abstrusiores, ad quas alia occasione reveniemus, valde sane est memorabilis.

9.

Secundo loco considerabimus progressionem hancce

$$1+x^{\frac {1}{2}}{\frac {1-x^{m}}{1-x}}+x{\frac {(1-x^{m})}{(1-x)}}{\frac {(1-x^{m-1})}{(1-xx)}}+x^{\frac {3}{2}}{\frac {(1-x^{m})(1-x^{m-1})(1-x^{m-2})}{(1-x)(1-xx)(1-x^{3})}}+{\text{etc.}}$$

sive

$$1+x^{\frac {1}{2}}(m,1)+x(m,2)+x^{\frac {3}{2}}(m,3)+xx(m,4)+{\text{etc.}}$$

quam per ${\textstyle F(x,m)}$ denotabimus. Restringemus hanc disquisitionem ad casum eum, ubi ${\textstyle m}$ est integer positivus, ita ut haec quoque series semper abrumpatur