Third memoir on the expansion of certain infinite products. (Q1524912)

Es handelt sich um Beziehungen der folgenden Functionen zu einander: \[ (x) \equiv (1-x)(1-xq)(1-xq^2)\cdots, \] \[ P(x) = 1/(xe^{\theta i})(xe^{-\theta i}) \equiv 1 + \frac{A_1(\theta)}{q_1}x + \frac{A_2(\theta)}{q_2!}x^2 +\cdots, \] wo \[ q_r \equiv 1-q^r,\quad q_r! \equiv (1-q)(1-q^2)\dots(1-q^r) \] ist; \[ \begin{aligned} P(x\lambda)/P(x) &= 1 + \frac{L_1(\theta)}{q_1}x +\cdots,\\ P(x\mu)/P(x) &= 1 + \frac{M_1(\theta)}{q_1}x +\cdots,\end{aligned} \] \[ \begin{alignedat}{2} \lambda_r &\equiv 1 - \lambda q^{r-1},&\quad \lambda_r! &\equiv \lambda_1\lambda_2\dots \lambda_r,\\ \mu_r &\equiv 1 - \mu q^{r-1},&\quad \mu_r! &\equiv \mu_1\mu_2\dots \mu_r,\end{alignedat} \] \[ \lambda_r^2! \equiv (1 - \lambda^2)(1 - \lambda^2q)\dots(1 - \lambda^2q^{r-1}),\text{ u. s. w.} \] \[ \varphi\{a,b,c,q,x\}\equiv 1 + \frac{a_1b_1}{q_1c_1}x + \frac{a_2!b_2!}{q_2!c_2!}x^2 +\cdots, \] \[ f_r \equiv 1 - 2q^{r-\frac12}\cos2\varphi + q^{2r-1}. \] Wir heben aus den Resultaten folgende hervor: \[ \begin{multlined} \frac{L_r}{q_r!} = \frac{M_r\mu_{r+1}}{q_r!}\frac{\lambda_r!}{\mu_{r+1}!} + \frac{M_{r-2}\mu_{r-1}}{q_{r-2}!}\frac{(\mu-\lambda)\lambda_{r-1}!}{q_1!\mu_r!} +\cdots\\ + \frac{M_{r-2s}\mu_{r-2s+1}}{q_{r-2s}} \frac{(\mu-\lambda)(\mu-\lambda q)\dots(\mu-\lambda q^{s-1})}{q_s!\mu_{r-s+1}!}\lambda_{r-s}! +\cdots,\end{multlined} \] \[ \begin{multlined} \frac{P(\sqrt{\mu q})}{P(x)} = \frac{(x\sqrt{\mu q})(\mu x\sqrt{\mu q})}{(\mu)(\mu x^2)}\\ \times \left\{\mu_1 + \frac{M_1\mu_2}{q_1!} \frac{x-\sqrt{\mu q}}{1-x\mu\sqrt{\mu q}} + \frac{M_2\mu_3}{q_2!} \frac{(x-\sqrt{\mu q})(x-\sqrt{\mu q^3})}{(1-x\mu\sqrt{\mu q})(1-x\mu\sqrt{\mu q^3})} +\cdots\right\},\end{multlined} \] \[ \begin{split} 1 &+ \frac{(u-q^\frac12)(v-q^\frac12)}{q_1q_2}A_2 + \frac{(u-q^\frac12)(u-q^\frac32)(v-q^\frac12)(v-q^\frac32)}{q_4!}A_4 +\cdots\\ &\quad= \frac{(q^\frac12\mu u)(q^\frac12\mu v)}{(\mu)(\mu uv)} \left\{\mu_1 + \frac{\mu_3M_2}{q_2!} \frac{(u-q^\frac12)(v-q^\frac12)}{(1-\mu uq^\frac12)(1-\mu vq^\frac12)}\right.\\ &\quad\quad+ \left. \frac{\mu_5M_4(u-q^\frac12)(u-q^\frac32)(v-q^\frac12)(v-q^\frac32)}{q_4!(1-\mu uq^\frac12)(1-\mu uq^\frac32)(1-\mu vq^\frac12)(1-\mu vq^\frac32)} +\cdots\right\},\end{split} \] \[ \frac{A_r}{\lambda_{r+1}!} + \frac{A_{r+2}}{q_1\lambda_{r+2}!}\lambda + \frac{A_{r+4}}{q_2!\lambda_{r+3}!}\lambda^2 +\cdots= \frac{(\lambda^2)}{(\lambda)^2P(\lambda,2\theta)} \frac{L_r}{(1-\lambda^2q^{r-1})!}, \] \[ \begin{split} 1 &+ \sum \frac{A_r(\theta)A_r(\varphi)}{q_r!}x^r =\text{ Vielfachen von }(x^2-1)\\ &+ \frac{(\lambda^2)}{(\lambda)^2P(\lambda,2\varphi)}\left\{\lambda_1 + \frac{\lambda_2L_1(\theta)L_1(\varphi)}{q_1(1-\lambda^2)}x + \frac{\lambda_3L_2(\theta)L_2(\varphi)}{q_2!(1-\lambda^2)(1-\lambda^2q)}x^2 +\cdots\right\},\end{split} \] \[ \begin{split} \frac{P(\lambda xe^{\varphi i})P(\lambda xe^{-\varphi i})}{P(xe^{\varphi i})P(xe^{-\varphi i})} &= \varphi\{\lambda,\lambda^2,\lambda q,q,x^2\}\\ &+ x\frac{L_1(\theta)L_1(\varphi)}{q_1\lambda_1} \varphi\{\lambda,\lambda^2,\lambda q^2,q,x^2\}\\ &+ x^2\frac{L_2(\theta)L_2(\varphi)}{q_2!\lambda_2!} \varphi\{\lambda,\lambda^2q^2,\lambda q^3,q,x^2\} +\cdots.\end{split} \]