samples/texts/1085779/page_10.md

ratio and exponential constant) therefore

$$\mathcal{H}(\frac{1}{\pi },\frac{1}{\varphi },\frac{1}{e};2r(3);1)=\pi \varphi e(\frac{{\tanh }^{-1}\left(\frac{1}{\sqrt{\pi }}\right)}{\sqrt{\pi }(\pi -\varphi )(\pi -e)}+\frac{{\tanh }^{-1}\left(\frac{1}{\sqrt{\varphi }}\right)}{\sqrt{\varphi }(\varphi -\pi )(\varphi -e)}+\frac{{\tanh }^{-1}\left(\frac{1}{\sqrt{e}}\right)}{\sqrt{e}(e-\pi )(e-\varphi )})\backslash mathcal\{H\}\backslash left(\backslash frac\{1\}\{\backslash pi\},\; \backslash frac\{1\}\{\backslash phi\},\; \backslash frac\{1\}\{e\};\; 2r(3);\; 1\backslash right)\; =\; \backslash pi\backslash phi\; e\; \backslash left(\; \backslash frac\{\backslash tanh^\{-1\}\backslash left(\backslash frac\{1\}\{\backslash sqrt\{\backslash pi\}\}\backslash right)\}\{\backslash sqrt\{\backslash pi\}(\backslash pi\; -\; \backslash phi)(\backslash pi\; -\; e)\}\; +\; \backslash frac\{\backslash tanh^\{-1\}\backslash left(\backslash frac\{1\}\{\backslash sqrt\{\backslash phi\}\}\backslash right)\}\{\backslash sqrt\{\backslash phi\}(\backslash phi\; -\; \backslash pi)(\backslash phi\; -\; e)\}\; +\; \backslash frac\{\backslash tanh^\{-1\}\backslash left(\backslash frac\{1\}\{\backslash sqrt\{e\}\}\backslash right)\}\{\backslash sqrt\{e\}(e\; -\; \backslash pi)(e\; -\; \backslash phi)\}\; \backslash right)$$

Theorem 3.

$$\mathcal{H}(a_{r(m)};p_{r(m)};\eta )=\frac{1}{\eta }{}_2{F}_1(m,\frac{\eta }{p};\frac{\eta }{p}+1;a);a\in (0,1),m\in \mathbb{N}\backslash mathcal\{H\}(a\_\{r(m)\};\; p\_\{r(m)\};\; \backslash eta)\; =\; \backslash frac\{1\}\{\backslash eta\}\; \{\}\_2F\_1\backslash left(m,\; \backslash frac\{\backslash eta\}\{p\};\; \backslash frac\{\backslash eta\}\{p\}\; +\; 1;\; a\backslash right);\; \backslash quad\; a\; \backslash in\; (0,\; 1),\; m\; \backslash in\; \backslash mathbb\{N\}$$

where ${}_2F_1$ is a Hypergeometric Series[3]

Proof.

\begin{align*} \mathcal{H}(a_{r(m)}; p_{r(m)}; \eta) &= \sum_{n_1,n_2,n_3,\ldots,n_m \ge 0} \frac{a^{n_1+n_2+\cdots+n_m}}{(pn_1+pn_2+\cdots+pn_m+\eta)} \\ &= \frac{1}{a^{\frac{\eta-1}{p}}} \int_0^1 \sum_{n_1,n_2,n_3,\ldots,n_m \ge 0} (ax^p)^{n_1+n_2+n_3+\cdots+n_m+\frac{\eta-1}{p}} dx \\ &= \int_0^1 \frac{x^{(\eta-1)}}{(1-ax^p)^m} dx \tag*{\text{(Let } t=x^p\text{)}} \\ &= \frac{1}{p} \int_0^1 \frac{t^{\frac{\eta}{p}-1}}{(1-at)^m} dt \end{align*}

Since

$$\frac{\mathrm{\Gamma }(\beta )\mathrm{\Gamma }(\gamma -\beta )}{\mathrm{\Gamma }(\gamma )}{}_2{F}_1(\alpha ,\beta ;\gamma ;z)={\int }_0^1{t}^{\beta -1}(1-t{)}^{\gamma -\beta -1}(1-zt{)}^{-\alpha }dt(\text{Euler Integral})\backslash frac\{\backslash Gamma(\backslash beta)\backslash Gamma(\backslash gamma-\backslash beta)\}\{\backslash Gamma(\backslash gamma)\}\; \{\}\_2F\_1(\backslash alpha,\; \backslash beta;\; \backslash gamma;\; z)\; =\; \backslash int\_0^1\; t^\{\backslash beta-1\}(1-t)^\{\backslash gamma-\backslash beta-1\}(1-zt)^\{-\backslash alpha\}dt\; \backslash quad\; (\backslash text\{Euler\; Integral\})$$

Therefore

$$\begin{array}{c}{\displaystyle \frac{1}{p}{\int }_0^1\frac{t^{\frac{\eta }{p}-1}}{(1-at{)}^m}dt=\frac{1}{p}\frac{\mathrm{\Gamma }\left(\frac{\eta }{p}\right)\mathrm{\Gamma }(1)}{\mathrm{\Gamma }(\frac{\eta }{p}+1)}{}_2{F}_1(m,\frac{\eta }{p};\frac{\eta }{p}+1;a)}\\ {\displaystyle \Rightarrow }\\ & {\displaystyle \mathcal{H}(a_{r(m)};p_{r(m)};\eta )=\frac{1}{\eta }{}_2{F}_1(m,\frac{\eta }{p};\frac{\eta }{p}+1;a)}& & \end{array}\backslash begin\{gather*\}\; \backslash frac\{1\}\{p\}\; \backslash int\_0^1\; \backslash frac\{t^\{\backslash frac\{\backslash eta\}\{p\}-1\}\}\{(1-at)^m\}\; dt\; =\; \backslash frac\{1\}\{p\}\; \backslash frac\{\backslash Gamma\backslash left(\backslash frac\{\backslash eta\}\{p\}\backslash right)\; \backslash Gamma(1)\}\{\backslash Gamma\backslash left(\backslash frac\{\backslash eta\}\{p\}+1\backslash right)\}\; \{\}\_2F\_1\backslash left(m,\; \backslash frac\{\backslash eta\}\{p\};\; \backslash frac\{\backslash eta\}\{p\}+1;\; a\backslash right)\; \backslash \backslash \; \backslash Rightarrow\; \backslash \backslash \; \backslash mathcal\{H\}(a\_\{r(m)\};\; p\_\{r(m)\};\; \backslash eta)\; =\; \backslash frac\{1\}\{\backslash eta\}\; \{\}\_2F\_1\backslash left(m,\; \backslash frac\{\backslash eta\}\{p\};\; \backslash frac\{\backslash eta\}\{p\}+1;\; a\backslash right)\; \backslash tag\{9\}\; \backslash end\{gather*\}$$

$\square$