Student of the Polytechnic of Turin who spends time to make calculations

My favorite formulas:

$$z!_{(\infty)}=\prod_{k=1}^{\infty}k^{\text{sinc}(z-k)}=\exp\left(-\frac{1}{2\pi i}\oint_{\Gamma}\frac{\text{Li}'_0(t)}{t^{z+1}}\mathrm{d}t\right)$$

$$\Delta_{\alpha}(z)=\frac{1}{\alpha}\sum_{\omega:\,\omega^\alpha=1}\omega^z\qquad\arg(\omega)\in(-\pi,\pi]$$

$$z!_{(\alpha)}=\frac{\displaystyle\alpha^{1+\frac{z}{\alpha}}\Gamma\left(1+\frac{z}{\alpha}\right)}{\displaystyle\prod_{j=1}^{\alpha}\left[\alpha^{\frac{j}{\alpha}}\Gamma\left(\frac{j}{\alpha}\right)\right]^{C_{\alpha}(z-j)}}\qquad \text{where }C_{\alpha}(z)=\Re[\Delta_{\alpha}](z)$$

$$\sum_{n=0}^{\infty}n!_{(\alpha)}\overset{\mathcal{R}}{=}\frac{1}{\alpha\sqrt[\alpha]{e}}\text{Ei}\left(\frac{1}{\alpha}\right)+\sum_{k=1}^{\alpha-1}\left[\frac{\cos\left(\frac{k\pi}{\alpha}\right)}{\sqrt[\alpha]{e\alpha^k}}\Gamma\left(1-\frac{k}{\alpha}\right)+\frac{1}{k}{}_1F_1\left(\left.{1 \atop 1+\frac{k}{\alpha}}\right|-\frac{1}{\alpha}\right)-1\right]$$

$$\displaystyle(-1)^m\sum_{n=1}^{x}n^s\ln(n)^m\propto\zeta^{(m)}(-s)+m!\sum_{k=0}^{m}\frac{(-\ln(x))^{m-k}}{(m-k)!}\left(\frac{x^{s+1}}{(s+1)^{k+1}}-\sum_{n=k}^{s}\frac{B_{n+1}}{(n+1)!}\left[n\atop k\right]^{(s-n+1)}x^{s-n}\right)$$

$$\int \tan^{-1}(\alpha\cdot\cos(x))x^n\mathrm{d}x=2n!\sum_{k=0}^{n}\frac{x^{n-k}}{(n-k)!}\Im\left[i^k\cdot\text{Ti}_{k+2}\left(\frac{\sqrt{\alpha^2+1}-1}{\alpha}\cdot e^{ix}\right)\right]$$

$$\int_0^x \frac{t^{m-1}}{(1\pm t^n)^p}\mathrm{d}t=\frac{x^{m}}{n}\sum_{j=1}^{p-1}\frac{\displaystyle\left[\prod_{k=j+1}^{p-1}\frac{nk-m}{nk}\right]}{j\left(1\pm x^{n}\right)^{j}}+(I(x)-J(x))\left[\prod_{k=1}^{p-1}\frac{nk-m}{nk}\right]$$

$$ I(x)=\frac{1}{n}\sum_{k=1}^{n}\Re\left[e^{m\theta}\ln(1-xe^{-i\theta})\right]\quad \begin{cases}\text{When }+:&J(x)=\displaystyle\sum_{k=1}^{\left\lfloor{\frac{m-1}{n}}\right\rfloor}(-1)^{k}\frac{x^{m-nk}}{m-nk}\qquad\theta=\frac{2k-1}{n}\pi\\ \text{When }-:&\displaystyle J(x)=\sum_{k=1}^{\left\lfloor{\frac{m-1}{n}}\right\rfloor}\frac{x^{m-nk}}{m-nk}\quad\qquad\qquad\theta=\frac{2k}{n}\pi \end{cases}$$

$$\Psi_n(t,\nu):=B_n\left(\ln(t)-\psi(\nu),-\psi^{(1)}(\nu),...,-\psi^{(n-1)}(\nu)\right)$$ $$\text{Li}^{(n,0)}_{\nu}(z)=\int_{0}^{\infty}\frac{t^{\nu+s-1}}{\Gamma(\nu+s)}\text{Li}_{-s}\left(ze^{-t}\right)\Psi_n(t,\nu+s)\mathrm{d}t\qquad {\Re(\nu)+s>0\atop s=1,2,3,...}$$

$$\begin{align*} \text{Li}^{(1,0)}_{-\nu}(z)=&\int_{0}^{\infty}\text{Li}_{-\nu-1}\left(ze^{-t}\right)(\gamma+\ln(t))\mathrm{d}t\\ =&\int_{0}^{1}\left[\text{Li}_{-\nu}(z)-\frac{\text{Li}_{-\nu}(zs)}{s}\right]\frac{1}{\ln(s)}\mathrm{d}s&\nu\geq 0 \end{align*}$$