Definition:Zeraoulia Function

Zeraoulia Function

Definition of Zeraoulia new Special function
Zeraoulia function is a new Special function see,  proposed by Zeraoulia Rafik in 17/02/2017 and have been studied by Zeraoulia Rafik,Alvaro Humberto Salas  Davide L.Ocampo  and published in International Journal of Mathematics and Mathematical Sciences , ,  it behave like more than error function and it is defined as :


 * $\displaystyle \map T a = \int_0^a \paren {e^{-x^2} }^{\map {\operatorname {erf} } x} \rd x$

Mathematica gives for the first $100$ digits:


 * $\map T \infty = 0 \cdotp 9721069927691785931510778754423911755542721833855699009722910408441888759958220033410678218401258734$

with $\map T 0 = 0$

Series expansion of Zeraoulia Function
Series expansion of zeraoulia function defined by $T(x)=\int_{0}^{x}e^{-\xi^2 \text{erf}(\xi)}d\xi $ defined by this identity (for proof see 4-page 5)  :

\begin{equation}\label{eq_4} \int\limits_{0}^{x}e^{-\xi ^{2}\text{erf}(\xi )d\xi }=\sum\limits_{n=0}^{\infty }\lim_{\varepsilon ->0}\left( \sum\limits _{\substack{ k_{1}+2k_{2}+\cdots +nk_{n}=n \\ k_{1}\geq 0,k_{2}\geq 0,...,k_{n}\geq 0}}\prod\limits_{j=1}^{n}\frac{^{A_{j,\varepsilon }^{k_{j}}}% }{k_{j}!}\right) \frac{x^{n+1}}{n+1} \end{equation} where: \begin{eqnarray*} A_{j,\epsilon } &=&\frac{2(-1)^{(j-1)/2}}{(j-2)(\frac{1}{2}(j-3))!\sqrt{\pi }% }\text{ if \ }j\geq 3\text{ and }j\text{ an odd integer;} \\ A_{j,\epsilon } &=&\varepsilon \text{ otherwise \ }(0<\varepsilon <1)\text{. } \end{eqnarray*}