Definition:Zeraoulia Function

Definition
The 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 :


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

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 $\ds \map T x = \int_0^x e^{-\xi^2 \map \erf \xi} \rd \xi$ defined by this identity (for proof see 4-page 5)


 * $\ds \int_0^x e^{-\xi^e \map \erf \xi} \rd \xi = \sum {n \mathop = 0}^\infty \lim_{\varepsilon \mathop \to 0} \paren {\sum_{\substack {k_1 + 2 k_2 + \cdots + n k_n = n \\ k_1 \ge 0, k_2 \ge 0, \ldots, k_n \ge 0} } \prod_{j \mathop = 1}^n \frac {A_{j, \varepsilon}^{k_j} } {k_j!} } \frac {x^{n + 1} } {n + 1}$

where:
 * $A_{j, \epsilon} = \begin {cases} \dfrac {2 \paren {-1}^{\paren {j - 1} / 2} } {\paren {j - 2} \paren {\frac 1 2 \paren {j - 3} }! \sqrt \pi} & : j \ge 3 \text { and } j \text { odd} \\

\varepsilon & : \text{ otherwise} \paren {0 < \varepsilon < 1} \end {cases}$