*1.2. The* ℵ *Function*

The ℵ function was introduced by the authors during the examination of fractional differential equations particularly the drift less Fokker-Planck equation [17,18]. The function is a generalization of Fox *H* function and allows to handle different initial conditions. Today, the ℵ function is well established and in use in several applications [19–23].

An <sup>ℵ</sup> function <sup>ℵ</sup>*m*,*<sup>n</sup> <sup>p</sup>*,*<sup>q</sup>* (*z*) is defined via a Mellin-Barnes type integral using integers *<sup>m</sup>*, *<sup>n</sup>*, *pk*, *qk* such that 0 ≤ *m* ≤ *qk*, 0 ≤ *n* ≤ *pk*, for *ai*, *bj*, *ai*,*k*, *bi*,*<sup>k</sup>* ∈ C with C, the set of complex numbers, and for *<sup>α</sup>i*, *<sup>β</sup>j*, *<sup>α</sup>i*,*k*, *<sup>β</sup>i*,*<sup>k</sup>* <sup>∈</sup> <sup>R</sup><sup>+</sup> = (0, <sup>∞</sup>) (*<sup>i</sup>* <sup>=</sup> 1, 2, . . . , *pk*; *<sup>j</sup>* <sup>=</sup> 1, 2, . . . , *qk*), *<sup>τ</sup><sup>k</sup>* <sup>R</sup> for k = 1,...,r. The integration path C extends from *γ* − i∞ to *γ* + i∞ , and is such that the poles of the gamma functions in the numerator Γ 1 − *aj* − *αjs* , *j* = 1, ... , *n* do not coincide with the poles of the gamma functions Γ *bj* + *βjs* , *j* = 1, ... , *m*. The parameters *pk* and *qk* are non-negative integers satisfying 0 ≤ *n* ≤ *pk*, 0 ≤ *m* ≤ *qk*. All the poles of the integrand (8) are often assumed to be simple, and the empty product is interpreted as unity. The ℵ function is defined as follows

$$\mathcal{N}^{m,n}\_{p\_k,q\_k,\mathbf{r}\_k;r}\left(z \Big|\begin{array}{c} (a\_i,a\_i)\_{1,p} \\ (b\_j,\beta\_j)\_{1,q} \end{array}\right) = \frac{1}{2\pi i} \int\_{\mathcal{C}} \mathcal{A}^{m,n}\_{p\_k,q\_k,\mathbf{r}\_k;r}(s) \, z^{-s} ds\tag{8}$$

with the Mellin representation of the kernel <sup>A</sup>*m*,*<sup>n</sup> pk*,*qk*,*τk*;*r*(*s*)

$$\mathcal{A}\_{p\_{\mathbb{L}},q\_{\mathbb{K}},\pi\_{\mathbb{K}}r}^{\mathfrak{m},\mathfrak{n}}\left(\begin{array}{c}\left(a\_{i},a\_{i}\right)\_{1,\mathfrak{p}}\\\left(b\_{j},\beta\_{j}\right)\_{1,\mathfrak{q}}\end{array}\bigg|s\right) = \frac{\prod\_{j=1}^{m}\Gamma\left(b\_{j}+\beta\_{j}s\right)\prod\_{j=1}^{n}\Gamma\left(1-a\_{j}-a\_{j}s\right)}{\sum\_{k=1}^{r}\pi\_{k}\prod\_{j=m+1}^{\mathfrak{k}}\Gamma\left(1-b\_{j,k}-\beta\_{j,k}s\right)\prod\_{j=n+1}^{p\_{k}}\Gamma\left(a\_{j,k}+a\_{j,k}s\right)}.\tag{9}$$

Here

$$z^{-s} = \exp\left[-s\{\log|z| + i\arg z\}\right], \; z \neq 0, \; i = \sqrt{-1}. \tag{10}$$

Note that the Fox *H* function follows from the ℵ function in case when *r* = 1 and *τ<sup>k</sup>* = 1. If *τ<sup>k</sup>* = 1 for *k* = 1, ... ,*r* the Aleph function reduces to a Saxena *I* function [10]. According to the definition of C we are numerically dealing with a Bromwich integral in the form

$$\begin{split} \mathcal{R}^{m,n}\_{p\_k,q\_k,\mathbf{r}\_k;\mathcal{T}}\left(z\bigg|\begin{array}{c} \left(a\_i,a\_i\right)\_{1,p} \\ \left(b\_j,\beta\_j\right)\_{1,q} \end{array}\right) &=& \frac{1}{2\pi i} \int\_{\mathcal{C}} \mathcal{A}^{m,n}\_{p\_k,q\_k,\mathbf{r}\_k;\mathcal{T}}(s) \, z^{-s} ds \\ &=& \frac{1}{2\pi i} \int\_{\gamma-i\infty}^{\gamma+i\infty} \mathcal{A}^{m,n}\_{p\_k,q\_k,\mathbf{r}\_k;\mathcal{T}}(s) \, z^{-s} ds \\ &=& \frac{1}{2\pi} \int\_{-\infty}^{\infty} \mathcal{A}^{m,n}\_{p\_k,q\_k,\mathbf{r}\_k;\mathcal{T}}(\gamma+i\sigma) \, z^{-\gamma-i\sigma} d\sigma. \end{split} \tag{11}$$

The Aleph function in the present form is the result of solving integral and differential equations using linear transform techniques and is now considered the most generalized special function of a function representation [22].
