**Appendix A. Lauricella Function**

In 1893, G. Lauricella [39] investigated the properties of four series *F*(*n*) *<sup>A</sup>* , *<sup>F</sup>*(*n*) *<sup>B</sup>* , *<sup>F</sup>*(*n*) *<sup>C</sup>* , *<sup>F</sup>*(*n*) *D* of *n* variables. When *n* = 2, these functions coincide with Appell's *F*2, *F*3, *F*4, *F*1, respectively. When *n* = 1, they all coincide with Gauss' <sup>2</sup>*F*1. We present here only the Lauricella series *F*(*n*) *<sup>D</sup>* given as follows

$$F\_D^{(n)}(a, b\_1, \dots, b\_n; c; \mathbf{x}\_1, \dots, \mathbf{x}\_n) = \sum\_{m\_1=0}^{\infty} \dots \sum\_{m\_n=0}^{\infty} \frac{(a)\_{m\_1 + \dots + m\_n} (b\_1)\_{m\_1} \dots (b\_n)\_{m\_n}}{(c)\_{m\_1 + \dots + m\_n}} \frac{\mathbf{x}\_1^{m\_1}}{m\_1!} \dots \frac{\mathbf{x}\_n^{m\_n}}{m\_n!} \tag{A1}$$

where |*x*1|,..., |*xn*| < 1. The Pochhammer symbol (*q*)*<sup>i</sup>* indicates the *i*-th rising factorial of *q*, i.e.,

$$\dot{q}(q)\_i = q(q+1)\dots(q+i-1) = \frac{\Gamma(q+i)}{\Gamma(q)}\quad\text{if}\quad i = 1,2,\dots \tag{A2}$$

If *<sup>i</sup>* <sup>=</sup> 0, (*q*)*<sup>i</sup>* <sup>=</sup> 1. Function *<sup>F</sup>*(*n*) *<sup>D</sup>* (.) can be expressed in terms of multiple integrals as follows [42]

$$F\_D^{(n)}(a, b\_1, \ldots, b\_n; c; \mathbf{x}\_1, \ldots, \mathbf{x}\_n) = \frac{\Gamma(c)}{\Gamma(c - \sum\_{i=1}^n b\_i) \prod\_{i=1}^n \Gamma(b\_i)} \times$$

$$\int\_{\Omega} \dots \int \prod\_{i=1}^n u\_i^{b\_i - 1} (1 - \sum\_{i=1}^n u\_i)^{c - \sum\_{i=1}^n b\_i - 1} (1 - \sum\_{i=1}^n \mathbf{x}\_i u\_i)^{-a} \prod\_{i=1}^n \mathbf{d} u\_i \tag{A3}$$

where Ω = {(*u*1, *u*2, ... , *un*); 0 ≤ *ui* ≤ 1, *i* = 1, ... , *n*, and 0 ≤ *u*<sup>1</sup> + *u*<sup>2</sup> + ... + *un* ≤ 1}, Real(*bi*) > 0 for *i* = 1, ... , *n* and Real(*c* − *b*<sup>1</sup> − ... − *bn*) > 0. Lauricella's *FD* can be written as a one-dimensional Euler-type integral for any number *n* of variables. The integral form of *F*(*n*) *<sup>D</sup>* (.) is given as follows when Real(*a*) > 0 and Real(*c* − *a*) > 0

$$F\_D^{(n)}(a, b\_1, \dots, b\_n; c; \mathbf{x}\_1, \dots, \mathbf{x}\_n) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)} \int\_0^1 u^{a-1} (1-u)^{c-x-1} (1-u\mathbf{x}\_1)^{-b\_1} \dots (1-u\mathbf{x}\_n)^{-b\_n} d\mathbf{u}.\tag{A4}$$

Lauricella has given several transformation formulas, from which we use the two following relationships. More details can be found in Exton's book [43] on hypergeometric equations.

$$\begin{aligned} &F\_D^{(n)}(a, b\_1, \dots, b\_n; c; \mathbf{x}\_1, \dots, \mathbf{x}\_n) \\ &= \prod\_{i=1}^n (1 - \mathbf{x}\_i)^{-b\_i} F\_D^{(n)}\left(c - a, b\_1, \dots, b\_n; c; \frac{\mathbf{x}\_1}{\mathbf{x}\_1 - 1}, \dots, \frac{\mathbf{x}\_n}{\mathbf{x}\_n - 1}\right) \end{aligned} \tag{A5}$$

$$\mathbf{x} = (\mathbf{1} - \mathbf{x}\_1)^{-a} F\_D^{(n)} \left( a\_r \mathbf{c} - \sum\_{i=1}^n b\_{ji} b\_{2r}, \dots, b\_{ni} \mathbf{c}; \frac{\mathbf{x}\_1}{\mathbf{x}\_1 - 1}, \frac{\mathbf{x}\_1 - \mathbf{x}\_2}{\mathbf{x}\_1 - 1}, \dots, \frac{\mathbf{x}\_1 - \mathbf{x}\_n}{\mathbf{x}\_1 - 1} \right) \tag{A6}$$

$$\mathbf{x} = (\mathbf{1} - \mathbf{x}\_n)^{-a} F\_D^{(n)} \left( a, b\_1, \dots, b\_{n-1}, c - \sum\_{i=1}^n b\_{i\cdot}; c\_\cdot \frac{\mathbf{x}\_n - \mathbf{x}\_1}{\mathbf{x}\_n - 1}, \frac{\mathbf{x}\_n - \mathbf{x}\_2}{\mathbf{x}\_n - 1}, \dots, \frac{\mathbf{x}\_n - \mathbf{x}\_{n-1}}{\mathbf{x}\_n - 1}, \frac{\mathbf{x}\_n}{\mathbf{x}\_n - 1} \right) \tag{A7}$$

$$\mathbf{x} = (\mathbf{1} - \mathbf{x}\_1)^{\mathbf{c} - \mathbf{a}} \prod\_{i=1}^{n} (\mathbf{1} - \mathbf{x}\_i)^{-b\_i} \mathbf{F}\_D^{(n)} \left( \mathbf{c} - a\_r \mathbf{c} - \sum\_{i=1}^{n} b\_i, b\_2, \dots, b\_{\text{nl}}; \mathbf{c}; \mathbf{x}\_1, \mathbf{x}\_1, \mathbf{x}\_1, \frac{\mathbf{x}\_2 - \mathbf{x}\_1}{\mathbf{x}\_2 - 1}, \dots, \frac{\mathbf{x}\_n - \mathbf{x}\_1}{\mathbf{x}\_n - 1} \right) \tag{A8}$$

$$\mathbf{x} = (\mathbf{1} - \mathbf{x}\_{\mathrm{n}})^{c-a} \prod\_{i=1}^{n} (\mathbf{1} - \mathbf{x}\_{i})^{-b\_{i}} \mathbf{F}\_{\mathrm{D}}^{(n)} \left( \mathbf{c} - a\_{r} b\_{1}, \dots, b\_{n-1}, \mathbf{c} - \sum\_{i=1}^{n} b\_{i}; \mathbf{c}\_{r}^{\*} \frac{\mathbf{x}\_{1} - \mathbf{x}\_{\mathrm{n}}}{\mathbf{x}\_{1} - \mathbf{1}}, \dots, \frac{\mathbf{x}\_{n-1} - \mathbf{x}\_{\mathrm{n}}}{\mathbf{x}\_{n-1} - \mathbf{1}}, \mathbf{x}\_{\mathrm{n}} \right). \tag{A9}$$

## **Appendix B. Demonstration of Derivative**
