**3. Definitions and Propositions**

This section presents some definitions and exposes some propositions related to the multiple power series used to derive the closed-form expression of the expectation *<sup>E</sup>***X**<sup>1</sup> {ln[<sup>1</sup> <sup>+</sup> **<sup>X</sup>***T***Σ**−<sup>1</sup> <sup>1</sup> **<sup>X</sup>**]} and *<sup>E</sup>***X**<sup>1</sup> {ln[<sup>1</sup> <sup>+</sup> **<sup>X</sup>***T***Σ**−<sup>1</sup> <sup>2</sup> **X**]}, and as a consequence the KLD between two central MCDs.

**Definition 1.** *The Humbert series of <sup>n</sup> variables, denoted* <sup>Φ</sup>(*n*) <sup>2</sup> *, is defined for all xi* <sup>∈</sup> <sup>C</sup>, *<sup>i</sup>* <sup>=</sup> 1, . . . , *n, by the following multiple power series (Section 1.4 in [33])*

$$\Phi\_2^{(n)}(b\_1, \dots, b\_n; c; \mathbf{x}\_1, \dots, \mathbf{x}\_n) = \sum\_{m\_1=0}^{\infty} \dots \sum\_{m\_n=0}^{\infty} \frac{(b\_1)\_{m\_1} \dots (b\_n)\_{m\_n}}{(c)\_{\sum\_{i=1}^n m\_i}} \prod\_{i=1}^n \frac{\mathbf{x}\_i^{m\_i}}{m\_i!}.\tag{6}$$

The Pochhammer symbol (*q*)*<sup>i</sup>* indicates the *i*-th rising factorial of *q*, i.e., for an integer *i* > 0

$$(q)\_i = q(q+1)\dots(q+i-1) = \prod\_{k=0}^{i-1} (q+k) = \frac{\Gamma(q+i)}{\Gamma(q)}\tag{7}$$

*3.1. Integral Representation for* Φ(*n*) 2

**Proposition 1.** *The following integral representation is true for Real*{*c*} <sup>&</sup>gt; *Real*{∑*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *bi*} > 0 *and Real*{*bi*} > 0 *where Real*{.} *denotes the real part of the complex coefficients*

$$\int \dots \int\_{\Lambda} \left(1 - \sum\_{i=1}^{n} u\_{i}\right)^{c - \sum\_{i=1}^{n} b\_{i} - 1} \prod\_{i=1}^{n} u\_{i}^{b\_{i} - 1} e^{x\_{i} \mu\_{i}} \text{d}u\_{i} = B\left(b\_{1}, \dots, b\_{n}, c - \sum\_{i=1}^{n} b\_{i}\right) \Phi\_{2}^{(n)}\left(b\_{1}, \dots, b\_{n}; c; \mathbf{x}\_{1}, \dots, \mathbf{x}\_{n}\right) \tag{8}$$

*where* Δ = {(*u*1, ... , *un*)|0 ≤ *ui* ≤ 1, *i* = 1, ... , *n*; 0 ≤ *u*<sup>1</sup> + ... + *un* ≤ 1} *and the multivariate beta function B is the extension of beta function to more than two arguments (called also Dirichlet function) defined as (Section 1.6.1 in [34])*

$$B(b\_1, \ldots, b\_n, b\_{n+1}) = \frac{\prod\_{i=1}^{n+1} \Gamma(b\_i)}{\Gamma(\sum\_{i=1}^{n+1} b\_i)}.\tag{9}$$

**Proof.** The power series of exponential function is given by

$$e^{\mathbf{x}\_i u\_i} = \sum\_{m\_i=0}^{\infty} \frac{\mathbf{x}\_i^{m\_i}}{m\_i!} u\_i^{m\_i}. \tag{10}$$

By substituting the expression of the exponential into the multiple integrals we have

$$\begin{split} & \int \dots \int\_{\Delta} \left( 1 - \sum\_{i=1}^{n} u\_{i} \right)^{c - \sum\_{i=1}^{n} b\_{i} - 1} \prod\_{i=1}^{n} u\_{i}^{b\_{i} - 1} e^{\mathbf{x}\_{i} \boldsymbol{\mu}\_{i}} \mathbf{d} \boldsymbol{u}\_{i} \\ & \quad = \int \dots \int\_{\Delta} \left( 1 - \sum\_{i=1}^{n} u\_{i} \right)^{c - \sum\_{i=1}^{n} b\_{i} - 1} \left( \prod\_{i=1}^{n} \sum\_{m\_{i}=0}^{\infty} \frac{\boldsymbol{\mathcal{X}}\_{i}^{m\_{i}}}{m\_{i}!} \boldsymbol{u}\_{i}^{m\_{i} + b\_{i} - 1} \mathbf{d} \boldsymbol{u}\_{i} \right) \\ & \quad = \sum\_{m\_{1} = 0}^{\infty} \dots \sum\_{m\_{n} = 0}^{\infty} \left( \prod\_{i=1}^{n} \frac{\boldsymbol{\mathcal{X}}\_{i}^{m\_{i}}}{m\_{i}!} \right) \times \mathbf{I}\_{D} \end{split} \tag{11}$$

where the multivariate integral **I***D*, which is a generalization of a beta integral, is the type-1 Dirichlet integral (Section 1.6.1 in [34]) given by

$$\begin{split} \mathbf{I}\_{D} &= \int \dots \int\_{\Delta} \left( 1 - \sum\_{i=1}^{n} u\_{i} \right)^{c - \sum\_{i=1}^{n} b\_{i} - 1} \prod\_{i=1}^{n} u\_{i}^{m\_{i} + b\_{i} - 1} \mathbf{d}u\_{i} \\ &= \frac{\prod\_{i=1}^{n} \Gamma(b\_{i} + m\_{i}) \Gamma(c - \sum\_{i=1}^{n} b\_{i})}{\Gamma(c + \sum\_{i=1}^{n} m\_{i})} . \end{split} \tag{12}$$

Knowing that Γ(*bi* + *mi*) = Γ(*bi*)(*bi*)*mi* , the expression of **I***<sup>D</sup>* can be written otherwise

$$\mathbf{I}\_D = \frac{\prod\_{i=1}^n \Gamma(b\_i)\Gamma(c - \sum\_{i=1}^n b\_i)}{\Gamma(c)} \frac{\prod\_{i=1}^n (b\_i)\_{m\_i}}{(c)\_{\sum\_{i=1}^n m\_i}}.\tag{13}$$

Finally, plugging (13) back into (12) leads to the final result

$$\frac{\Gamma(c-\sum\_{i=1}^{n}b\_{i})\prod\_{i=1}^{n}\Gamma(b\_{i})}{\Gamma(c)}\sum\_{\begin{subarray}{c}m\_{1},\ldots,m\_{i}\\m\_{n}=0\\\blacksquare\end{subarray}}^{+\infty}\frac{\prod\_{i=1}^{n}(b\_{i})\_{m\_{i}}}{(c)\_{\sum\_{i=1}^{n}m\_{i}}}\prod\_{i=1}^{n}\frac{\chi\_{i}^{m\_{i}}}{m\_{i}!} = B\left(b\_{1},\ldots,b\_{n},c-\sum\_{i=1}^{n}b\_{i}\right)\Phi\_{2}^{(n)}\left(b\_{1},\ldots,b\_{n};c;\mathbf{x}\_{1},\ldots,\mathbf{x}\_{n}\right)\tag{14}$$

Given Proposition 1, we consider the particular cases *n* = {1, 2} one by one as follows: Case *n* = 1

$$\frac{1}{B(b\_1, c - b\_1)} \int\_0^1 u\_1^{b\_1 - 1} e^{x\_1 u\_1} (1 - u\_1)^{c - b\_1 - 1} \mathrm{d}u\_1 = \sum\_{m\_1 = 0}^{\infty} \frac{(b\_1)\_{m\_1}}{(c)\_{m\_1}} \frac{x\_1^{m\_1}}{m\_1!} = \Phi\_2^{(1)}(b\_1; c; \mathbf{x}\_1) = {}\_1F\_1(b\_1, c; \mathbf{x}\_1) \tag{15}$$

where <sup>1</sup>*F*1(.) is the confluent hypergeometric function of the first kind (Section 9.21 in [35]). Case *n* = 2

$$\frac{1}{B(b\_1, b\_2, c - b\_1 - b\_2)} \int \int\_{\substack{u\_1 \ge 0, u\_2 \ge 0 \\ u\_1 + u\_2 \le 1}} u\_1^{b\_1 - 1} u\_2^{b\_2 - 1} e^{x\_1 u\_1 + x\_2 u\_2} (1 - u\_1 - u\_2)^{c - b\_1 - b\_2 - 1} \mathrm{d}u\_1 \mathrm{d}u\_2$$

$$\frac{1}{B} = \sum\_{m\_1 = 0}^{\infty} \sum\_{m\_2 = 0}^{\infty} \frac{(b\_1)\_{m\_1} (b\_2)\_{m\_2}}{(c)\_{m\_1 + m\_2}} \frac{x\_1^{m\_1}}{m\_1!} \frac{x\_2^{m\_2}}{m\_2!} = \Phi\_2^{(2)}(b\_1, b\_2; c; \mathbf{x}\_1, \mathbf{x}\_2) = \Phi\_2(b\_1, b\_2; c; \mathbf{x}\_1, \mathbf{x}\_2) \tag{16}$$

where the double series Φ<sup>2</sup> is one of the components of the Humbert series of two variables [36] that generalize Kummer's confluent hypergeometric series <sup>1</sup>*F*<sup>1</sup> of one variable. The double series <sup>Φ</sup><sup>2</sup> converges absolutely at any *<sup>x</sup>*1, *<sup>x</sup>*<sup>2</sup> <sup>∈</sup> <sup>C</sup>.

*3.2. Multiple Power Series F*(*n*) *N*

**Definition 2.** *We define a new multiple power series, denoted by F*(*n*) *<sup>N</sup> and given by*

$$\begin{split} &F\_{N}^{(n)}\left(a;b\_{1},\ldots,b\_{n};c,c\_{n};\mathbf{x}\_{1},\ldots,\mathbf{x}\_{n}\right) \\ &=\mathbf{x}\_{n}^{-a}\sum\_{\begin{subarray}{c}m\_{1},\ldots\\m\_{n}=0\end{subarray}}^{+\infty}\frac{\left(a\right)\_{\sum\_{i=1}^{n}m\_{i}}(a-c\_{n}+1)\_{\sum\_{i=1}^{n}m\_{i}}}{(a+b\_{n}-c\_{n}+1)\_{\sum\_{i=1}^{n}m\_{i}}}\frac{\prod\_{i=1}^{n-1}(b\_{i})\_{m\_{i}}}{(c)\_{\sum\_{i=1}^{n}m\_{i}}}\prod\_{i=1}^{n-1}\left(\frac{\mathbf{x}\_{i}}{\mathbf{x}\_{n}}\right)^{m\_{i}}\frac{1}{m\_{i}!}\frac{(1-\mathbf{x}\_{n}^{-1})^{m\_{n}}}{m\_{n}!}.\end{split} \tag{17}$$

*The multiple power series (17) is absolutely convergent on the region* <sup>|</sup>*xix*−<sup>1</sup> *<sup>n</sup>* <sup>|</sup> <sup>+</sup> <sup>|</sup><sup>1</sup> <sup>−</sup> *<sup>x</sup>*−<sup>1</sup> *<sup>n</sup>* <sup>|</sup> <sup>&</sup>lt; <sup>1</sup> *in* <sup>C</sup>*n*, <sup>∀</sup> *<sup>i</sup>* ∈ {1, . . . , *<sup>n</sup>* <sup>−</sup> <sup>1</sup>}*.*

The multiple power series *F*(*n*) *<sup>N</sup>* (.) can also be transformed into two other expressions as follows

$$\begin{split} &F\_{N}^{(n)}(a;b\_{1},\ldots,b\_{n};c,c\_{n};\mathbf{x}\_{1},\ldots,\mathbf{x}\_{n}) \\ &=\sum\_{\substack{m\_{1},\ldots,n\\m\_{n}=0}}^{+\infty}\frac{(a-c\_{n}+1)\_{\sum\_{i=1}^{n-1}m\_{i}}(b\_{n})\_{m\_{n}}(a)\_{\sum\_{i=1}^{n}m\_{i}}}{(a+b\_{n}-c\_{n}+1)\_{\sum\_{i=1}^{n}m\_{i}}}\frac{\prod\_{i=1}^{n-1}(b\_{i})\_{m\_{i}}}{(c)\_{\sum\_{i=1}^{n-1}m\_{i}}}\prod\_{i=1}^{n}\frac{\mathbf{x}\_{i}^{m\_{i}}}{m\_{i}!}\frac{(1-\mathbf{x}\_{n})^{m\_{n}}}{m\_{n}!},\tag{18}$$

$$=\mathbf{x}\_{n}^{1-\mathbf{c}\_{n}}\sum\_{\substack{m\_{1},\ldots,\\m\_{n}=0}}^{+\infty}\frac{(a-\mathbf{c}\_{n}+1)\_{\sum\_{i=1}^{n}m\_{i}}(b\_{n}-\mathbf{c}\_{n}+1)\_{m\_{n}}(a)\_{\sum\_{i=1}^{n-1}m\_{i}}}{(a+b\_{n}-\mathbf{c}\_{n}+1)\_{\sum\_{i=1}^{n}m\_{i}}}\frac{\prod\_{i=1}^{n-1}(b\_{i})\_{m\_{i}}}{(c)\_{\sum\_{i=1}^{n-1}m\_{i}}}\prod\_{i=1}^{n}\frac{\mathbf{x}\_{i}^{m\_{i}}}{m\_{i}!}\frac{(1-\mathbf{x}\_{n})^{m\_{n}}}{m\_{n}!}.\tag{19}$$

By Horn's rule for the determination of the convergence region (see [37], Section 5.7.2), the multiple power series (18) and (19) are absolutely convergent on region |*xi*| < 1, ∀ *i* ∈ {1, . . . , *<sup>n</sup>* <sup>−</sup> <sup>1</sup>}, <sup>|</sup><sup>1</sup> <sup>−</sup> *xn*<sup>|</sup> <sup>&</sup>lt; 1 in <sup>C</sup>*n*.

Equation (18) can then be deduced from (17) by using the following development where the *<sup>F</sup>*(*p*) *<sup>N</sup>* function can be written as

$$F\_N^{(n)}\left(a;b\_1,\ldots,b\_n;c,c\_n;x\_1,\ldots,x\_n\right) = x\_n^{-a} \sum\_{\substack{m\_1,\ldots,m\_n\\m\_{n-1}=0}}^{+\infty} \frac{(a)\_{\sum\_{i=1}^{n-1}m\_i}(a-c\_n+1)\_{\sum\_{i=1}^{n-1}m\_i}}{(a+b\_n-c\_n+1)\_{\sum\_{i=1}^{n-1}m\_i}} \frac{\prod\_{i=1}^{n-1}(b\_i)\_{m\_i}}{(c)\_{\sum\_{i=1}^{n-1}m\_i}}$$

$$\times \prod\_{i=1}^{n-1} \left(\frac{\chi\_i}{\chi\_n}\right)^{m\_i} \frac{1}{m\_i!} \sum\_{m\_n=0}^{\infty} \frac{(a)\_{m\_n}(a-c\_n+1)\_{m\_n}}{(a+b\_n-c\_n+1)\_{m\_n}} \frac{(1-\chi\_n^{-1})^{m\_n}}{m\_n!} \tag{20}$$

and *α* = *a* + ∑*n*−<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> *mi* is used here to alleviate writing equations. Using the definition of Gauss' hypergeometric series <sup>2</sup>*F*1(.) [34] and the Pfaff transformation [38], we can write

$$\sum\_{m\_n=0}^{\infty} \frac{(a)\_{m\_n}(a - c\_n + 1)\_{m\_n}}{(a + b\_n - c\_n + 1)\_{m\_n}} \frac{(1 - x\_n^{-1})^{m\_n}}{m\_n!} = \,\_2F\_1\left(a, a - c\_n + 1; a + b\_n - c\_n + 1; 1 - x\_n^{-1}\right) \tag{21}$$

$$\mathbf{x} = \mathbf{x}\_n^a \,\_2F\_1\left(\mathbf{a}, b\_n; \mathbf{a} + b\_n - \mathbf{c}\_n + 1; 1 - \mathbf{x}\_n\right) \tag{22}$$

$$\mathbf{x} = \mathbf{x}\_n^\alpha \sum\_{m\_p=0}^\infty \frac{(\boldsymbol{\alpha})\_{m\_n} (\boldsymbol{b}\_n)\_{m\_n}}{(\boldsymbol{a} + \boldsymbol{b}\_n - \boldsymbol{c}\_n + 1)\_{m\_n}} \frac{(1 - \boldsymbol{x}\_n)^{m\_n}}{m\_n!}. \tag{23}$$

By substituting (23) into (20), and using the following two relations:

$$(a)\_{\sum\_{i=1}^{n-1} m\_i} (a)\_{m\_n} = (a)\_{\sum\_{i=1}^n m\_i} \, \, \, \tag{24}$$

$$(a + b\_n - c\_n + 1)\_{\sum\_{i=1}^{n-1} m\_i} (a + b\_n - c\_n + 1)\_{m\_n} = (a + b\_n - c\_n + 1)\_{\sum\_{i=1}^{n} m\_i} \tag{25}$$

we can get (18).

The second transformation is given as follows

$$\,\_2F\_1\left(\mathfrak{a}, \mathfrak{a} - \mathfrak{c}\_n + 1; b\_n - \mathfrak{c}\_n + \mathfrak{a} + 1; 1 - \mathfrak{x}\_n^{-1}\right)$$

$$\mathbf{x} = \mathbf{x}\_n^{\mathbf{a} - \mathbf{c}\_n + 1} \,\_2F\_1\left( b\_n - \mathbf{c}\_n + 1, \mathbf{a} - \mathbf{c}\_n + 1; \mathbf{a} + b\_n - \mathbf{c}\_n + 1; 1 - \mathbf{x}\_n \right) \tag{26}$$

$$\mathbf{x} = \mathbf{x}\_n^{a - c\_n + 1} \sum\_{m\_n = 0}^{\infty} \frac{(a - c\_n + 1)\_{m\_n} (b\_n - c\_n + 1)\_{m\_n}}{(a + b\_n - c\_n + 1)\_{m\_n}} \frac{(1 - \chi\_n)^{m\_n}}{m\_n!}. \tag{27}$$

By substituting (27) into (20), we get (19).

**Lemma 1.** *The multiple power series <sup>F</sup>*(*n*) *<sup>N</sup> is equal to the Lauricella D-hypergeometric function F*(*n*) *<sup>D</sup> (see Appendix A) [39] when a* − *cn* + 1 = *c and is given as follows*

$$F\_N^{(n)}(a; b\_1, \ldots, b\_n; c, c\_n; \mathbf{x}\_1, \ldots, \mathbf{x}\_n) = \sum\_{\substack{m\_1, \ldots, m\_n\\m\_n = 0}}^{+\infty} \frac{(a)\_{\sum\_{i=1}^n m\_i} \prod\_{l=1}^n (b\_l)\_{m\_l}}{(a + b\_n - c\_n + 1)\_{\sum\_{i=1}^n m\_i}} \prod\_{i=1}^{n-1} \frac{\mathbf{x}\_i^{m\_i}}{m\_i!} \frac{(1 - \mathbf{x}\_n)^{m\_n}}{m\_n!} \tag{28}$$
 
$$= F\_D^{(n)}(a, b\_1, \ldots, b\_n; a + b\_n - c\_n + 1; \mathbf{x}\_1, \ldots, \mathbf{x}\_{n-1}, 1 - \mathbf{x}\_n) \tag{29}$$

**Proof.** By using Equation (18) of the multiple power series *<sup>F</sup>*(*n*) *<sup>N</sup>* and after having simplified (*a* − *cn* + 1) ∑*n*−<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> *mi* to the numerator and (*c*) ∑*n*−<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> *mi* to the denominator, we can get the result.

*3.3. Integral Representation for F*(*n*+1) *N*

**Proposition 2.** *The following integral representation is true for Real*{*a*} > 0,*Real*{*a* − *cn*+<sup>1</sup> + 1} > 0, *and Real*{*a* − *cn*+<sup>1</sup> + *bn*+<sup>1</sup> + 1} > 0

$$\frac{\Gamma(a)\Gamma(a-c\_{n+1}+1)}{\Gamma(a-c\_{n+1}+b\_{n+1}+1)}F\_N^{(n+1)}(a;b\_1,\dots,b\_{n+1};c,c\_{n+1};x\_1,\dots,x\_{n+1})$$

$$\begin{split} \Gamma = \int\_0^\infty e^{-r}r^{a-1} \Phi\_2^{(n)}(b\_1,\dots,b\_n;c;rx\_1,\dots,rx\_n) \mathrm{II}(b\_{n+1},c\_{n+1};rx\_{n+1}) \mathrm{d}r \end{split} \tag{30}$$

*where U*(·) *is the confluent hypergeometric function of the second kind (Section 9.21 in [35]) defined for Real*{*b*} > 0*, Real*{*z*} > 0 *by the following integral representation*

$$\mathrm{d}I(b,c;z) = \frac{1}{\Gamma(b)} \int\_0^\infty e^{-zt} t^{b-1} (1+t)^{c-b-1} \mathrm{d}t \tag{31}$$

*and* Φ(*n*) <sup>2</sup> (·) *is defined by Equation (6).*

**Proof.** The multiple power series <sup>Φ</sup>(*n*) <sup>2</sup> and the confluent hypergeometric function *U*(·) are absolutely convergent on [0, +∞]. Using these functions in the above integral and changing the order of integration and summation, which is easily justified by absolute convergence, we get

$$\int\_0^\infty e^{-r} r^{a-1} \Phi\_2^{(n)}(b\_1, \dots, b\_n; c; r\chi\_1, \dots, r\chi\_n) \mathcal{U}(b\_{n+1}, c\_{n+1}; r\chi\_{n+1}) \mathrm{d}r$$

$$= \sum\_{m\_1=0}^\infty \dots \sum\_{m\_n=0}^\infty \frac{(b\_1)\_{m\_1} \dots (b\_n)\_{m\_n}}{(c)\_{\sum\_{i=1}^n m\_i}} \left(\prod\_{i=1}^n \frac{\chi\_i^{m\_i}}{m\_i!} \right) \mathcal{I} \tag{32}$$

where integral **I** is defined as follows

$$\mathbf{I} = \int\_0^\infty e^{-r} r^{a-1 + \sum\_{i=1}^n m\_i} \mathcal{U}(b\_{n+1}, c\_{n+1}; r\mathbf{x}\_{n+1}) \mathrm{d}r. \tag{33}$$

Substituting the integral expression of *U*(·) in the previous equation and replacing *α* = *a* + ∑*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *mi* to alleviate writing equations, we have

$$\mathbf{I} = \frac{1}{\Gamma(b\_{n+1})} \int\_0^\infty \int\_0^\infty \frac{e^{-(1+x\_{n+1}t)r} r^{a-1} t^{b\_{n+1}-1}}{(1+t)^{-(c\_{n+1}-b\_{n+1}-1)}} \mathrm{d}r \mathrm{d}t. \tag{34}$$

Knowing that [35]

$$\int\_0^\infty e^{-(1+\chi\_{n+1}t)r} r^{a-1} \mathrm{d}r = \frac{\Gamma(\alpha)}{(1+\chi\_{n+1}t)^a} \tag{35}$$

and

$$\int\_0^\infty \frac{t^{b\_{n+1}-1}(1+t)^{c\_{n+1}-b\_{n+1}-1}}{(1+x\_{n+1}t)^a} dt = \frac{\Gamma(b\_{n+1})\Gamma(a-c\_{n+1}+1)}{\Gamma(a+b\_{n+1}-c\_{n+1}+1)} \,\_2F\_1(a,b\_{n+1};a+b\_{n+1}-c\_{n+1}+1;1-x\_{n+1}) \tag{36}$$

the new expression of **I** is then given by

$$\mathbf{I} = \frac{\Gamma(a)\Gamma(a - c\_{n+1} + 1)}{\Gamma(a + b\_{n+1} - c\_{n+1} + 1)} \sum\_{m\_{n+1} = 0}^{+\infty} \frac{(a)\_{m\_{n+1}} (b\_{n+1})\_{m\_{n+1}}}{(a + b\_{n+1} - c\_{n+1} + 1)\_{m\_{n+1}}} \frac{(1 - \chi\_{n+1})^{m\_{n+1}}}{m\_{n+1}!} . \tag{37}$$

Using the fact that Γ(*α*) = Γ(*a*)(*a*)∑*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *mi* and (*a*)∑*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *mi* (*α*)*mn*+<sup>1</sup> = (*a*) ∑*n*+<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> *mi* , and developing the same method to Γ(*α* + *bn*+<sup>1</sup> − *cn*+<sup>1</sup> + 1), the final complete expression of the integral is then given by

$$\frac{\Gamma(a)\Gamma(a-c\_{n+1}+1)}{\Gamma(a+b\_{n+1}-c\_{n+1}+1)}\sum\_{m\_1=0}^{\infty}\dots\sum\_{m\_{n+1}=0}^{\infty}\frac{(b\_1)\_{m\_1}\dots\dots(b\_n)\_{m\_n}}{(c)\_{\sum\_{i=1}^n m\_i}}\frac{(a-c\_{n+1}+1)\sum\_{i=1}^n m\_i(b\_{n+1})m\_{n+1}(a)\sum\_{i=1}^{n+1} m\_i}{(a+b\_{n+1}-c\_{n+1}+1)\sum\_{i=1}^{n+1} m\_i}\prod\_{i=1}^n \frac{\Gamma\left(\frac{n}{2}\right)}{m\_i!}$$

$$\times\frac{(1-x\_{n+1})^{m\_{n+1}}}{m\_{n+1}!} = \frac{\Gamma(a)\Gamma(a-c\_{n+1}+1)}{\Gamma(a-c\_{n+1}+b\_{n+1}+1)}F\_N^{(n+1)}(a;b\_1,\dots,b\_{n+1};c,c\_{n+1};x\_1,\dots,x\_{n+1}). \tag{38}$$

#### **4. Expression of** *<sup>E</sup>***X<sup>1</sup>***{***ln[<sup>1</sup> <sup>+</sup> <sup>X</sup>***T***Σ***−***<sup>1</sup> <sup>1</sup> X]***}*

**Proposition 3.** *Let X*<sup>1</sup> *be a random vector that follows a central MCD with pdf given by f <sup>X</sup>*<sup>1</sup> (*x*|**Σ**1, *p*)*. Expectation EX*<sup>1</sup> {ln[<sup>1</sup> <sup>+</sup> *<sup>X</sup>T***Σ**−<sup>1</sup> <sup>1</sup> *X*]} *is given as follows*

$$E\_{\mathbf{X}^1} \left\{ \ln[1 + \mathbf{X}^T \Sigma\_1^{-1} \mathbf{X}] \right\} = \psi \left( \frac{1+p}{2} \right) - \psi \left( \frac{1}{2} \right) \tag{39}$$

*where ψ*(.) *is the digamma function defined as the logarithmic derivative of the Gamma function (Section 8.36 in [35]).*

**Proof.** Expectation *<sup>E</sup>***X**<sup>1</sup> {ln[<sup>1</sup> <sup>+</sup> **<sup>X</sup>***T***Σ**−<sup>1</sup> <sup>1</sup> **X**]} is developed as follows

$$E\_{\mathbf{X}^1} \{ \ln[1 + \mathbf{X}^T \boldsymbol{\Sigma}\_1^{-1} \mathbf{X}] \} = \frac{A}{|\boldsymbol{\Sigma}\_1|^{\frac{1}{2}}} \int\_{\mathbb{R}^p} \frac{\ln[1 + \mathbf{x}^T \boldsymbol{\Sigma}\_1^{-1} \mathbf{x}]}{[1 + \mathbf{x}^T \boldsymbol{\Sigma}\_1^{-1} \mathbf{x}]^{\frac{1+p}{2}}} d\mathbf{x} \tag{40}$$

where *A* = Γ( <sup>1</sup>+*<sup>p</sup>* <sup>2</sup> )*π*<sup>−</sup> <sup>1</sup>+*<sup>p</sup>* <sup>2</sup> . Utilizing the following property log(*x*)*f*(*x*)d*x* = *<sup>∂</sup> ∂a x<sup>a</sup> f*(*x*) d*x* + + *<sup>a</sup>*=0, as a consequence the expectation is given as follows

$$E\_{\mathbf{X}^1} \{ \ln[1 + \mathbf{X}^T \Sigma\_1^{-1} \mathbf{X}] \} = \frac{A}{|\Sigma\_1|^{\frac{1}{2}}} \frac{\partial}{\partial a} \int\_{\mathbb{R}^p} [1 + \mathbf{x}^T \Sigma\_1^{-1} \mathbf{x}]^{a - \frac{1+p}{2}} \mathbf{dx} \bigg|\_{a=0} \tag{41}$$

Consider the transformation *y* = **Σ**−1/2 <sup>1</sup> *x* where *y* = [*y*1, *y*2, ... , *yp*] *<sup>T</sup>*. The Jacobian determinant is given by d*y* = |**Σ**1| <sup>−</sup>1/2d*x* (Theorem 1.12 in [40]). The new expression of the expectation is given by

$$E\_{\mathbf{X}^1} \{ \ln[1 + \mathbf{X}^T \Sigma\_1^{-1} \mathbf{X}] \} = A \frac{\partial}{\partial a} \int\_{\mathbb{R}^p} [1 + \mathbf{y}^T \mathbf{y}]^{a - \frac{1+p}{2}} \mathbf{d} \mathbf{y} \Big|\_{a=0}. \tag{42}$$

Let *u* = *yTy* be a transformation where the Jacobian determinant is given by (Lemma 13.3.1 in [41])

$$\mathrm{d}y = \frac{\pi^{\frac{p}{2}}}{\Gamma(\frac{p}{2})} u^{\frac{p}{2}-1} \mathrm{d}u.\tag{43}$$

The new expectation is as follows

$$E\_{\mathbf{X}^1}\{\ln[1+\mathbf{X}^T\Sigma\_1^{-1}\mathbf{X}]\} = \frac{\Gamma(\frac{1+p}{2})}{\pi^{1/2}\Gamma(\frac{p}{2})}\frac{\partial}{\partial a}\int\_0^{+\infty}u^{\frac{p}{2}-1}(1+u)^{a-\frac{1+p}{2}}\mathrm{d}u\Big|\_{a=0}\tag{44}$$

Using the definition of beta function, we can write that

$$\int\_0^{+\infty} u^{\frac{p}{2}-1} (1+u)^{a-\frac{1+p}{2}} \mathrm{d}u = \frac{\Gamma(\frac{p}{2})\Gamma(\frac{1}{2}-a)}{\Gamma(\frac{1+p}{2}-a)}.\tag{45}$$

The derivative of the last integral w.r.t *a* is given by

$$\frac{\partial}{\partial a} \int\_0^{+\infty} u^{\frac{p}{2}-1} (1+u)^{a-\frac{1+p}{2}} \mathrm{d}u \Big|\_{a=0} = \frac{\Gamma(\frac{p}{2})\Gamma(\frac{1}{2})}{\Gamma(\frac{1+p}{2})} \left[ \psi(\frac{1+p}{2}) - \psi(\frac{1}{2}) \right] \tag{46}$$

Finally, the expression of *<sup>E</sup>***X**<sup>1</sup> {ln[<sup>1</sup> <sup>+</sup> **<sup>X</sup>***T***Σ**−<sup>1</sup> <sup>1</sup> **X**]} is given by

$$E\_{\mathbf{X}^1} \{ \ln[1 + \mathbf{X}^T \Sigma\_1^{-1} \mathbf{X}] \} = \psi \left( \frac{1+p}{2} \right) - \psi \left( \frac{1}{2} \right). \tag{47}$$

#### **5. Expression of** *<sup>E</sup>***X<sup>1</sup>***{***ln[<sup>1</sup> <sup>+</sup> <sup>X</sup>***T***Σ***−***<sup>1</sup> <sup>2</sup> X]***}*

**Proposition 4.** *Let X*<sup>1</sup> *and X*<sup>2</sup> *be two random vectors that follow central MCDs with pdfs given, respectively, by f <sup>X</sup>*<sup>1</sup> (*x*|**Σ**1, *p*) *and f <sup>X</sup>*<sup>2</sup> (*x*|**Σ**2, *<sup>p</sup>*)*. Expectation <sup>E</sup>X*<sup>1</sup> {ln[<sup>1</sup> <sup>+</sup> *<sup>X</sup>T***Σ**−<sup>1</sup> <sup>2</sup> *X*]} *is given as follows*

$$\begin{split} &E\_{\mathbf{X}^{l}}\{\ln[1+\mathbf{X}^{T}\boldsymbol{\Sigma}\_{2}^{-1}\mathbf{X}]\} = \boldsymbol{\Psi}\left(\frac{1+p}{2}\right) - \boldsymbol{\Psi}\left(\frac{1}{2}\right) + \ln\lambda\_{p} \\ &- \frac{\partial}{\partial a}\left\{F\_{D}^{(p)}\left(a,\underbrace{\frac{1}{2},\frac{1}{2},\dots,\frac{1}{2},a+\frac{1}{2};a+\frac{1+p}{2};1-\frac{\lambda\_{1}}{\lambda\_{p}},\dots,1-\frac{\lambda\_{p-1}}{\lambda\_{p}},1-\frac{1}{\lambda\_{p}}\right)\right\}\Bigg|\_{a=0}. \end{split} \tag{48}$$

*where λ*1*,. . . , λ<sup>p</sup> are the eigenvalues of the real matrix* **Σ**1**Σ**−<sup>1</sup> <sup>2</sup> *, and <sup>F</sup>*(*p*) *<sup>D</sup>* (.) *represents the Lauricella D-hypergeometric function defined for p variables.*

**Proof.** To prove Proposition 4, different steps are necessary. They are described in the following:

#### *5.1. First Step: Eigenvalue Expression*

Expectation *<sup>E</sup>***X**<sup>1</sup> {ln[<sup>1</sup> <sup>+</sup> **<sup>X</sup>***T***Σ**−<sup>1</sup> <sup>2</sup> **X**]} is computed as follows

$$E\_{\mathbf{X}^1} \{ \ln[1 + \mathbf{X}^T \Sigma\_2^{-1} \mathbf{X}] \} = \frac{A}{|\Sigma\_1|^{\frac{1}{2}}} \int\_{\mathbb{R}^{\mathcal{V}}} \frac{\ln[1 + \mathbf{x}^T \Sigma\_2^{-1} \mathbf{x}]}{[1 + \mathbf{x}^T \Sigma\_1^{-1} \mathbf{x}]^{\frac{1+\mathcal{V}}{2}}} d\mathbf{x} \tag{49}$$

where *A* = Γ( <sup>1</sup>+*<sup>p</sup>* <sup>2</sup> )*π*<sup>−</sup> <sup>1</sup>+*<sup>p</sup>* <sup>2</sup> . Consider transformation *y* = **Σ**−1/2 <sup>1</sup> *x* where *y* = [*y*1, *y*2, ... , *yp*] *T*. The Jacobian determinant is given by d*y* = |**Σ**1| <sup>−</sup>1/2d*x* (Theorem 1.12 in [40]) and matrix **Σ** = **Σ** 1 2 <sup>1</sup> **<sup>Σ</sup>**−<sup>1</sup> <sup>2</sup> **Σ** 1 2 <sup>1</sup> is a real symmetric matrix since **Σ**<sup>1</sup> and **Σ**<sup>2</sup> are real symmetric matrixes. Then, the expectation is evaluated as follows

$$E\_{\mathbf{X}^1} \{ \ln[1 + \mathbf{X}^T \Sigma\_2^{-1} \mathbf{X}] \} = A \int\_{\mathbb{R}^p} \frac{\ln[1 + \mathbf{y}^T \Sigma \mathbf{y}]}{[1 + \mathbf{y}^T \mathbf{y}]^{\frac{1+p}{2}}} \mathrm{d}\mathbf{y}.\tag{50}$$

Matrix **Σ** can be diagonalized by an orthogonal matrix **P** with **P**−<sup>1</sup> = **P***<sup>T</sup>* and **Σ** = **PDP**−<sup>1</sup> where **D** is a diagonal matrix composed of the eigenvalues of **Σ**. Considering that *yT***Σ***y* = tr(**Σ***yyT*) = tr(**PDP***TyyT*) = tr(**DP***TyyT***P**), the expectation can be written as

$$E\_{\mathbf{X}^1} \{ \ln[1 + \mathbf{X}^T \Sigma\_2^{-1} \mathbf{X}] \} = A \int\_{\mathbb{R}^r} \frac{\ln[1 + \text{tr}(\mathbf{D} \mathbf{P}^T \mathbf{y} y^T \mathbf{P})]}{[1 + \mathbf{y}^T \mathbf{y}]^{\frac{1+p}{2}}} \text{d}y. \tag{51}$$

Let *z* = **P***Ty* with *z* = [*z*1, *z*2, ... , *zp*] *<sup>T</sup>* be a transformation where the Jacobian determinant is given by d*<sup>z</sup>* <sup>=</sup> <sup>|</sup>**P***T*|d*<sup>y</sup>* <sup>=</sup> <sup>d</sup>*y*. Using the fact that tr(**DP***TyyT***P**) = tr(**D***zzT*) = *<sup>z</sup>T***D***<sup>z</sup>* and *yTy* = *zT***P***T***P***z* = *zTz*, then the previous expectation (51) is given as follows

$$E\_{\mathbf{X}^1} \{ \ln[1 + \mathbf{X}^T \Sigma\_2^{-1} \mathbf{X}] \} = A \int\_{\mathbb{R}^r} \frac{\ln[1 + \mathbf{z}^T \mathbf{D} \mathbf{z}]}{[1 + \mathbf{z}^T \mathbf{z}]^{\frac{1+p}{2}}} \mathbf{d} \mathbf{z} \tag{52}$$

$$=A\int\_{\mathbb{R}}\dots\int\_{\mathbb{R}}\frac{\ln\left[1+\sum\_{i=1}^{p}\lambda\_{i}z\_{i}^{2}\right]}{\left[1+\sum\_{i=1}^{p}z\_{i}^{2}\right]^{\frac{1+p}{2}}}\text{d}z\_{1}\dots\text{d}z\_{p}\tag{53}$$

where *λ*1,. . . , *λ<sup>p</sup>* are the eigenvalues of **Σ**1**Σ**−<sup>1</sup> <sup>2</sup> .
