*5.4. Final Step*

The last integral is related to the confluent hypergeometric function of the second kind *U*(.) as follows

$$\int\_0^{+\infty} u^{a-\frac{1}{2}} (1+u)^{-\frac{1+p}{2}} e^{-ru} \mathrm{d}u = \Gamma(a+\frac{1}{2}) \mathcal{U}(a+\frac{1}{2}, a+1-\frac{p}{2}, r). \tag{85}$$

As a consequence, the new expression is

$$\begin{split} &E\_{\mathbf{X}^{1}}\{\ln\left[1+\mathbf{X}^{T}\Sigma\_{2}^{-1}\mathbf{X}\right]\}=-\frac{\partial}{\partial a}\left\{A\frac{\Gamma(a+\frac{1}{2})}{\Gamma(a)}B\left(\frac{1}{2},\dots,\frac{1}{2}\right) \\ &\times\int\_{0}^{+\infty}r^{a-1}e^{-\lambda\_{F}r}\Phi\_{2}^{(p-1)}\left(\frac{1}{2},\dots,\frac{1}{2};\frac{p}{2};1;(\lambda\_{p}-\lambda\_{1})r,\dots,(\lambda\_{p}-\lambda\_{p-1})r\right)\mathcal{U}(a+\frac{1}{2},a+1-\frac{p}{2},r)\mathrm{d}r\right\}\Big|\_{a=0}.\end{split} \tag{86}$$

Using the transformation *r* = *λpr* and the Proposition 2, and taking into account the expression of *A*, the new expression becomes

$$\begin{split} &E\_{\mathbf{X}^{1}}\{\ln[1+\mathbf{X}^{T}\Sigma\_{2}^{-1}\mathbf{X}]\} = -\frac{\partial}{\partial a}\left\{\frac{B(a+\frac{1}{2},\frac{p}{2})}{B\left(\frac{1}{2},\frac{p}{2}\right)}\lambda\_{p}^{-a} \\ &\times F\_{N}^{(p)}\left(a;\underbrace{\frac{1}{2},\dots,\frac{1}{2},a+\frac{1}{2}}\_{p};\frac{p}{2},a-\frac{p}{2}+1;1-\frac{\lambda\_{1}}{\lambda\_{p}},\dots,1-\frac{\lambda\_{p-1}}{\lambda\_{p}},\lambda\_{p}^{-1}\right)\right\}\bigg|\_{a=0} \end{split} \tag{87}$$

Knowing that

$$\left. \frac{\partial}{\partial a} \left\{ \frac{B\left(\frac{p}{2}, a + \frac{1}{2}\right)}{B\left(\frac{p}{2}, \frac{1}{2}\right)} \right\} \right|\_{a=0} = \psi\left(\frac{1}{2}\right) - \psi\left(\frac{1+p}{2}\right), \text{ and} \tag{88}$$

$$\left.F\_N^{(p)}\left(a; \frac{1}{2}, \dots, \frac{1}{2}, a + \frac{1}{2}; \frac{p}{2}, a - \frac{p}{2} + 1; 1 - \frac{\lambda\_1}{\lambda\_p}, \dots, 1 - \frac{\lambda\_{p-1}}{\lambda\_p}, \lambda\_p^{-1}\right)\right|\_{a=0} = 1,\tag{89}$$

the new 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**]} becomes

$$\begin{split} E\_{\mathbf{X}^{l}}\{\ln[1+\mathbf{X}^{T}\Sigma\_{2}^{-1}\mathbf{X}]\}&=\psi\left(\frac{1+p}{2}\right)-\psi\left(\frac{1}{2}\right) \\ &-\frac{\partial}{\partial a}\left\{\lambda\_{p}^{-a}F\_{N}^{(p)}\left(a;\underbrace{\frac{1}{2},\dots,\frac{1}{2},a+\frac{1}{2}}\_{p};\frac{p}{2},a-\frac{p}{2}+1;1-\frac{\lambda\_{1}}{\lambda\_{p}},\dots,1-\frac{\lambda\_{p-1}}{\lambda\_{p}},\lambda\_{p}^{-1}\right)\right\}\Bigg|\_{a=0}.\end{split}\tag{90}$$

Applying the expression given by (18) of Definition 2 and relying on Lemma 1, the final result corresponds to the D-hypergeometric function of Lauricella *<sup>F</sup>*(*p*) *<sup>D</sup>* (.) given by

$$\begin{split} &E\_{\mathbf{X}^{l}}\{\ln[1+\mathbf{X}^{T}\Sigma\_{2}^{-1}\mathbf{X}]\} = \psi\left(\frac{1+p}{2}\right) - \psi\left(\frac{1}{2}\right) \\ & - \frac{\partial}{\partial a}\Big\{\lambda\_{p}^{-s} \sum\_{m\_{1},\cdots,p}^{+\infty} \frac{(a)\_{\sum\_{i=1}^{p}m\_{i}}^{\operatorname{(}}(a+\frac{1}{2})\_{m\_{i}}\prod\_{i=1}^{p-1}(\frac{1}{2})\_{m\_{i}}}{(a+\frac{1+p}{2})\_{\sum\_{i=1}^{p}m\_{i}}} \prod\_{i=1}^{p-1} \left(1-\frac{\lambda\_{i}}{\lambda\_{p}}\right)^{m\_{i}} \frac{1}{m\_{i}!} \frac{(1-\lambda\_{p}^{-1})^{m\_{p}}}{m\_{p}!} \right\}\Big|\_{a=0} \\ & \qquad \qquad (1+n) \end{split} \tag{91}$$

$$\Psi = \Psi\left(\frac{1+p}{2}\right) - \Psi\left(\frac{1}{2}\right) - \frac{\partial}{\partial a}\left\{\lambda\_p^{-a} F\_D^{(p)}\left(a, \underbrace{1, \ldots, \frac{1}{2}}\_{p}, a + \frac{1}{2}; a + \frac{1+p}{2}; 1 - \frac{\lambda\_1}{\lambda\_p}, \ldots, 1 - \frac{\lambda\_{p-1}}{\lambda\_p}, 1 - \frac{1}{\lambda\_p}\right)\right\}\bigg|\_{a=0}.\tag{92}$$

The final development of the previous expression is as follows

$$\begin{split} E\_{\mathbf{X}^{1}}\{\ln[1+\mathbf{X}^{T}\boldsymbol{\Sigma}\_{2}^{-1}\mathbf{X}]\}&=\psi\left(\frac{1+p}{2}\right)-\psi\left(\frac{1}{2}\right)+\ln\lambda\_{p}\\ &-\frac{\partial}{\partial a}\left\{F\_{D}^{(p)}\left(a,\underbrace{\frac{1}{2},\frac{1}{2},\dots,\frac{1}{2},a+\frac{1}{2}}\_{p};a+\frac{1+p}{2};1-\frac{\lambda\_{1}}{\lambda\_{p}},\dots,1-\frac{\lambda\_{p-1}}{\lambda\_{p}},1-\frac{1}{\lambda\_{p}}\right)\right\}\Bigg|\_{a=0}.\end{split}\tag{93}$$

In this section, we presented the exact 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**]}. In addition, the multiple power series *<sup>F</sup>*(*p*) *<sup>D</sup>* which appears to be a special case of *<sup>F</sup>*(*p*) *<sup>N</sup>* provides many properties and numerous transformations (see Appendix A) that make easier the convergence of the multiple power series. In the next section, we establish the KLD closed-form expression based on the expression of the latter expectation.
