Appendix A: Proofs
1.1 A.1. Proof of (5.1)–(5.4)
Proof
When the dimension n > 1, Ωn(∥t∥2) defined as
$$ {\Omega}_{n}(\|t\|^{2})=\frac{\Gamma(\frac{n}{2})}{\sqrt{\pi}}\sum\limits_{k=0}^{\infty}\frac{(-1)^{k}\|t\|^{2k}}{(2k)!} \frac{\Gamma(\frac{2k+1}{2})}{\Gamma(\frac{n+2k}{2})}, $$
we have the characteristic function as follows.
$$ \begin{array}{@{}rcl@{}} \psi_{\textbf{X}}(\textbf{t})&=&E(e^{i\textbf{t}^{\textit{T}}\textbf{X}}) =e^{i\textbf{t}^{\textit{T}}\boldsymbol{\mu}}E(e^{i\textbf{t}^{\textit{T}}R\boldsymbol{\Sigma}^{\frac{1}{2}}\textbf{U}^{(n)}}) =e^{i\textbf{t}^{\textit{T}}\boldsymbol{\mu}}E[{\Omega}_{n}(R^{2}\textbf{t}^{\textit{T}}\boldsymbol{\Sigma}\textbf{t})]\\ &=&e^{i\textbf{t}^{\textit{T}}\boldsymbol{\mu}}{\int}_{0}^{\infty}{\Omega}_{n}(v\textbf{t}^{\textit{T}}\boldsymbol{\Sigma}\textbf{t}) \frac{1}{{\int}_{0}^{\infty}t^{\frac{n}{2}-1}g_{n}(t)dt}v^{\frac{n}{2}-1}\frac{v^{N-1}\exp(-av^{s})}{(1+\exp(-bv^{s}))^{2r}}dv\\ &=&e^{i\textbf{t}^{\textit{T}}\boldsymbol{\mu}}\left[{\int}_{0}^{\infty}\frac{t^{\frac{n}{2}+N-2}\exp(-at^{s})}{(1+ \exp(-bt^{s}))^{2r}} \right]^{-1} {\int}_{0}^{\infty}{\Omega}_{n}(v\textbf{t}^{\textit{T}}\boldsymbol{\Sigma}\textbf{t}) \frac{v^{\frac{n}{2}+N-2}\exp(-av^{s})}{(1+\exp(-bv^{s}))^{2r}}dv\\ &=&e^{i\textbf{t}^{\textit{T}}\boldsymbol{\mu}}\left[\frac{\Gamma(\frac{1}{s}(\frac{n}{2}+N-1))}{b^{\frac{1}{s} (\frac{n}{2}+N-1)}s} {\Phi}_{2r}^{*}(-1,\frac{1}{s}(\frac{n}{2}+N-1),\frac{a}{b})\right]^{-1}\\ &&\times{\int}_{0}^{\infty}\frac{\Gamma(\frac{n}{2})}{\pi^{\frac{1}{2}}}\sum\limits_{k=0}^{\infty} \frac{(-1)^{k}(\textbf{t}^{\textit{T}}\boldsymbol{\Sigma}\textbf{t})^{k}v^{k}{\Gamma}(\frac{2k+1}{2})}{(2k)!{\Gamma}(\frac{n+2k}{2})} \frac{v^{\frac{n}{2}+N-2}\exp(-av^{s})}{(1+\exp(-bv^{s}))^{2r}}dv\\ &=&e^{i\textbf{t}^{\textit{T}}\boldsymbol{\mu}}\left[\frac{\Gamma(\frac{1}{s}(\frac{n}{2}+N-1))}{b^{\frac{1}{s} (\frac{n}{2}+N-1)}s} {\Phi}_{2r}^{*}(-1,\frac{1}{s}(\frac{n}{2}+N-1),\frac{a}{b})\right]^{-1}\\ &&\times\sum\limits_{k=0}^{\infty}\frac{\Gamma(\frac{n}{2})}{\pi^{\frac{1}{2}}} \frac{(-1)^{k}(\textbf{t}^{\textit{T}}\boldsymbol{\Sigma}\textbf{t})^{k}{\Gamma}(\frac{2k+1}{2})}{(2k)!{\Gamma}(\frac{n+2k}{2})} {\int}_{0}^{\infty}\frac{v^{\frac{n}{2}+N+k-2}\exp(-av^{s})}{(1+\exp(-bv^{s}))^{2r}}dv\\ &=&e^{it^{\textit{T}}\boldsymbol{\mu}}\left[\frac{\Gamma(\frac{1}{s}(\frac{n}{2}+N-1))}{b^{\frac{1}{s}(\frac{n}{2}+N-1)}s} {\Phi}_{2r}^{*}(-1,\frac{1}{s}(\frac{n}{2}+N-1),\frac{a}{b})\right]^{-1}\\ &&\times\sum\limits_{k=0}^{\infty}\frac{\Gamma(\frac{n}{2})}{\pi^{\frac{1}{2}}} \frac{(-1)^{k}(\textbf{t}^{\textit{T}}\boldsymbol{\Sigma}\textbf{t})^{k}{\Gamma}(\frac{2k+1}{2})}{(2k)!{\Gamma}(\frac{n+2k}{2})} \frac{\Gamma(\frac{1}{s}(\frac{n}{2}+N - 1))}{b^{\frac{1}{s}(\frac{n}{2}+N-1)}s} {\Phi}_{2r}^{*}(-1,\frac{1}{s}(\frac{n}{2}+N - 1),\frac{a}{b})\\ &=&e^{i\textbf{t}^{\textit{T}}\boldsymbol{\mu}}\frac{\Gamma(\frac{n}{2})}{\pi^{\frac{1}{2}}}\sum\limits_{k=0}^{\infty} \frac{(-1)^{k}}{(2k)!}\frac{\Gamma(k+\frac{1}{2}){\Gamma}(\frac{1}{s}(\frac{n}{2}+N+k-1))}{\Gamma(k+\frac{n}{2}) {\Gamma}(\frac{1}{s}(\frac{n}{2}+N-1))b^{\frac{1}{s}(\frac{n}{2}+N-1)}}\\ &&\times\frac{{\Phi}_{2r}^{*}(-1,\frac{1}{s}(\frac{n}{2}+N+k-1),\frac{a}{b})}{{\Phi}_{2r}^{*}(-1,\frac{1}{s}(\frac{n}{2}+N-1), \frac{a}{b})} (\textbf{t}^{\textit{T}}\boldsymbol{\Sigma}\textbf{t})^{k}\\ &=&e^{i\textbf{t}^{\textit{T}}\boldsymbol{\mu}}\sum\limits_{k=0}^{\infty}\frac{(-1)^{k}}{(2k)!}\gamma_{k} \frac{{\Phi}_{2r}^{*}(-1,\frac{1}{s}(\frac{n}{2}+N+k-1),\frac{a}{b})}{{\Phi}_{2r}^{*}(-1,\frac{1}{s}(\frac{n}{2}+N-1),\frac{a}{b})} (\textbf{t}^{\textit{T}}\boldsymbol{\Sigma}\textbf{t})^{k}, \end{array} $$
where
$$ \gamma_{k}(N,n,b,s)=\frac{\Gamma(\frac{n}{2})}{\pi^{\frac{1}{2}}b^{\frac{1}{s}(\frac{n}{2}+N-1)}} \frac{\Gamma(k+\frac{1}{2}){\Gamma}(\frac{1}{s}(\frac{n}{2}+N+k-1))}{\Gamma(k+\frac{n}{2}){\Gamma}(\frac{1}{s}(\frac{n}{2}+N-1))}. $$
Here \({\Phi }_{2r}^{*}\) is the generalized Hurwitz-Lerch zeta function.
When the dimension n > 1, Ωn(∥t∥2) defined as
$$ {\Omega}_{n}(\|t\|^{2})=_{0}F_{1}(\frac{n}{2};-\frac{1}{4}\|t\|^{2}), $$
we have the characteristic function as follows.
$$ \begin{array}{@{}rcl@{}} \psi_{\textbf{X}}(\textbf{t})&=&e^{i\textbf{t}^{\textit{T}}\boldsymbol{\mu}}E[{\Omega}_{n}(R^{2}\textbf{t}^{\prime} \boldsymbol{\Sigma}\textbf{t})]\\ &=&e^{i\textbf{t}^{\textit{T}}\boldsymbol{\mu}}\left[{\int}_{0}^{\infty}\frac{t^{\frac{n}{2}+N-2}\exp(-at^{s})}{(1+ \exp(-bt^{s}))^{2r}}\right]^{-1} {\int}_{0}^{\infty}{\Omega}_{n}(v\textbf{t}^{\textit{T}}\boldsymbol{\Sigma}\textbf{t})\frac{v^{\frac{n}{2}+N-2}\exp(-av^{s})} {(1+\exp(-bv^{s}))^{2r}}dv\\ &=&e^{i\textbf{t}^{\textit{T}}\boldsymbol{\mu}}\left[\frac{\Gamma(\frac{1}{s}(\frac{n}{2}+N-1))}{b^{\frac{1}{s} (\frac{n}{2}+N-1)}s} {\Phi}_{2r}^{*}(-1,\frac{1}{s}(\frac{n}{2}+N-1),\frac{a}{b})\right]^{-1}\\ &&\times{\int}_{0}^{\infty} {~}_{0}F_{1}(\frac{n}{2};\frac{1}{4}v\textbf{t}^{\textit{T}}{\Sigma}\textbf{t})\frac{v^{\frac{n}{2}+N-2}\exp(-av^{s})} {(1+\exp(-bv^{s}))^{2r}}dv\\ &=&e^{i\textbf{t}^{\textit{T}}\boldsymbol{\mu}}\left[\frac{\Gamma(\frac{1}{s}(\frac{n}{2}+N-1))}{b^{\frac{1}{s} (\frac{n}{2}+N-1)}s} {\Phi}_{2r}^{*}(-1,\frac{1}{s}(\frac{n}{2}+N-1),\frac{a}{b})\right]^{-1}\\ &&\times\sum\limits_{k=0}^{\infty}\frac{\Gamma(\frac{1}{s}(\frac{n}{2}+N-1))}{b^{\frac{1}{s}(\frac{n}{2}+N-1)}s} \frac{(\textbf{t}^{\textit{T}}{\Sigma} \textbf{t})^{k}}{4^{k}(\frac{n}{2})^{[k]}k!}{\Phi}_{2r}^{*}(-1,\frac{1}{s}(\frac{n}{2}+N+k-1),\frac{a}{b})\\ &=&e^{i\textbf{t}^{\textit{T}}\boldsymbol{\mu}}\sum\limits_{k=0}^{\infty} \frac{\Gamma(\frac{1}{s}(\frac{n}{2} + N+k-1))}{\Gamma(\frac{1}{s}(\frac{n}{2} + N - 1))b^{\frac{k}{s}}4^{k}(\frac{n}{2})^{[k]}k!} \frac{{\Phi}_{2r}^{*}(-1,\frac{1}{s}(\frac{n}{2}+N+k-1),\frac{a}{b})}{{\Phi}_{2r}^{*}(-1,\frac{1}{s}(\frac{n}{2}+N-1),\frac{a}{b})} (\textbf{t}^{\textit{T}}\boldsymbol{\Sigma} \textbf{t})^{k}. \end{array} $$
Thus, we have characteristic generators as follows:
$$ \begin{array}{@{}rcl@{}} \phi_{\textbf{X}}(\|\textbf{u}_{n}\|^{2}) &=&\sum\limits_{k=0}^{\infty} \frac{(-1)^{k}}{(2k)!}\gamma_{k}(N,n,b,s) \frac{{\Phi}_{2r}^{*}(-1,\frac{1}{s}(\frac{n}{2}+N+k-1),\frac{a}{b})}{{\Phi}_{2r}^{*}(-1,\frac{1}{s}(\frac{n}{2}+N-1),\frac{a}{b})} (\|\textbf{u}_{n}\|)^{2k},\\ \phi_{\textbf{X}}(\|\textbf{u}_{n}\|^{2}) &=&\sum\limits_{k=0}^{\infty} \frac{\Gamma(\frac{1}{s}(\frac{n}{2}+N+k-1))}{\Gamma(\frac{1}{s}(\frac{n}{2}+N-1))b^{\frac{k}{s}}4^{k}(\frac{n}{2})^{[k]}k!} \frac{{\Phi}_{2r}^{*}(-1,\frac{1}{s}(\frac{n}{2}+N+k-1),\frac{a}{b})}{{\Phi}_{2r}^{*}(-1,\frac{1}{s}(\frac{n}{2}+N-1),\frac{a}{b})} (\|\textbf{u}_{n}\|)^{2k}, \end{array} $$
wher
$$\gamma_{k}(N,n,b,s)=\frac{\Gamma(\frac{n}{2})}{\pi^{\frac{1}{2}}b^{\frac{1}{s}(\frac{n}{2}+N-1)}} \frac{\Gamma(k+\frac{1}{2}){\Gamma}(\frac{1}{s}(\frac{n}{2}+N+k-1))}{\Gamma(k+\frac{n}{2}){\Gamma}(\frac{1}{s}(\frac{n}{2}+N-1))}.$$
□
1.2 A.2. Proof of (7.1)–(7.4)
Proof
When the dimension n > 1, Ωn(∥t∥2) defined as
$$ {\Omega}_{n}(\|t\|^{2})=\frac{\Gamma(\frac{n}{2})}{\sqrt{\pi}}\sum\limits_{k=0}^{\infty}\frac{(-1)^{k}\|t\|^{2k}}{(2k)!} \frac{\Gamma(\frac{2k+1}{2})}{\Gamma(\frac{n+2k}{2})}, $$
we have the characteristic function as follows.
$$ \begin{array}{@{}rcl@{}} \psi_{\textbf{Y}}(\textbf{t})&=&E(e^{i\textbf{t}^{\textit{T}}\textbf{Y}}) =e^{i\textbf{t}^{\textit{T}}\boldsymbol{\mu}_{\textbf{Y}}}\frac{1}{{\int}_{0}^{\infty}t^{\frac{m}{2}-1}g_{m}(t)dt}\\ &&\times{\int}_{0}^{\infty}{\Omega}_{n}(v\textbf{t}^{\textit{T}}\boldsymbol{{\Sigma}_{\textbf{Y}}}\textbf{t}) {\sum}_{j=0}^{N-1}\frac{(N-1)_{j}}{j!}\frac{\Gamma(\frac{n-m}{2}+j)}{b^{\frac{n-m}{2}+j}}v^{N+\frac{m}{2}-j-2}e^{-av} {\Phi}_{2r}^{*}(-e^{-bv},\frac{n-m}{2}+j,\frac{a}{b})dv\\ &=&e^{it^{\textit{T}}\boldsymbol{\mu}_{\textbf{Y}}}\left[\sum\limits_{k=0}^{\infty}{\sum}_{j=0}^{N-1}\alpha_{k}^{*}{\upbeta}_{j} {\Gamma}(\frac{m}{2}+N-j-1)\right]^{-1}I_{3}, \end{array} $$
where
$$ {\upbeta}_{j}=\frac{(N-1)_{j}}{j!}\frac{\Gamma(\frac{n-m}{2}+j)}{b^{\frac{n-m}{2}+j}{\Gamma}(2r)},~ \alpha_{k}=\frac{\Gamma(2r+k)(-1)^{k}}{k!},~ \alpha_{k}^{*}=\frac{\alpha_{k}}{(k+\frac{a}{b})^{\frac{m}{2}+N-1}}. $$
$$ \begin{array}{@{}rcl@{}} I_{3}&=&{\int}_{0}^{\infty}\frac{\Gamma(\frac{m}{2})}{\pi^{\frac{1}{2}}}{\sum}_{l=0}^{\infty} \frac{(-1)^{l}(\textbf{t}^{\textit{T}} \boldsymbol{{\Sigma}_{\textbf{Y}}}\textbf{t})^{l}v^{l}{\Gamma}(\frac{2l+1}{2})}{(2l)!{\Gamma}(\frac{m+2l}{2})}\\ &&\times\sum\limits_{k=0}^{N-1}\frac{(N-1)_{k}}{k!}\frac{\Gamma(\frac{n-m}{2}+k)}{b^{\frac{n-m}{2}+k}}v^{N+\frac{m}{2}-k-2}e^{-av} {\Phi}_{2r}^{*}(-e^{-bv},\frac{n-m}{2}+k,\frac{a}{b})dv\\ &=&\frac{\Gamma(\frac{m}{2})}{\pi^{\frac{1}{2}}}{\sum}_{l=0}^{\infty}\frac{(-1)^{l}{\Gamma}(\frac{2l+1}{2})}{(2l)!{\Gamma}(\frac{m+2l}{2})} (\textbf{t}^{\textit{T}}\boldsymbol{{\Sigma}_{\textbf{Y}}}\textbf{t})^{l}\sum\limits_{k=0}^{N-1}{\upbeta}_{k}{\int}_{0}^{\infty}v^{N+\frac{m}{2}-k-2+l}e^{-av} {\sum}_{p=0}^{\infty}\frac{\Gamma(2r + p)}{p!}\frac{(-1)^{p}e^{-bpv}}{(p+\frac{a}{b})^{\frac{n-m}{2}+k}}dv\\ &=&{\sum}_{l=0}^{\infty}\sum\limits_{k=0}^{N-1} {\sum}_{p=0}^{\infty} \frac{(-1)^{l}}{(2l)!}\frac{\Gamma(\frac{m}{2}){\Gamma}(\frac{2l+1}{2})}{\Gamma(\frac{1}{2}){\Gamma}(\frac{m+2l}{2})}{\upbeta}_{k}\alpha_{p} \frac{\Gamma(\frac{m}{2}+N-k+l-1)}{b^{N-k+l}(p+\frac{a}{b})^{\frac{n-m}{2}+N+l}}(\textbf{t}^{\textit{T}}\boldsymbol{{\Sigma}_{\textbf{Y}}}\textbf{t})^{l}\\ &=&{\sum}_{l=0}^{\infty}\sum\limits_{k=0}^{N-1} {\sum}_{p=0}^{\infty} \frac{(-1)^{l}}{(2l)!}\alpha_{p}^{*}{\upbeta}_{k}A_{l}{\Gamma}(\frac{m}{2}+N-k+l-1)(\textbf{t}^{\textit{T}}\boldsymbol{{\Sigma}_{\textbf{Y}}}\textbf{t})^{l}, \end{array} $$
$$ \begin{array}{@{}rcl@{}} \psi_{\textbf{Y}}(\textbf{t})\!&=&\!e^{i\textbf{t}^{\textit{T}}\boldsymbol{\mu}_{\textbf{Y}}} \frac{{\sum}_{l=0}^{\infty} \sum\limits_{k=0}^{N-1} {\sum}_{p=0}^{\infty} \frac{(-1)^{l}}{(2l)!}\alpha_{p}^{*}{\upbeta}_{k}A_{l}{\Gamma}(\frac{m}{2}+N - k+l - 1)} {\sum\limits_{k=0}^{\infty}{\sum}_{j=0}^{N-1}\alpha_{k}^{*}{\upbeta}_{j}{\Gamma}(\frac{m}{2}+N-j-1)}(\textbf{t}^{\textit{T}} \boldsymbol{{\Sigma}_{\textbf{Y}}}\textbf{t})^{l},\\ \phi_{(m),y}(\|\boldsymbol{\xi}_{(m)}\|^{2}) \!&=&\!\frac{{\sum}_{l=0}^{\infty}\sum\limits_{k=0}^{N-1} {\sum}_{p=0}^{\infty} \frac{(-1)^{l}}{(2l)!}\alpha_{p}^{*}{\upbeta}_{k}A_{l}{\Gamma}(\frac{m}{2}+N-k+l-1)} {\sum\limits_{k=0}^{\infty}{\sum}_{j=0}^{N-1}\alpha_{k}^{*}{\upbeta}_{j}{\Gamma}(\frac{m}{2}+N-j-1)}\|\boldsymbol{\xi}_{(m)}\|^{2l}, \end{array} $$
where
$$ A_{x}\triangleq A_{x}(N,m,a,b,p,k)=\frac{B(\frac{m}{2},x+\frac{1}{2})}{B(\frac{m}{2}+x,\frac{1}{2})b^{N-k+x}(p+\frac{a}{b})^{x-\frac{m}{2}+1}}, $$
$$ {\upbeta}_{x}\triangleq{\upbeta}_{x}(N,n,m,b,r)=\frac{(N-1)_{x}}{x!}\frac{\Gamma(\frac{n-m}{2}+x)}{b^{\frac{n-m}{2}+x}{\Gamma}(2r)}, $$
$$ \alpha_{x}(r)\triangleq\alpha_{x}(r)=\frac{\Gamma(2r+x)(-1)^{x}}{x!},~\alpha_{x}^{*}\triangleq\alpha_{x}^{*}(N,n,a,b,r)= \frac{\alpha_{x}}{(x+\frac{a}{b})^{\frac{n}{2}+N-1}}. $$
When the dimension n > 1, Ωn(∥t∥2) defined as
$$ {\Omega}_{n}(\|t\|^{2})=_{0}F_{1}(\frac{n}{2};-\frac{1}{4}\|t\|^{2}), $$
we have the characteristic function as follows.
$$ \begin{array}{@{}rcl@{}} \psi_{\textbf{Y}}(\textbf{t})\!&=&e^{i\textbf{t}^{\textit{T}}\boldsymbol{\mu}_{\textbf{Y}}}\frac{1}{{\int}_{0}^{\infty}t^{\frac{m}{2}-1}g_{m}(t)dt} {\int}_{0}^{\infty}{\Omega}_{n}(v\textbf{t}^{\textit{T}}\boldsymbol{{\Sigma}_{\textbf{Y}}}\textbf{t})\\ &&\times{\sum}_{j=0}^{N-1}\frac{(N-1)_{j}}{j!}\frac{\Gamma(\frac{n-m}{2}+j)}{b^{\frac{n-m}{2}+j}}v^{N+\frac{m}{2}-j-2}e^{-av} {\Phi}_{2r}^{*}(-e^{-bv},\frac{n-m}{2}+j,\frac{a}{b})dv\\ &=&e^{i\textbf{t}^{\textit{T}}\boldsymbol{\mu}_{\textbf{Y}}}\left[\sum\limits_{k=0}^{\infty}{\sum}_{j=0}^{N-1}\alpha_{k}^{*}{\upbeta}_{j} {\Gamma}(\frac{m}{2}+N-j-1)\right]^{-1}{\int}_{0}^{\infty}\frac{\Gamma(\frac{m}{2})}{\pi^{\frac{1}{2}}} {\sum}_{l=0}^{\infty}\frac{(\textbf{t}^{\textit{T}}\boldsymbol{\Sigma}_{\textbf{Y}}\textbf{t})^{l}}{4^{l}(\frac{m}{2})^{[l]}}\\ &&\times{\sum}_{j=0}^{N-1}\frac{(N-1)_{j}}{j!}\frac{\Gamma(\frac{n-m}{2}+j)}{b^{\frac{n-m}{2}+j}}v^{N+\frac{m}{2}-j-2}e^{-av} {\Phi}_{2r}^{*}(-e^{-bv},\frac{n-m}{2}+j,\frac{a}{b})dv\\ &=&\!e^{i\textbf{t}^{\textit{T}}\boldsymbol{\mu}_{\textbf{Y}}}\frac{{\sum}_{l=0}^{\infty}{\sum}_{j=0}^{N - 1}\frac{\Gamma(\frac{m}{2})}{\pi^{\frac{1}{2}}} \frac{(\textbf{t}^{\textit{T}}\boldsymbol{\Sigma}_{\textbf{Y}}\textbf{t})^{l}}{4^{l}(\frac{m}{2})^{[l]}} \sum\limits_{k=0}^{\infty}{\upbeta}_{j}\alpha_{k}\frac{\Gamma(N+\frac{m}{2}-j - 1)}{b^{N+\frac{m}{2}-j-1}} {\Phi}_{2r}^{*}(-e^{-bv},\frac{n-m}{2} + j,\frac{a}{b})}{\sum\limits_{k=0}^{\infty}{\sum}_{j=0}^{N-1}\alpha_{k}^{*}{\upbeta}_{j} {\Gamma}(\frac{m}{2}+N-j-1)}\\ &=&e^{i\textbf{t}^{\textit{T}}\boldsymbol{\mu}_{\textbf{Y}}}\frac{{\sum}_{l=0}^{\infty}{\sum}_{j=0}^{N-1}\sum\limits_{k=0}^{\infty}\alpha_{k}{\upbeta}_{j}^{*} {\Phi}_{2r}^{*}(-e^{-bv},\frac{n-m}{2}+j,\frac{a}{b})}{\sum\limits_{k=0}^{\infty}{\sum}_{j=0}^{N-1}\alpha_{k}^{*}{\upbeta}_{j}{\Gamma}(\frac{m}{2}+N-j-1)} \frac{\Gamma(\frac{m}{2})}{\pi^{\frac{1}{2}}4^{l}(\frac{m}{2})^{[l]}}(\textbf{t}^{\textit{T}}\boldsymbol{\Sigma}_{\textbf{Y}}\textbf{t})^{l}, \end{array} $$
$$ \phi_{(m),y}(\|\boldsymbol{\xi}_{(m)}\|^{2}) = \frac{{\sum}_{l=0}^{\infty}{\sum}_{j=0}^{N-1}\sum\limits_{k=0}^{\infty}\alpha_{k}{\upbeta}_{j}^{*} {\Phi}_{2r}^{*}(-e^{-bv},\frac{n-m}{2} + j,\frac{a}{b})}{\sum\limits_{k=0}^{\infty}{\sum}_{j=0}^{N-1}\alpha_{k}^{*}{\upbeta}_{j}{\Gamma}(\frac{m}{2}+N-j-1)} \frac{\Gamma(\frac{m}{2})}{\pi^{\frac{1}{2}}4^{l}(\frac{m}{2})^{[l]}}\|\boldsymbol{\xi}_{(m)}\|^{2l}, $$
where
$$ {\upbeta}_{x}\triangleq{\upbeta}_{x}(N,n,m,r) = \frac{(N - 1)_{x}}{x!}\frac{\Gamma(\frac{n-m}{2}+x)}{\Gamma(2r)b^{\frac{n-m}{2}+x}},~ {\upbeta}_{x}^{*}\triangleq{\upbeta}_{x}^{*}(N,n,m,b,r) = \frac{{\upbeta}_{x}{\Gamma}(N+\frac{m}{2}-x-1)}{b^{N+\frac{m}{2}-x-1}}, $$
$$ \alpha_{x}\triangleq\alpha_{x}(r)=\frac{(-1)^{x}{\Gamma}(2r+x)}{x!},~ \alpha_{x}^{*}\triangleq\alpha_{x}^{*}(N,m,a,b,r)=\frac{\alpha_{x}}{(x+\frac{a}{b})^{\frac{n}{2}+N-1}}. $$
Here \({\Phi }_{2r}^{*}\) is the generalized Hurwitz-Lerch zeta function. □
