#### 期刊菜单

The Unified (r, s)-Relative Differential Entropy Based on Joint Distribution of Random Density Matrix
DOI: 10.12677/APP.2019.96037, PDF, HTML, XML, 下载: 602  浏览: 1,681  国家自然科学基金支持

Abstract: The unified (r, s)-relative differential entropy of the joint distribution of eigenvalues of random density matrices is studied by Laplace transform and Laplace inverse transform. On the one hand, the unified (r, s)-relative differential entropy of the joint distribution of the eigenvalues to diagonal entries of random density matrices induced by partial tracing (the diagonal entries of random density matrices induced by partial tracing to joint distribution of the eigenvalues) over Haar-distributed bipartite pure states is defined. On the other hand, the unified (r, s)-relative differential entropy in the three cases is calculated. The range of differential entropy is generalized.

1. 引言

2. 统一(r, s)相对微分熵的定义

1) 在 $\mathrm{Re}\left(z\right)>0$ 上的伽马函数为 $\Gamma \left(z\right)={\int }_{0}^{\infty }{t}^{z-1}{\text{e}}^{-t}\text{d}t$$\Gamma \left(x\right),\psi \left(x\right)$ 分别代表Gamma 函数和Digamma函数。

$\psi \left(x\right)=\frac{\text{d}\mathrm{ln}\Gamma \left(x\right)}{\text{d}x}=\frac{{\Gamma }^{\prime }\left(x\right)}{\Gamma \left(x\right)}$$\psi \left(k+1\right)={H}_{k}-\gamma$ ，其中 ${H}_{k}={\sum }_{j=1}^{k}\frac{1}{j}$$\gamma \approx 0.57721$ 为欧拉常数。

2) 若Wishart矩阵W的构成矩阵Z是复的均值为0，方差为 ${\sigma }^{\text{2}}$ 的独立同分布的高斯变量， $Z{Z}^{+}$ 联合概率密度的相应特征值 $\left({\mu }_{j}\in \left[0,\infty \right),j=1,2,\cdots ,m\right)$ 的联合概率函数为

$q\left({\mu }_{1},\cdots ,{\mu }_{m}\right)={C}_{q}\mathrm{exp}\left(-\underset{j=1}{\overset{m}{\sum }}{\mu }_{j}\right)\underset{1\le i ，其中 $\underset{j=1}{\overset{m}{\sum }}{\mu }_{j}=1$ (1)

${C}_{\text{q}}=1/\underset{1\le j\le m}{\prod }\left[j!\left(n-j\right)!\right]$ (2)

3) 在Haar分布的双体纯态上取部分迹所诱导的随机密度矩阵的特征值 $\left({\lambda }_{j}\in \left[0,1\right],j=1,2,\cdots ,m\right)$ 的联合概率函数为

$p\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)={C}_{p}\underset{1\le i ，其中 $\underset{j=1}{\overset{m}{\sum }}{\lambda }_{j}=1$ (3)

${C}_{p}=\Gamma \left(mn\right){C}_{q}$ (4)

4) 在Haar分布的双体纯态上其对角元的联合概率函数为

$\psi \left({\rho }_{11},{\rho }_{22},\cdots ,{\rho }_{mm}\right)={C}_{\psi }\underset{j=1}{\overset{m}{\prod }}{\rho }_{jj}^{n-1}$ ，其中 $\underset{j=1}{\overset{m}{\sum }}{\rho }_{jj}=1$ (5)

${C}_{\Psi }=\frac{\Gamma \left(mn\right)}{\Gamma {\left(n\right)}^{m}}$ (6)

5) ${I}_{m}\left(\alpha ,r\right)={\int }_{0}^{\infty }\cdot \cdot \cdot {\int }_{0}^{\infty }\mathrm{exp}\left(-\underset{j=1}{\overset{m}{\sum }}{\mu }_{j}\right)\underset{1\le i (7)

${I}_{m}\left(n-m,1\right)=\frac{\text{1}}{{C}_{q}}$ (8)

${I}_{m}\left(n-\text{1},\text{0}\right)={\Gamma }^{m}\left(n\right)$ (9)

$\begin{array}{c}\frac{\partial }{\partial r}\partial {I}_{m}\left(\alpha \left(r\right),\beta \left(r\right)\right)=\frac{\partial {I}_{m}\left(\alpha \left(r\right),\beta \left(r\right)\right)}{\partial \alpha }×\frac{\text{d}\alpha }{\text{d}r}+\frac{\partial {I}_{m}\left(\alpha \left(r\right),\beta \left(r\right)\right)}{\partial \beta }×\frac{\text{d}\beta }{\text{d}r}\\ ={I}_{m}\left(\alpha \left(r\right),\beta \left(r\right)\right)\left\{{\alpha }^{\prime }\left(r\right)×\underset{k=0}{\overset{m-1}{\sum }}\psi \left(\alpha \left(r\right)+1+k\beta \left(r\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\beta }^{\prime }\left(r\right)×\left[\underset{k=1}{\overset{m-1}{\sum }}k\psi \left(\alpha \left(r\right)+1+k\beta \left(r\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{k=1}{\overset{m}{\sum }}k\psi \left(1+k\beta \left(r\right)\right)-m\psi \left(1+\beta \left(r\right)\right)\right]\right\}\end{array}$ (10)

${E}_{r}^{s}\left(p||\psi \right)=\left\{\begin{array}{l}{h}_{r}^{s}\left(p||\psi \right),r\ne 1,s\ne 0\hfill \\ {h}_{r}\left(p||\psi \right),r\ne 1,s=0\hfill \\ {h}^{r}\left(p||\psi \right),r\ne 1,s=1\hfill \\ {}_{r}h\left(p||\psi \right),r\ne 1,s={r}^{-1}\hfill \\ h\left(p||\psi \right),r=1\hfill \end{array}$

${h}_{r}^{s}\left(p||\psi \right)=-{\left[\left(1-r\right)s\right]}^{-1}\left[{\left(\int p\left(\lambda \right){\left(p\left(\lambda \right)/\psi \left(\lambda \right)\right)}^{r-1}\text{d}\lambda \right)}^{s}-1\right],r>0,r\ne 1,s\ne 0$

${h}_{r}\left(p||\psi \right)=-{\left(1-r\right)}^{-1}\mathrm{ln}\left(\int p\left(\lambda \right){\left(p\left(\lambda \right)/\psi \left(\lambda \right)\right)}^{r-1}\text{d}\lambda \right),r>0,r\ne 1$

${h}^{r}\left(p||\psi \right)=-{\left(1-r\right)}^{-1}\left(\int p\left(\lambda \right){\left(p\left(\lambda \right)/\psi \left(\lambda \right)\right)}^{r-1}\text{d}\lambda -1\right),r>0,r\ne 1$

${}_{r}h\left(p||\psi \right)=-{\left(r-1\right)}^{-1}\left[{\left(\int p\left(\lambda \right){\left(p\left(\lambda \right)/\psi \left(\lambda \right)\right)}^{1/r-1}\text{d}\lambda \right)}^{r}-1\right],r>0,r\ne 1$

$h\left(p||\psi \right)=\int p\left(\lambda \right)\mathrm{ln}\left(p\left(\lambda \right)/\psi \left(\lambda \right)\right)\text{d}\lambda$

${E}_{r}^{s}\left(\psi ||p\right)=\left\{\begin{array}{l}{h}_{r}^{s}\left(\psi ||p\right),r\ne 1,s\ne 0\hfill \\ {h}_{r}\left(\psi ||p\right),r\ne 1,s=0\hfill \\ {h}^{r}\left(\psi ||p\right),r\ne 1,s=1\hfill \\ {}_{r}h\left(\psi ||p\right),r\ne 1,s={r}^{-1}\hfill \\ h\left(\psi ||p\right),r=1\hfill \end{array}$

${h}_{r}^{s}\left(\psi ||p\right)=-{\left[\left(1-r\right)s\right]}^{-1}\left[{\left(\int \psi \left(\lambda \right){\left(\psi \left(\lambda \right)/p\left(\lambda \right)\right)}^{r-1}\text{d}\lambda \right)}^{s}-1\right],r>0,r\ne 1,s\ne 0$

${h}_{r}\left(\psi ||p\right)=-{\left(1-r\right)}^{-1}\mathrm{ln}\left(\int \psi \left(\lambda \right){\left(\psi \left(\lambda \right)/p\left(\lambda \right)\right)}^{r-1}\text{d}\lambda \right),r>0,r\ne 1$

${h}^{r}\left(\psi ||p\right)=-{\left(1-r\right)}^{-1}\left(\int \psi \left(\lambda \right){\left(\psi \left(\lambda \right)/p\left(\lambda \right)\right)}^{r-1}\text{d}\lambda -1\right),r>0,r\ne 1$

${}_{r}h\left(\psi ||p\right)=-{\left(r-1\right)}^{-1}\left[{\left(\int \psi \left(\lambda \right){\left(\psi \left(\lambda \right)/p\left(\lambda \right)\right)}^{1/r-1}\text{d}\lambda \right)}^{r}-1\right],r>0,r\ne 1$

$h\left(\psi ||p\right)=\int \psi \left(\lambda \right)\mathrm{ln}\left(\psi \left(\lambda \right)/p\left(\lambda \right)\right)\text{d}\lambda$

3. 统一(r, s)相对微分熵的计算

${h}_{r}\left(p||\psi \right)={\left(r-1\right)}^{-1}\left[r\mathrm{ln}{C}_{p}+\left(1-r\right)\mathrm{ln}{C}_{\psi }+\mathrm{ln}{I}_{m}\left(\left(1-m\right)r+n-1,r\right)-\mathrm{ln}\Gamma \left(mn\right)\right]$

$\begin{array}{c}{h}_{r}\left(p||\psi \right)=-{\left(1-r\right)}^{-1}\mathrm{ln}\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }p\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){\left(p\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)/\psi \left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\right)}^{r-1}\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}\right)\\ ={\left(r-1\right)}^{-1}\mathrm{ln}\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }{p}^{r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){\psi }^{1-r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}\right)\end{array}$

${F}_{r}\left(t\right)={\left(r-1\right)}^{-1}\mathrm{ln}\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\delta \left(t-\underset{j=1}{\overset{m}{\sum }}{\lambda }_{j}\right){p}^{r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){\psi }^{1-r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}\right)$

${F}_{r}\left(t\right)$ 应用Laplace变换 $\left(t\to s\right)$ ，则有

${\stackrel{˜}{F}}_{r}\left(s\right)={\left(r-1\right)}^{-1}\mathrm{ln}\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\mathrm{exp}\left(-s\underset{j=1}{\overset{m}{\sum }}{\lambda }_{j}\right){p}^{r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){\psi }^{1-r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}\right)$

${\lambda }_{k}=\frac{{\mu }_{k}}{s},k=1,2,\cdots ,m$

$\frac{{C}_{p}}{{C}_{q}}\mathrm{exp}\left(\underset{i=1}{\overset{m}{\sum }}{\mu }_{m}\right)q\left({\mu }_{1},\cdots {\mu }_{m}\right)=p\left({\mu }_{1},\cdots {\mu }_{m}\right)$$p\left(\frac{{\mu }_{1}}{s},\cdots ,\frac{{\mu }_{m}}{s}\right)={s}^{m-nm}p\left({\mu }_{1},\cdots ,{\mu }_{m}\right)$

$p\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)={s}^{m-mn}{C}_{p}\underset{1\le i (11)

$\psi \left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)={s}^{m-mn}{C}_{\psi }\underset{j=1}{\overset{m}{\prod }}{\mu }_{j}^{n-1}$ (12)

$\begin{array}{c}{\stackrel{˜}{F}}_{r}\left(s\right)={\left(r-1\right)}^{-1}\mathrm{ln}\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\mathrm{exp}\left(-s\underset{j=1}{\overset{m}{\sum }}{\lambda }_{j}\right){\left({s}^{m-mn}{C}_{p}\underset{1\le i

${L}^{-1}\left\{{s}^{-m}\right\}\left(t\right)=\frac{{t}^{m-1}}{\Gamma \left(m\right)}$ (13)

${L}^{-1}\left\{{s}^{-mn}\right\}\left(t\right)=\frac{{t}^{mn-1}}{\Gamma \left(mn\right)}$ (14)

${F}_{r}\left(t\right)={\left(r-1\right)}^{-1}\mathrm{ln}\left(\frac{{t}^{mn-1}}{\Gamma \left(mn\right)}{C}_{p}^{r}{C}_{\psi }^{1-r}{I}_{m}\left(\left(1-m\right)r+n-1,r\right)\right)$

$\begin{array}{c}{h}_{r}\left(p||\psi \right)={F}_{r}\left(1\right)={\left(r-1\right)}^{-1}\mathrm{ln}\left(\frac{{C}_{p}^{r}{C}_{\psi }^{1-r}}{\Gamma \left(mn\right)}{I}_{m}\left(\left(1-m\right)r+n-1,r\right)\right)\\ ={\left(r-1\right)}^{-1}\left[r\mathrm{ln}{C}_{p}+\left(1-r\right)\mathrm{ln}{C}_{\psi }+\mathrm{ln}{I}_{m}\left(\left(1-m\right)r+n-1,r\right)-\mathrm{ln}\Gamma \left(mn\right)\right]\end{array}$ (15)

$\begin{array}{c}\underset{r\to 1}{\mathrm{lim}}{h}_{r}\left(p||\psi \right)=-\mathrm{ln}\left(\underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!\right)-m\psi \left(2\right)+m\mathrm{ln}\Gamma \left(n\right)\\ -\underset{k=n-m}{\overset{n-1}{\sum }}\left(n-k-1\right)\psi \left(k+1\right)+\underset{k=1}{\overset{m}{\sum }}k\psi \left(k+1\right)\end{array}$

$\begin{array}{l}{\frac{\partial \mathrm{ln}{I}_{m}\left(\left(1-m\right)r+n-1,r\right)}{\partial r}|}_{r=1}=\frac{1}{{I}_{m}\left(\left(1-m\right)r+n-1,r\right)}{\frac{\partial {I}_{m}\left(\left(1-m\right)r+n-1,r\right)}{\partial r}|}_{r=1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left(1-m\right)×\underset{k=0}{\overset{m-1}{\sum }}\psi \left(\left(1-m\right)r+n+kr\right)-m\psi \left(1+r\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{k=1}{\overset{m-1}{\sum }}k\psi \left(\left(1-m\right)r+n+kr\right)+{\underset{k=1}{\overset{m}{\sum }}k\psi \left(1+kr\right)|}_{r=1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\underset{k=0}{\overset{m-1}{\sum }}\left(1-m+n+k\right)×\psi \left(1-m+n+k\right)-m\psi \left(2\right)+\underset{k=1}{\overset{m}{\sum }}k\psi \left(1+k\right)\end{array}$ (16)

$\begin{array}{c}\underset{r\to 1}{\mathrm{lim}}{h}_{r}\left(p||\psi \right)={\left(r-1\right)}^{-1}\left[r\mathrm{ln}{C}_{p}+\left(1-r\right)\mathrm{ln}{C}_{\psi }+\mathrm{ln}{I}_{m}\left(\left(1-m\right)r+n-1,r\right)-\mathrm{ln}\Gamma \left(mn\right)\right]\\ =\mathrm{ln}{C}_{p}-\mathrm{ln}{C}_{\psi }+{\frac{\partial \mathrm{ln}{I}_{m}\left(\left(1-m\right)r+n-1,r\right)}{\partial r}|}_{r=1}\\ =-\mathrm{ln}\left(\underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!\right)-m\psi \left(2\right)+m\mathrm{ln}\Gamma \left(n\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{k=n-m}{\overset{n-1}{\sum }}\left(n-k-1\right)\psi \left(k+1\right)+\underset{k=1}{\overset{m}{\sum }}k\psi \left(k+1\right)\end{array}$

${h}^{r}\left(p||\psi \right)={\left(r-1\right)}^{-1}\left(\frac{{C}_{p}^{r}{C}_{\psi }^{1-r}}{\Gamma \left(mn\right)}{I}_{m}\left(\left(1-m\right)r+n-1,r\right)-1\right)$

$\begin{array}{c}{h}^{r}\left(p||\psi \right)=-{\left(1-r\right)}^{-1}\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }p\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){\left(p\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)/\psi \left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\right)}^{r-1}\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}-1\right)\\ ={\left(r-1\right)}^{-1}\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }{p}^{r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){\psi }^{1-r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}-1\right)\end{array}$

${F}^{r}\left(t\right)={\left(r-1\right)}^{-1}\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\delta \left(t-\underset{j=1}{\overset{m}{\sum }}{\lambda }_{j}\right){p}^{r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){\psi }^{1-r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}-1\right)$

${F}^{r}\left(t\right)$ 应用Laplace变换 $\left(t\to s\right)$ ，则有

${\stackrel{˜}{F}}^{r}\left(s\right)={\left(r-1\right)}^{-1}\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\mathrm{exp}\left(-s\underset{j=1}{\overset{m}{\sum }}{\lambda }_{j}\right){p}^{r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){\psi }^{1-r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}-1\right)$

$\begin{array}{c}{\stackrel{˜}{F}}^{r}\left(s\right)={\left(r-1\right)}^{-1}\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\mathrm{exp}\left(-s\underset{j=1}{\overset{m}{\sum }}{\lambda }_{j}\right){\left({s}^{m-mn}{C}_{p}\underset{1\le i

${F}^{r}\left(t\right)={\left(r-1\right)}^{-1}\left(\frac{{t}^{mn-1}}{\Gamma \left(mn\right)}{C}_{p}^{r}{C}_{\psi }^{1-r}{I}_{m}\left(\left(1-m\right)r+n-1,r\right)-1\right)$

${h}^{r}\left(p||\psi \right)={F}^{r}\left(1\right)={\left(r-1\right)}^{-1}\left(\frac{{C}_{p}^{r}{C}_{\psi }^{1-r}}{\Gamma \left(mn\right)}{I}_{m}\left(\left(1-m\right)r+n-1,r\right)-1\right)$ (17)

$\begin{array}{c}\underset{r\to 1}{\mathrm{lim}}{h}^{r}\left(p||\psi \right)=-\mathrm{ln}\left(\underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!\right)-m\psi \left(2\right)+m\mathrm{ln}\Gamma \left(n\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{k=n-m}{\overset{n-1}{\sum }}\left(n-k-1\right)\psi \left(k+1\right)+\underset{k=1}{\overset{m}{\sum }}k\psi \left(k+1\right)\end{array}$

$\begin{array}{l}{\frac{\partial {I}_{m}\left(\left(1-m\right)r+n-1,r\right)}{\partial r}|}_{r=1}\\ ={I}_{m}\left(\left(1-m\right)r+n-1,r\right)\left[\left(1-m\right)×\underset{k=0}{\overset{m-1}{\sum }}\psi \left(\left(1-m\right)r+n+kr\right)-m\psi \left(1+r\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{k=1}{\overset{m-1}{\sum }}k\psi \left(\left(1-m\right)r+n+kr\right)+{\underset{k=1}{\overset{m}{\sum }}k\psi \left(1+kr\right)\right]|}_{r=1}\\ ={I}_{m}\left(n-m,1\right)\left[\underset{k=0}{\overset{m-1}{\sum }}\left(1-m+n+k\right)×\psi \left(1-m+n+k\right)-m\psi \left(2\right)+\underset{k=1}{\overset{m}{\sum }}k\psi \left(1+k\right)\right]\end{array}$ (18)

$\begin{array}{c}\underset{r\to 1}{\mathrm{lim}}{h}^{r}\left(p||\psi \right)=\left[\mathrm{ln}{C}_{p}{C}_{p}^{r}{C}_{\psi }^{1-r}\frac{{I}_{m}\left(\left(1-m\right)r+n-1,r\right)}{\Gamma \left(mn\right)}-\mathrm{ln}{C}_{\psi }{C}_{p}^{r}{C}_{\psi }^{1-r}\frac{{I}_{m}\left(\left(1-m\right)r+n-1,r\right)}{\Gamma \left(mn\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\frac{{C}_{p}^{r}{C}_{\psi }^{1-r}}{\Gamma \left(mn\right)}\frac{\partial {I}_{m}\left(\left(1-m\right)r+n-1,r\right)}{\partial r}\right]|}_{r=1}\\ =\mathrm{ln}{C}_{p}-\mathrm{ln}{C}_{\psi }+\frac{{C}_{p}}{\Gamma \left(mn\right)}{\frac{\partial {I}_{m}\left(\left(1-m\right)r+n-1,r\right)}{\partial r}|}_{r=1}\\ =-\mathrm{ln}\left(\underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!\right)-m\psi \left(2\right)+m\mathrm{ln}\Gamma \left(n\right)-\underset{k=n-m}{\overset{n-1}{\sum }}\left(n-k-1\right)\psi \left(k+1\right)+\underset{k=1}{\overset{m}{\sum }}k\psi \left(k+1\right)\end{array}$

${}_{r}h\left(p||\psi \right)={\left(1-r\right)}^{-1}\left[\frac{{C}_{p}{C}_{\psi }^{r-1}}{\Gamma \left(mnr\right)}{\left({I}_{m}\left(n-1-\frac{m-1}{r},\frac{1}{r}\right)\right)}^{r}-1\right]$

$\begin{array}{c}{}_{r}h\left(p||\psi \right)=-{\left(r-1\right)}^{-1}\left[{\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }p\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){\left(p\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)/\psi \left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\right)}^{1/r-1}\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}\right)}^{r}-1\right]\\ ={\left(1-r\right)}^{-1}\left[{\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }{p}^{1/r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){\psi }^{1-1/r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}\right)}^{r}-1\right]\end{array}$

${}_{r}F\left(t\right)={\left(1-r\right)}^{-1}\left[{\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\delta \left(t-\underset{j=1}{\overset{m}{\sum }}{\lambda }_{j}\right){p}^{1/r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){\psi }^{1-1/r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}\right)}^{r}-1\right]$

${}_{r}F\left(t\right)$ 应用Laplace变换 $\left(t\to s\right)$ ，则有

${}_{r}\stackrel{˜}{F}\left(s\right)={\left(1-r\right)}^{-1}\left[{\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\mathrm{exp}\left(-s\underset{j=1}{\overset{m}{\sum }}{\lambda }_{j}\right){p}^{1/r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){\psi }^{1-1/r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}\right)}^{r}-1\right]$

$\begin{array}{c}{}_{r}\stackrel{˜}{F}\left(s\right)={\left(1-r\right)}^{-1}\left[{\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\mathrm{exp}\left(-s\underset{j=1}{\overset{m}{\sum }}{\lambda }_{j}\right){\left({s}^{m-mn}{C}_{p}\underset{1\le i

${L}^{-1}\left\{{s}^{-mnr}\right\}\left(t\right)=\frac{{t}^{mnr-1}}{\Gamma \left(mnr\right)}$ (19)

${}_{r}F\left(t\right)={\left(1-r\right)}^{-1}\left[\frac{{t}^{mnr-1}}{\Gamma \left(mnr\right)}{C}_{p}{C}_{\psi }^{r-1}{\left({I}_{m}\left(n-1+\left(1-m\right)/r,1/r\right)\right)}^{r}-1\right]$

${}_{r}h\left(p||\psi \right)={}_{r}F\left(1\right)={\left(1-r\right)}^{-1}\left[\frac{{C}_{p}{C}_{\psi }^{r-1}}{\Gamma \left(mnr\right)}{\left({I}_{m}\left(n-1+\left(1-m\right)/r,1/r\right)\right)}^{r}-1\right]$ (20)

$\begin{array}{c}\underset{r\to 1}{\mathrm{lim}}{}_{r}h\left(p||\psi \right)=-\mathrm{ln}\left(\underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!\right)-m\psi \left(2\right)+m\mathrm{ln}\Gamma \left(n\right)-\mathrm{ln}\Gamma \left(mn\right)-mn\psi \left(mn\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{k=n-m}{\overset{n-1}{\sum }}\left(n-k-1\right)\psi \left(k+1\right)+\underset{k=1}{\overset{m}{\sum }}k\psi \left(k+1\right)\end{array}$

${I}_{m}{\left(\alpha \left(r\right),\beta \left(r\right)\right)}^{r}=y$

$r\mathrm{ln}{I}_{m}\left(\alpha \left(r\right),\beta \left(r\right)\right)=\mathrm{ln}y$

$\begin{array}{l}\mathrm{ln}{I}_{m}\left(\alpha \left(r\right),\beta \left(r\right)\right)+\frac{r}{{I}_{m}\left(\alpha \left(r\right),\beta \left(r\right)\right)}\frac{\partial {I}_{m}\left(\alpha \left(r\right),\beta \left(r\right)\right)}{\partial r}=\frac{{y}^{\prime }}{y}\\ {y}^{\prime }=y\left[\mathrm{ln}{I}_{m}\left(\alpha \left(r\right),\beta \left(r\right)\right)+\frac{r}{{I}_{m}\left(\alpha \left(r\right),\beta \left(r\right)\right)}\frac{\partial {I}_{m}\left(\alpha \left(r\right),\beta \left(r\right)\right)}{\partial r}\right]\end{array}$ (21)

$\begin{array}{l}\frac{\partial {\left({I}_{m}\left(n-1+\left(1-m\right)/r,1/r\right)\right)}^{r}}{\partial r}={\left({I}_{m}\left(n-1+\left(1-m\right)/r,1/r\right)\right)}^{r}\\ \left[\mathrm{ln}{I}_{m}\left(n-1+\left(1-m\right)/r,1/r\right)\begin{array}{c}\text{ }\\ \text{ }\end{array}+\frac{r}{{I}_{m}\left(n-1+\left(1-m\right)/r,1/r\right)}×\frac{\partial {I}_{m}\left(n-1+\left(1-m\right)/r,1/r\right)}{\partial r}\right]\end{array}$ (22)

$\begin{array}{l}{\frac{\partial {I}_{m}\left(n-1+\left(1-m\right)/r,1/r\right)}{\partial r}|}_{r=1}\\ ={I}_{m}\left(n-1+\left(1-m\right)/r,1/r\right)\left[\frac{m-1}{{r}^{2}}\underset{k=0}{\overset{m-1}{\sum }}\psi \left(n+\left(1-m\right)/r+k/r\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{1}{{r}^{2}}\underset{k=1}{\overset{m-1}{\sum }}k\psi \left(n+\left(1-m\right)/r+k/r\right)-\frac{1}{{r}^{2}}\underset{k=1}{\overset{m}{\sum }}k\psi \left(1+k/r\right)+{\frac{1}{{r}^{2}}m\psi \left(1+1/r\right)\right]|}_{r=1}\\ ={I}_{m}\left(n-m,1\right)\left[\underset{k=n-m}{\overset{n-1}{\sum }}\left(1+k-n\right)\psi \left(k+1\right)-\underset{k=1}{\overset{m}{\sum }}k\psi \left(1+k\right)+m\psi \left(2\right)\right]\end{array}$ (23)

$\begin{array}{c}\underset{r\to 1}{\mathrm{lim}}{}_{r}h\left(p||\psi \right)=-\left[\frac{{C}_{p}{C}_{\psi }^{r-1}\mathrm{ln}{C}_{\psi }}{\Gamma \left(mnr\right)}{\left({I}_{m}\left(n-1+\left(1-m\right)/r,1/r\right)\right)}^{r}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{C}_{p}{C}_{\psi }^{r-1}\psi \left(mnr\right)mn}{\Gamma \left(mnr\right)}{\left({I}_{m}\left(n-1+\left(1-m\right)/r,1/r\right)\right)}^{r}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\frac{{C}_{p}{C}_{\psi }^{r-1}}{\Gamma \left(mnr\right)}\frac{\partial {\left({I}_{m}\left(n-1+\left(1-m\right)/r,1/r\right)\right)}^{r}}{\partial r}\right]|}_{r=1}\end{array}$

$\begin{array}{l}=-{\left[\mathrm{ln}{C}_{p}-\mathrm{ln}{C}_{\psi }+\frac{{C}_{p}}{\Gamma \left(mn\right)}\frac{\partial {\left({I}_{m}\left(n-1+\left(1-m\right)/r,1/r\right)\right)}^{r}}{\partial r}\right]|}_{r=1}\\ =-\mathrm{ln}\left(\underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!\right)-m\psi \left(2\right)+m\mathrm{ln}\Gamma \left(n\right)-\mathrm{ln}\Gamma \left(mn\right)-mn\psi \left(mn\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{k=n-m}{\overset{n-1}{\sum }}\left(n-k-1\right)\psi \left(k+1\right)+\underset{k=1}{\overset{m}{\sum }}k\psi \left(k+1\right)\end{array}$

${h}_{r}\left(\psi ||p\right)={\left(r-1\right)}^{-1}\left[r\mathrm{ln}{C}_{\psi }+\left(1-r\right)\mathrm{ln}{C}_{p}+\mathrm{ln}{I}_{m}\left(\left(m-1\right)r-m+n,1-r\right)-\mathrm{ln}\Gamma \left(mn\right)\right]$

$\begin{array}{c}{h}_{r}\left(\psi ||p\right)=-{\left(1-r\right)}^{-1}\mathrm{ln}\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\psi \left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){\left(\psi \left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)/p\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\right)}^{r-1}\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}\right)\\ ={\left(r-1\right)}^{-1}\mathrm{ln}\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }{\psi }^{r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){p}^{1-r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}\right)\end{array}$

${{F}^{\prime }}_{r}\left(t\right)={\left(r-1\right)}^{-1}\mathrm{ln}\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\delta \left(t-\underset{j=1}{\overset{m}{\sum }}{\lambda }_{j}\right){\psi }^{r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){p}^{1-r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}\right)$

${{F}^{\prime }}_{r}\left(t\right)$ 应用Laplace变换 $\left(t\to s\right)$ ，则有

${\stackrel{˜}{{F}^{\prime }}}_{r}\left(s\right)={\left(r-1\right)}^{-1}\mathrm{ln}\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\mathrm{exp}\left(-s\underset{j=1}{\overset{m}{\sum }}{\lambda }_{j}\right){\psi }^{r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){p}^{1-r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}\right)$

$\begin{array}{c}{\stackrel{˜}{{F}^{\prime }}}_{r}\left(s\right)={\left(r-1\right)}^{-1}\mathrm{ln}\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\mathrm{exp}\left(-s\underset{j=1}{\overset{m}{\sum }}{\lambda }_{j}\right){\left({s}^{m-mn}{C}_{p}\underset{1\le i

${{F}^{\prime }}_{r}\left(t\right)={\left(r-1\right)}^{-1}\mathrm{ln}\left(\frac{{t}^{mn-1}}{\Gamma \left(mn\right)}{C}_{\psi }^{r}{C}_{p}^{1-r}{I}_{m}\left(\left(m-1\right)r+n-m,1-r\right)\right)$

$\begin{array}{c}{h}_{r}\left(\psi ||p\right)={{F}^{\prime }}_{r}\left(1\right)={\left(r-1\right)}^{-1}\mathrm{ln}\left(\frac{{C}_{\psi }^{r}{C}_{p}^{1-r}}{\Gamma \left(mn\right)}{I}_{m}\left(\left(m-1\right)r+n-m,1-r\right)\right)\\ ={\left(r-1\right)}^{-1}\left[r\mathrm{ln}{C}_{\psi }+\left(1-r\right)\mathrm{ln}{C}_{p}+\mathrm{ln}{I}_{m}\left(\left(m-1\right)r+n-m,1-r\right)-\mathrm{ln}\Gamma \left(mn\right)\right]\end{array}$ (24)

$\underset{r\to 1}{\mathrm{lim}}{h}_{r}\left(\psi ||p\right)=\mathrm{ln}\left(\underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!\right)-m\mathrm{ln}\Gamma \left(n\right)+\frac{{m}^{2}-m}{2}\psi \left(n\right)+\frac{m-{m}^{2}}{2}\psi \left(1\right)$

$\begin{array}{l}{\frac{\partial \mathrm{ln}{I}_{m}\left(\left(m-1\right)r+n-m,1-r\right)}{\partial r}|}_{r=1}=\frac{1}{{I}_{m}\left(\left(m-1\right)r+n-m,1-r\right)}{\frac{\partial {I}_{m}\left(\left(m-1\right)r+n-m,1-r\right)}{\partial r}|}_{r=1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left(m-1\right)×\underset{k=0}{\overset{m-1}{\sum }}\psi \left(\left(m-1\right)r+n-m+1+k\left(1-r\right)\right)+m\psi \left(1+\left(1-r\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{k=1}{\overset{m-1}{\sum }}k\psi \left(\left(m-1\right)r+n-m+1+k\left(1-r\right)\right)-{\underset{k=1}{\overset{m}{\sum }}k\psi \left(1+k\left(1-r\right)\right)|}_{r=1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left(m-1\right)×m\psi \left(n\right)-\frac{{m}^{2}+m}{2}\psi \left(1\right)+m\psi \left(1\right)\end{array}$ (25)

$\begin{array}{c}\underset{r\to 1}{\mathrm{lim}}{h}_{r}\left(\psi ||p\right)={\left(r-1\right)}^{-1}\left[r\mathrm{ln}{C}_{p}+\left(1-r\right)\mathrm{ln}{C}_{\psi }+\mathrm{ln}{I}_{m}\left(\left(m-1\right)r+n-m,1-r\right)-\mathrm{ln}\Gamma \left(mn\right)\right]\\ =\mathrm{ln}{C}_{\psi }-\mathrm{ln}{C}_{p}+{\frac{\partial \mathrm{ln}{I}_{m}\left(\left(m-1\right)r+n-m,1-r\right)}{\partial r}|}_{r=1}\\ =\mathrm{ln}\left(\underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!\right)-m\mathrm{ln}\Gamma \left(n\right)+\frac{{m}^{2}-m}{2}\psi \left(n\right)+\frac{m-{m}^{2}}{2}\psi \left(1\right)\end{array}$

${h}^{r}\left(\psi ||p\right)={\left(r-1\right)}^{-1}\left[\frac{{C}_{\psi }^{r}{C}_{p}^{1-r}}{\Gamma \left(mn\right)}{I}_{m}\left(\left(m-1\right)r+n-m,1-r\right)-1\right]$

$\begin{array}{c}{h}^{r}\left(\psi ||p\right)=-{\left(1-r\right)}^{-1}\left[{\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\psi \left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){\left(\psi \left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)/p\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\right)}^{r-1}\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}-1\right]\\ ={\left(r-1\right)}^{-1}\left[{\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }{\psi }^{r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){p}^{1-r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}-1\right]\end{array}$

${F}^{r}{}^{\prime }\left(t\right)={\left(r-1\right)}^{-1}\left[{\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\delta \left(t-\underset{j=1}{\overset{m}{\sum }}{\lambda }_{j}\right){\psi }^{r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){p}^{1-r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}-1\right]$

${F}^{r}{}^{\prime }\left(t\right)$ 应用Laplace变换 $\left(t\to s\right)$ ，则有

${\stackrel{˜}{F}}^{r}{}^{\prime }\left(s\right)={\left(r-1\right)}^{-1}\left[{\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\mathrm{exp}\left(-s\underset{j=1}{\overset{m}{\sum }}{\lambda }_{j}\right){\psi }^{r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){p}^{1-r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}-1\right]$

$\begin{array}{c}{\stackrel{˜}{F}}^{r}{}^{\prime }\left(s\right)={\left(r-1\right)}^{-1}\left[{\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\mathrm{exp}\left(-s\underset{j=1}{\overset{m}{\sum }}{\lambda }_{j}\right){\left({s}^{m-mn}{C}_{p}\underset{1\le i

${F}^{r}{}^{\prime }\left(t\right)={\left(r-1\right)}^{-1}\left[\frac{{t}^{mn-1}}{\Gamma \left(mn\right)}{C}_{\psi }^{r}{C}_{p}^{1-r}{I}_{m}\left(\left(m-1\right)r+n-m,1-r\right)-1\right]$

${h}^{r}{}^{\prime }\left(\psi ||p\right)={F}^{r}{}^{\prime }\left(1\right)={\left(r-1\right)}^{-1}\left[\frac{{C}_{\psi }^{r}{C}_{p}^{1-r}}{\Gamma \left(mn\right)}{I}_{m}\left(\left(m-1\right)r+n-m,1-r\right)-1\right]$

$\underset{r\to 1}{\mathrm{lim}}{h}^{r}\left(\psi ||p\right)=\mathrm{ln}\left(\underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!\right)-m\mathrm{ln}\Gamma \left(n\right)+\frac{{m}^{2}-m}{2}\psi \left(n\right)+\frac{m-{m}^{2}}{2}\psi \left(1\right)$ (26)

$\begin{array}{l}{\frac{\partial {I}_{m}\left(\left(m-1\right)r+n-m,1-r\right)}{\partial r}|}_{r=1}\\ ={I}_{m}\left(\left(m-1\right)r+n-m,1-r\right)\left[\left(m-1\right)×\underset{k=0}{\overset{m-1}{\sum }}\psi \left(\left(m-1\right)r+n-m+1+k\left(1-r\right)\right)+m\psi \left(2-r\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{k=1}{\overset{m-1}{\sum }}k\psi \left(\left(m-1\right)r+n-m+1+k\left(1-r\right)\right)-{\underset{k=1}{\overset{m}{\sum }}k\psi \left(1+k\left(1-r\right)\right)\right]|}_{r=1}\\ ={I}_{m}\left(n-1,0\right)\left[\underset{k=0}{\overset{m-1}{\sum }}\left(m-1-k\right)×\psi \left(n\right)+m\psi \left(1\right)-\underset{k=1}{\overset{m}{\sum }}k\left(1\right)\right]\end{array}$ (27)

$\begin{array}{c}\underset{r\to 1}{\mathrm{lim}}{h}^{r}\left(\psi ||p\right)=\left[\mathrm{ln}{C}_{\psi }{C}_{\psi }^{r}{C}_{p}^{1-r}\frac{{I}_{m}\left(\left(m-1\right)r+n-m,1-r\right)}{\Gamma \left(mn\right)}-\mathrm{ln}{C}_{p}{C}_{\psi }^{r}{C}_{p}^{1-r}\frac{{I}_{m}\left(\left(m-1\right)r+n-m,1-r\right)}{\Gamma \left(mn\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\frac{{C}_{\psi }^{r}{C}_{p}^{1-r}}{\Gamma \left(mn\right)}\frac{\partial {I}_{m}\left(\left(m-1\right)r+n-m,1-r\right)}{\partial r}\right]|}_{r=1}\\ =\frac{\mathrm{ln}{C}_{\psi }{I}_{m}\left(n-1,0\right)}{{\Gamma }^{m}\left(n\right)}-\frac{\mathrm{ln}{C}_{p}{I}_{m}\left(n-1,0\right)}{{\Gamma }^{m}\left(n\right)}+{\frac{{I}_{m}\left(n-1,0\right)}{{\Gamma }^{m}\left(n\right)}\frac{\partial {I}_{m}\left(\left(m-1\right)r+n-m,1-r\right)}{\partial r}|}_{r=1}\\ =\mathrm{ln}\left(\underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!\right)-m\mathrm{ln}\Gamma \left(n\right)+\frac{{m}^{2}-m}{2}\psi \left(n\right)+\frac{m-{m}^{2}}{2}\psi \left(1\right)\end{array}$

${}_{r}h\left(\psi ||p\right)={\left(1-r\right)}^{-1}\left[\frac{{C}_{\psi }{C}_{p}^{r-1}}{\Gamma \left(mnr\right)}{\left({I}_{m}\left(n-m+\frac{m-1}{r},1-\frac{1}{r}\right)\right)}^{r}-1\right]$

$\begin{array}{c}{}_{r}h\left(\psi ||p\right)=-{\left(r-1\right)}^{-1}\left[{\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\psi \left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){\left(\psi \left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)/p\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\right)}^{1/r-1}\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}\right)}^{r}-1\right]\\ ={\left(1-r\right)}^{-1}\left[{\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }{\psi }^{1/r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){p}^{1-1/r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}\right)}^{r}-1\right]\end{array}$

${}_{r}{F}^{\prime }\left(t\right)={\left(1-r\right)}^{-1}\left[{\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\delta \left(t-\underset{j=1}{\overset{m}{\sum }}{\lambda }_{j}\right){\psi }^{1/r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){p}^{1-1/r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}\right)}^{r}-1\right]$

${}_{r}{F}^{\prime }\left(t\right)$ 应用Laplace变换 $\left(t\to s\right)$ ，则有

${}_{r}\stackrel{˜}{{F}^{\prime }}\left(s\right)={\left(1-r\right)}^{-1}\left[{\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\mathrm{exp}\left(-s\underset{j=1}{\overset{m}{\sum }}{\lambda }_{j}\right){\psi }^{1/r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right){p}^{1-1/r}\left({\lambda }_{1},\cdots ,{\lambda }_{m}\right)\underset{k=1}{\overset{m}{\prod }}\text{d}{\lambda }_{k}\right)}^{r}-1\right]$

$\begin{array}{c}{}_{r}\stackrel{˜}{{F}^{\prime }}\left(s\right)={\left(1-r\right)}^{-1}\left[{\left({\int }_{0}^{\infty }\cdots {\int }_{0}^{\infty }\mathrm{exp}\left(-s\underset{j=1}{\overset{m}{\sum }}{\lambda }_{j}\right){\left({s}^{m-mn}{C}_{p}\underset{1\le i

${}_{r}{F}^{\prime }\left(t\right)={\left(1-r\right)}^{-1}\left[\frac{{t}^{mnr-1}}{\Gamma \left(mnr\right)}{C}_{\psi }{C}_{p}^{r-1}{\left({I}_{m}\left(n-m+\left(m-1\right)/r,1-1/r\right)\right)}^{r}-1\right]$

${}_{r}h\left(\psi ||p\right)={}_{r}{F}^{\prime }\left(1\right)={\left(1-r\right)}^{-1}\left[\frac{{C}_{\psi }{C}_{p}^{r-1}}{\Gamma \left(mnr\right)}{\left({I}_{m}\left(n-m+\left(m-1\right)/r,1-1/r\right)\right)}^{r}-1\right]$ (28)

$\underset{r\to 1}{\mathrm{lim}}{}_{r}h\left(\psi ||p\right)=\mathrm{ln}\left(\underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!\right)-m\mathrm{ln}\Gamma \left(n\right)-\mathrm{ln}\Gamma \left(mn\right)-mn\psi \left(mn\right)+\frac{{m}^{2}-m}{2}\psi \left(n\right)+\frac{m-{m}^{2}}{2}\psi \left(1\right)$

$\begin{array}{l}\frac{\partial {\left({I}_{m}\left(n-m+\left(m-1\right)/r,1-1/r\right)\right)}^{r}}{\partial r}={\left({I}_{m}\left(n-m+\left(m-1\right)/r,1-1/r\right)\right)}^{r}\\ \left[\mathrm{ln}{I}_{m}\left(n-m+\left(m-1\right)/r,1-1/r\right)\begin{array}{c}\text{ }\\ \text{ }\end{array}+\frac{r}{{I}_{m}\left(n-m+\left(m-1\right)/r,1-1/r\right)}×\frac{\partial {I}_{m}\left(n-m+\left(m-1\right)/r,1-1/r\right)}{\partial r}\right]\end{array}$ (29)

${\frac{\partial {I}_{m}\left(n-m+\left(m-1\right)/r,1-1/r\right)}{\partial r}|}_{r=1}={I}_{m}\left(n-1,0\right)\left[\left(1-m\right)\underset{k=0}{\overset{m-1}{\sum }}\psi \left(n\right)+\underset{k=1}{\overset{m}{\sum }}k\psi \left(n\right)+\underset{k=1}{\overset{m}{\sum }}k\psi \left(1\right)-m\psi \left(1\right)\right]$ (30)

$\begin{array}{c}\underset{r\to 1}{\mathrm{lim}}{}_{r}h\left(\psi ||p\right)=-\left[\frac{{C}_{\psi }{C}_{p}^{r-1}\mathrm{ln}{C}_{p}}{\Gamma \left(mnr\right)}{\left({I}_{m}\left(n-m+\left(m-1\right)/r,1-1/r\right)\right)}^{r}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\frac{{C}_{\psi }{C}_{p}^{r-1}\psi \left(mnr\right)mn}{\Gamma \left(mnr\right)}{\left({I}_{m}\left(n-m+\left(m-1\right)/r,1-1/r\right)\right)}^{r}\\ \text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}\text{ }+\frac{{C}_{\psi }{C}_{p}^{r-1}}{\Gamma \left(mnr\right)}{\frac{\partial {\left({I}_{m}\left(n-m+\left(m-1\right)/r,1-1/r\right)\right)}^{r}}{\partial r}\right]|}_{r=1}\\ =-\mathrm{ln}\left(\underset{k=1}{\overset{m}{\prod }}k!\left(n-k\right)!\right)-m\psi \left(2\right)+m\mathrm{ln}\Gamma \left(n\right)-\mathrm{ln}\Gamma \left(mn\right)-mn\psi \left(mn\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-\underset{k=n-m}{\overset{n-1}{\sum }}\left(n-k-1\right)\psi \left(k+1\right)+\underset{k=1}{\overset{m}{\sum }}k\psi \left(k+1\right)\end{array}$

4. 总结

 [1] Rathie, P.N. and Taneja, I.J. (1991) Unified (r,s)-Entropy and Its Bivariate Measures. Information Sciences, 54, 23-39. https://doi.org/10.1016/0020-0255(91)90043-T [2] Furuichi, S., Yanagi, K. and Kuriyama, K. (2004) Funda-mental Properties of Tsallis Relative Entropy. Journal of Mathematical Physics, 45, 4868-4877. https://doi.org/10.1063/1.1805729 [3] Hu, X.H. and Ye, Z.X. (2006) Generalized Quantum Entropy. Journal of Mathematical Physics, 47, 109-120. https://doi.org/10.1063/1.2165794 [4] Wang, J.M., Wu, J.D. and Cho, M. (2011) Unified (r,s)-Relative Entropy. International Journal of Theoretical Physics, 50, 1282-1295. https://doi.org/10.1007/s10773-010-0583-z [5] Luo, L.Z., Wang, J.M., Zhang, L., et al. (2016) The Differential Entropy of the Joint Distribution of Eigenvalues of Random Density Matrices. MDPI, 18, 342. https://doi.org/10.3390/e18090342