MS 460 New Methods for Multivariate Normal Moments

1. Introduction

Suppose that $X\sim {\mathcal{N}}_p(\mathbf{0},V)$, the p-dimensional normal distribution with mean $\mathbf{0}\in N^p$ and covariance V. For $N=\{0,1,2,\cdots \},n\in N^p$, and $x\in R^p$, define

$$\begin{array}{c}\hfill \left|n\right|=\sum _{j=1}^p{n}_j,n!={\Pi }_{j=1}^p{n}_j!,x^n=x_1^{n_1}\cdots x_p^{n_p},V_{r_1\cdots r_p}=E{X}_{r_1}\cdots X_{r_p},\end{array}$$

(1)

$$\begin{array}{c}\hfill {\mu }_n={\mu }_{n_1\cdots n_p}=E{X}^n=E{X}_1^{n_1}\cdots X_p^{n_p}=V_{1^{n_1}\cdots p^{n_p}}\end{array}$$

(2)

$=q_{1^{n_1}\cdots p^{n_p}}$ in the notation of Isserlis. Since $-X$ has the same distribution as X, ${\mu }_n=0$ if its order, $\left|n\right|$, is odd. Also,

$$\begin{array}{c}p=V=1\Rightarrow {\mu }_{2n}={\nu }_{2n}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\hfill \\ {\nu }_{2n}=13\cdots (2n-1)=\left(2n\right)!/(2^{n}n!)=E{N}^n\mathrm{for}X\sim {\mathcal{N}}_1(0,1).\hfill \end{array}$$

(3)

The p-variate normal distribution and its moments play a central role in the Edgeworth expansions for the distribution of the standardized vector sample mean, and more generally for a wide class of vector estimates based on a sample of size n. See Withers (1984) for $p=1$, and Withers and Nadarajah (2014) and Withers (2024) for $p>1$. The first r terms of these Edgeworth expansions need most of the multivariate Hermite polynomials and normal moments of order $3r$. The Edgeworth expansions need the moments and Hermite polynomials of $Y=V^{-1}X\sim {\mathcal{N}}_p(0,V^{-1})$, that is, with $V=\left(V_{jk}\right)$ replaced by $V^{-1}=\left(V^{jk}\right)$.

Isserlis (1918) showed that the general $2r$th moment of order $2r$ is

$$\begin{array}{c}{V}_{12\cdots ,2r}=E{X}_1{X}_2\cdots X_{2r}=\sum ^{{\nu }_{2n}}{V}_{j_1{j}_2}\cdots V_{j_{2r-1}{j}_{2r}},\hfill \end{array}$$

(4)

where summation is over all ${\nu }_{2n}$ permutations $j_1,\cdots ,j_{2r}$ of $1,2,\cdots ,2r-1,2r$ giving distinct terms. This is a special case of the formula for a multivariate moment in terms of the cumulants of a general random vector $X\in R^p$,

$$E{X}_{i_1}{X}_{i_2}\cdots X_{i_r}=B^{i_1,i_2,\cdots ,i_r}\left(\kappa \right)=\sum _{k=1}^r{B}_k^{i_1,i_2,\cdots ,i_r}\left(\kappa \right)$$

with $i_1,i_2,\cdots ,i_r$ replaced by $1,2,\cdots ,2r$ where $B^{i_1,i_2,\cdots ,i_r}\left(\kappa \right)$ and $B_k^{i_1,i_2,\cdots ,i_r}\left(\kappa \right)$ are the multivariate complete and partial exponential Bell polynomials, as given in (3.1) of Withers and Nadarajah (2013).

Here we give several methods for deriving normal moments, and a number of new results. Section 2 uses the method of successive specialisation using the step down rule (1). This soon becomes unwieldy without writing software. Section 3 summarises the results of Isserlis and Soper for bivariate moments. Section 4 gives the method of successive generalisation using quasi-differential operators. This powerful method has been used to obtain multivariate moments in terms of multivariate cumulants from the univariate formulas: see 3.29 of Stuart and Ord (1987). But it can also be used to obtain multivariate from univariate moments for parametric distributions, as we demonstrate here.

Section 5 gives the multinomial method, but in detail only for bivariate moments. Section 6 and Section 7 give for the 1st time the general moments of dimensions 3 and 4. and illustrate how it can be extended to find moments with $p>4$. Section 8 shows that these results are easily extended to moments of the multivariate normal with non-zero mean.

Jinadasa and Tracy (1986) and Tracy and Sultan (1993) and Holmquist (2007). gave central moments by matrix differentiation of $E{e}^{t^{\prime }X}=e^{t^{\prime }Vt/2}$. However the results are in terms of vec, ⊗ and permutation matrices, so make interpretation difficult. Its differentiation with respect to t can be used to give moments in terms of multivariate Bell polynomials, but again these take some effort to understand. Phillips, K. (2010) gives software in R for moments. For applications to quantum mechanics and field theory, see Simon (1974, page 9).

$$\begin{array}{c}Set{\left[r\right]}_j=r!/(r-j)!=r\cdots (r-j+1),1/k!=0\mathrm{for}k=-1,-2,\cdots \hfill \end{array}$$

(5)

2. The Method of Successive Specialisation

From (4) follows the new and more useful step down recurrence rule,

$$\begin{array}{c}E{X}_1\cdots X_{2r}={\mu }_{1^{2r}}=V_{12\cdots ,2r}=\sum _{k=1}^{2r-1}{V}_{k,2r}{V}_{12\cdots ,2r-1}^{\left(k\right)}\hfill \end{array}$$

(1)

where $V_{12\cdots ,2r-1}^{\left(k\right)}$ is $V_{12\cdots ,2r}$ with k and $2r$ removed. This gives the normal moments of order $2r$ in terms of those of order $2r-2$. For example,

$$\begin{array}{c}E{X}_1\cdots X_4={\mu }_{1111}=V_{1234}=\sum _{k=1}^3{V}_{k4}{V}_{123}^{\left(k\right)}=V_{14}{V}_{23}+V_{24}{V}_{13}+V_{34}{V}_{12},\hfill \\ E{X}_1\cdots X_6={\mu }_{1^6}=V_{12\cdots 6}=\sum _{k=1}^5{V}_{k6}{V}_{12\cdots 5}^{\left(k\right)}\hfill \\ =V_{16}{V}_{2345}+V_{26}{V}_{1345}+V_{36}{V}_{1245}+V_{46}{V}_{1235}+V_{56}{V}_{1234},\hfill \\ E{X}_1\cdots X_8={\mu }_{1^8}=V_{12\cdots 8}=\sum _{k=1}^7{V}_{k8}{V}_{12\cdots 7}^{\left(k\right)}=V_{18}{V}_{23\cdots 7}+\cdots V_{78}{V}_{12\cdots 6}.\hfill \end{array}$$

By (6) with $2r$ replaced by 1, that is, with $X_{2r}=X_1$,

$$\begin{array}{c}E{X}_1^2{X}_2\cdots X_{2r-1}=V_{1^{2}23\cdots ,2r-1}={\mu }_{21^{2r-2}}=\sum _{k=1}^{2r-1}{V}_{k1}{V}_{123\cdots ,2r-1}^{\left(k\right)}.\hfill \end{array}$$

(2)

For example taking $r=3$ then replacing 5 by 1 then 4 by 1 gives

$$\begin{array}{c}E{X}_1^2{X}_2\cdots X_5=V_{1^{2}2345}={\mu }_{21111}=V_{11}{V}_{2345}+\sum _{2345}^4{V}_{12}{V}_{1345},\hfill \end{array}$$

(3)

$$\begin{array}{c}E{X}_1^3{X}_2{X}_3{X}_4={\mu }_{3111}=V_{111234}=3V_{11}{V}_{1234}+6V_{12}{V}_{13}{V}_{14},\hfill \end{array}$$

(4)

$$\begin{array}{c}E{X}_1^4{X}_2{X}_3=V_{1^{4}23}={\mu }_{411}=3V_{11}(4V_{12}{V}_{13}+V_{11}{V}_{23}),\hfill \end{array}$$

(5)

where ${\sum }_{2345}^4$ in (3) sums over the 4 distinct terms obtained by permuting 2345. This shows that there is an error in $q_{1^{4}23}$ p139 of Isserlis (1918): his 12 should be 4. His other formulas on p139 pass the $V_{ij}\equiv 1$ test: under this condition the moments of order $2r$ are ${\nu }_{2r}$. This provides a useful check on moment formulas.

We now give all ${\mu }_{r_1{r}_2\cdots }$ of (2) of order $n=r_1+r_2+\cdots$ up to $n=8$. These are obtained from the bottom up. For example $X_7=X_1$ in (20) gives (19), and $X_7=X_1$ in (19) gives (18).

Without loss of generality, we assume that

$$\begin{array}{c}{V}_{jj}\equiv 1.\hfill \end{array}$$

(6)

To emphasize this, when $V_{jj}\equiv 1$, we set ${\rho }_{12\cdots }$ for $V_{12\cdots }$ of (1), and ${\rho }_{1^{n_1}\cdots p^{n_p}}$ for $V_{1^{n_1}\cdots p^{n_p}}$.

Moments of order 2.

$$\begin{array}{c}E{X}_1^2={\mu }_2={\rho }_{11}=1,E{X}_1{X}_2={\mu }_{11}={\rho }_{12}.\hfill \end{array}$$

Moments of order 4.

$$\begin{array}{c}E{X}_1^4={\mu }_4={\nu }_4=3,E{X}_1^3{X}_2={\mu }_{31}=3{\rho }_{12},E{X}_1^2{X}_2^2={\mu }_{22}=1+2{\rho }_{12}^2,\hfill \end{array}$$

$$\begin{array}{c}E{X}_1^2{X}_2{X}_3={\mu }_{211}={\rho }_{23}+2{\rho }_{12}{\rho }_{13},\hfill \end{array}$$

(7)

$$\begin{array}{c}E{X}_1\cdots X_4={\mu }_{1111}={\rho }_{1234}={\rho }_{12}{\rho }_{34}+{\rho }_{13}{\rho }_{24}+{\rho }_{14}{\rho }_{23}.\hfill \end{array}$$

(8)

Moments of order 6 in terms of moments of order 2,4.

$$\begin{array}{c}E{X}_1^6={\mu }_6={\nu }_6=15,E{X}_1^5{X}_2={\mu }_{51}=15{\rho }_{12},\hfill \\ E{X}_1^4{X}_2^2={\rho }_{1^4{2}^2}={\mu }_{42}=3(1+4{\rho }_{12}^2),\hfill \end{array}$$

$$\begin{array}{c}E{X}_1^4{X}_2{X}_3={\rho }_{1^{4}23}={\mu }_{411}=3({\rho }_{23}+4{\rho }_{12}{\rho }_{13})\mathrm{by}\left(5\right),\hfill \end{array}$$

(9)

$$\begin{array}{c}E{X}_1^3{X}_2^3={\rho }_{1^3{2}^3}={\mu }_{33}=3(3{\rho }_{12}+2{\rho }_{12}^3),\hfill \end{array}$$

$$\begin{array}{c}E{X}_1^3{X}_2^2{X}_3={\mu }_{321}=3(1+2{\rho }_{12}^2){\rho }_{13}+6{\rho }_{12}{\rho }_{23}\mathrm{by}\left(11\right),\hfill \end{array}$$

(10)

$$\begin{array}{c}E{X}_1^3{X}_2{X}_3{X}_4={\mu }_{3111}=3({\rho }_{12}{\rho }_{34}+{\rho }_{13}{\rho }_{24}+{\rho }_{14}{\rho }_{23}+2{\rho }_{12}{\rho }_{13}{\rho }_{14}),\hfill \end{array}$$

(11)

$$\begin{array}{c}E{X}_1^2{X}_2^2{X}_3^2={\mu }_{222}=1+2{\rho }_{12}^2+2{\rho }_{13}^2+2{\rho }_{23}^2+8{\rho }_{12}{\rho }_{13}{\rho }_{23},\hfill \end{array}$$

(12)

$$\begin{array}{c}E{X}_1^2{X}_2^2{X}_3{X}_4={\mu }_{2211}=\left({\mu }_{{21}^4}at{X}_5=X_2\right)={\rho }_{2^{2}34}+2{\rho }_{12}{\rho }_{1234}+\sum _{34}^2{\rho }_{13}{\rho }_{{12}^{2}4}\hfill \\ ={\rho }_{34}(1+2{\rho }_{12}^2)+2\sum _{12}^2{\rho }_{13}{\rho }_{14}+4{\rho }_{12}\sum _{34}^2{\rho }_{13}{\rho }_{24},\hfill \end{array}$$

(13)

$$\begin{array}{c}E{X}_1^2{X}_2\cdots X_5={\mu }_{{21}^4}={\rho }_{1^{2}2345}={\rho }_{2345}+\sum _{2345}^4{\rho }_{12}{\rho }_{1345}\mathrm{of}\left(8\right)\mathrm{by}\left(3\right)\hfill \\ =\sum _{2345}^3{\rho }_{23}{\rho }_{45}+2\sum _{2345}^6{\rho }_{12}{\rho }_{13}{\rho }_{45},\hfill \end{array}$$

(14)

$$\begin{array}{c}E{X}_1\cdots X_6={\mu }_{1^6}={\rho }_{123456}=\sum ^{15}{\rho }_{12}{\rho }_{34}{\rho }_{56}.\hfill \end{array}$$

Of these moments, ${\mu }_{3111},{\mu }_{2211},{\mu }_{{21}^4}$ are new. There is an error in ${\mu }_{411}=q_{1^{4}23}$ of Isserlis but his $q_{1^{\lambda }23}$ is correct.

Moments of order 8 in terms of lower moments. Again we use ${\nu }_{2n}$ of (3) and ${\mu }_{n_1{n}_2\cdots }={\rho }_{1^{n_1}{2}^{n_2}\cdots }=E{X}_1^{n_1}{X}_2^{n_2}\cdots .$

$$\begin{array}{c}{\mu }_8={\rho }_{1^8}={\nu }_8=105,{\mu }_{71}={\rho }_{1^{7}2}=105{\rho }_{12},{\mu }_{62}={\rho }_{1^6{2}^2}=15(1+6{\rho }_{12}^2),\hfill \end{array}$$

$$\begin{array}{c}{\mu }_{611}=({\mu }_{{51}^3}\mathrm{at}{X}_4=X_2)={\rho }_{1^{6}23}=15({\rho }_{23}+6{\rho }_{12}{\rho }_{13}),\hfill \end{array}$$

(15)

$$\begin{array}{c}{\mu }_{53}={\rho }_{1^5{2}^3}=15(3{\rho }_{12}+4{\rho }_{12}^3),\hfill \end{array}$$

$$\begin{array}{c}{\mu }_{521}=({\mu }_{51^3}\mathrm{at}{X}_4=X_2)={\rho }_{1^5{2}^{2}3}=15({\rho }_{13}+2{\rho }_{12}{\rho }_{23}+4{\rho }_{12}^2{\rho }_{13}),\hfill \end{array}$$

(16)

$$\begin{array}{c}{\mu }_{{51}^3}=({\mu }_{41^4}\mathrm{at}{X}_5=X_1)={\rho }_{1^{5}234}=4{\rho }_{1^{3}234}+\sum _{234}^3{\rho }_{12}{\rho }_{1^{4}34}\mathrm{of}\left(9\right),\left(11\right),\left(13\right),\hfill \\ {\mu }_{44}={\rho }_{1^4{2}^4}=({\mu }_{431}\mathrm{at}{X}_3=X_2)=3(3+24{\rho }_{12}^2+8{\rho }_{12}^4),\hfill \\ {\mu }_{431}={\rho }_{1^4{2}^{3}3}=({\mu }_{4211}\mathrm{at}{X}_4=X_2)\hfill \\ =3{\rho }_{1^2{2}^{3}3}+3{\rho }_{12}{\rho }_{1^3{2}^{2}3}+{\rho }_{13}{\rho }_{1^3{2}^3}\mathrm{of}\left(26\right)\mathrm{and}\left(3\right)\hfill \\ =9{\rho }_{12}(1+4{\rho }_{13}+4{\rho }_{12}{\rho }_{23})+24{\rho }_{12}^3{\rho }_{13}\mathrm{by}\left(29\right),\hfill \\ {\mu }_{422}=({\mu }_{4211}\mathrm{at}{X}_4=X_2)=3{\rho }_{1^2{2}^{3}3}+3{\rho }_{12}{\rho }_{1^3{2}^{2}3}+{\rho }_{13}{\rho }_{1^3{2}^3}.\mathrm{See}\left(28\right).\hfill \end{array}$$

$$\begin{array}{c}{\mu }_{4211}={\rho }_{1^4{2}^{2}34}=({\mu }_{{41}^4}\mathrm{at}{X}_5=X_2)=3{\rho }_{1^2{2}^{2}34}+2{\rho }_{12}{\rho }_{1^{3}234}\hfill \\ +\sum _{34}^2{\rho }_{13}{\rho }_{1^3{2}^{2}4}\mathrm{of}\left(13\right),\left(11\right)\mathrm{and}\left(3\right):\mathrm{see}\left(15\right),\left(10\right);\hfill \end{array}$$

$$\begin{array}{c}{\mu }_{{41}^4}={\rho }_{1^{4}2345}=({\mu }_{{31}^5}\mathrm{at}{X}_6=X_1)\hfill \\ =3{\rho }_{112345}+\sum _{2345}^4{\rho }_{12}{\rho }_{1^{3}345}\mathrm{of}\left(14\right)\mathrm{and}\left(11\right),\hfill \end{array}$$

(17)

$$\begin{array}{c}{\mu }_{332}={\rho }_{1^3{2}^3{3}^2}=({\mu }_{3311}\mathrm{at}{X}_4=X_3)=2{\rho }_{{12}^3{3}^2}+3{\rho }_{12}{\rho }_{1^2{2}^2{3}^2}+2{\rho }_{13}{\rho }_{1^2{2}^{3}3},\hfill \\ {\mu }_{3311}={\rho }_{1^3{2}^{3}34}=({\mu }_{32111}\mathrm{at}{X}_5=X_2)=2{\rho }_{12^{3}34}+3{\rho }_{12}{\rho }_{1^2{2}^{2}34}+\sum _{34}^2{\rho }_{13}{\rho }_{1^2{2}^{3}4}:\hfill \\ \mathrm{see}\left(17\right).\hfill \\ {\mu }_{3221}={\rho }_{1^3{2}^2{3}^{2}4}=({\mu }_{321^3}\mathrm{at}{X}_5=X_4)=2{\rho }_{{12}^{2}34^2}+2{\rho }_{12}{\rho }_{1^{2}234^2}+{\rho }_{13}{\rho }_{1^2{2}^2{4}^2}\hfill \\ +2{\rho }_{14}{\rho }_{1^2{2}^{2}34}:\mathrm{See}\left(20\right).\hfill \\ {\mu }_{32111}={\rho }_{1^3{2}^{2}345}=({\mu }_{{31}^5}\mathrm{at}{X}_6=X_2)=2{\rho }_{12^{2}345}+2{\rho }_{12}{\rho }_{1^{2}2345}+\sum _{345}^3{\rho }_{13}{\rho }_{1^2{2}^{2}45},\hfill \end{array}$$

$$\begin{array}{c}{\mu }_{{31}^5}=({\mu }_{21^6}\mathrm{at}{X}_7=X_1)={\rho }_{1^{3}2\cdots 6}=2{\rho }_{1\cdots 6}+\sum _{2\cdots 6}^5{\rho }_{12}{\rho }_{113456},\hfill \end{array}$$

(18)

$$\begin{array}{c}{\mu }_{2^4}={\rho }_{1^2{2}^2{3}^2{4}^2}=({\mu }_{2^3{1}^2}\mathrm{at}{X}_5=X_4)={\rho }_{3^2{4}^2}+2\sum _{12}^2\sum _{12}^2{\rho }_{1344}+2{\rho }_{12}{\rho }_{123^2{4}^2}\hfill \\ +2{\rho }_{14}({\rho }_{24}+2{\rho }_{23}{\rho }_{34})+2{\rho }_{13}{\rho }_{2344}+2{\rho }_{12}\sum _{34}^2{\rho }_{13}{\rho }_{2344},\hfill \\ {\mu }_{2^3{1}^2}={\rho }_{1^2{2}^2{3}^{2}45}=({\mu }_{2^2{1}^4}\mathrm{at}{X}_6=X_3)\hfill \\ ={\rho }_{3^{2}45}+\sum _{12}^2(2{\rho }_{13}{\rho }_{1345}+\sum _{45}^2{\rho }_{14}{\rho }_{1335})+2{\rho }_{12}{\rho }_{123^{2}45}+\sum _{45}^2{\rho }_{14}({\rho }_{25}+2{\rho }_{23}{\rho }_{35})\hfill \\ +2{\rho }_{13}{\rho }_{2345}+{\rho }_{12}(2{\rho }_{13}{\rho }_{2345}+\sum _{45}^2{\rho }_{14}{\rho }_{2335}),\hfill \\ {\mu }_{2^2{1}^4}={\rho }_{1^2{2}^{2}3\cdots 6}=({\mu }_{{21}^6}\mathrm{at}{X}_7=X_2)={\rho }_{2^{2}3\cdots 6}+2{\rho }_{12}{\rho }_{1\cdots 6}+\sum _{3\cdots 6}^4{\rho }_{13}({\rho }_{1456}\hfill \\ +\sum _{1456}^4{\rho }_{12}{\rho }_{2456})={\rho }_{3\cdots 6}+\sum _{12}^2\sum _{3\cdots 6}^4{\rho }_{13}{\rho }_{1456}+2{\rho }_{12}{\rho }_{1\cdots 6}+\sum _{3\cdots 6}^{12}{\rho }_{13}{\rho }_{24}{\rho }_{1256}+{\rho }_{12}{\alpha }_2\hfill \\ \mathrm{where}{\alpha }_2=\sum _{3\cdots 6}^4{\rho }_{13}{\rho }_{2456}=\sum _{3\cdots 6}^{12}{\rho }_{13}{\rho }_{24}{\rho }_{56},\hfill \end{array}$$

$$\begin{array}{c}{\mu }_{{21}^6}={\rho }_{1^{2}234567}={\rho }_{234567}+\sum _{2\cdots 7}^6{\rho }_{12}{\rho }_{134567}\mathrm{by}\left(2\right),\hfill \end{array}$$

(19)

$$\begin{array}{c}{\mu }_{1^8}=\sum ^{105}{\rho }_{12}{\rho }_{34}{\rho }_{56}{\rho }_{78}=\sum _{k=1}^7{\rho }_{k8}{\rho }_{12\cdots 7}^k={\rho }_{18}{\rho }_{2\cdots 7}+\cdots +{\rho }_{78}{\rho }_{12\cdots 6},\hfill \end{array}$$

(20)

where ${\rho }_{12\cdots 7}^k$ is ${\rho }_{12\cdots 7}$ with k removed. These iterative expressions for ${\mu }_{{51}^3},{\mu }_{{41}^4},{\mu }_{{31}^5},$${\mu }_{2^2{1}^4},{\mu }_{{21}^6}$ are new. However without software, it is easy to make an error with these substitutions. For ${\mu }_{3221}$, see (20).

NOTE 2.1.

This relation giving moments in terms of the covariance is a special case of the following. For any random variable $X\in R$ with rth cumulant ${\kappa }_r,r\ge 1,$ its rth moment is given by

$$m_r=B_r\left(\kappa \right)=\sum _{k=1}^r{B}_{rk}\left(\kappa \right),r\ge 1,where\kappa =({\kappa }_1,{\kappa }_2,\cdots )$$

where $B_{rk}\left(\kappa \right)$ is the partial exponential Bell polynomial defined by (4) below. For any random vector $X\in R^p$ with rth cumulant ${\kappa }_r,r\in N^p,$ this becomes

$$m_r=B_r\left(\kappa \right)=\sum _{k=1}^{\left|r\right|}{B}_{rk}\left(\kappa \right)where\left|r\right|=\sum _{k=1}^p{r}_k$$

and $B_{rk}\left(\kappa \right)$ is the multivariate partial exponential Bell polynomial. It may be written down from the univariate form. See Withers and Nadarajah (2012, 2013, 2014).

3. Bivariate Moments of Soper and Isserlis

Soper (1915-16) and Isserlis (1918) gave the general bivariate normal moment with correlation $\rho ={\rho }_{12}$ when $V_{jj}\equiv 1$ in terms of ${\nu }_{2n}$ of (3), as

$$\begin{array}{cc}{\mu }_{rs}\hfill & =E{X}_1^r{X}_2^s=\sum _{0\le k\le s/2}{m}_{rsk}{(1-{\rho }^2)}^k{\rho }^{s-2k},m_{rsk}=\left({\displaystyle \genfrac{}{}{0pt}{}{s}{2k}}\right){\nu }_{2k}{\nu }_{r+s-2k}.\hfill \end{array}$$

(1)

Putting $s=0,1,2,3,4,5$ then expanding ${(1-{\rho }^2)}^k$ and simplifying, gives

$$\begin{array}{cc}{\mu }_{r0}\hfill & =E{X}_1^r={\nu }_r,{\mu }_{r1}=E{X}_1^r{X}_2={\nu }_{r+1}\rho ,\hfill \end{array}$$

(2)

$$\begin{array}{cc}{\mu }_{r2}\hfill & =E{X}_1^r{X}_2^2={\nu }_r(1+r{\rho }^2),{\mu }_{r3}=E{X}_1^r{X}_2^3={\nu }_{r+1}[3\rho +(r-1){\rho }^3],\hfill \end{array}$$

(3)

$$\begin{array}{cc}{\mu }_{r4}\hfill & =E{X}_1^r{X}_2^4={\nu }_r[3+6r{\rho }^2+r(r-2){\rho }^4],\hfill \end{array}$$

(4)

$$\begin{array}{cc}{\mu }_{r5}\hfill & =E{X}_1^r{X}_2^5={\nu }_{r+1}[15\rho +10(r-1){\rho }^3+(r-1)(r-3){\rho }^5].\hfill \end{array}$$

(5)

Special cases are

$$\begin{array}{c}{\mu }_{11}=\rho ,{\mu }_{31}=3\rho ,{\mu }_{51}=15\rho ,{\mu }_{71}=105\rho ,{\mu }_{91}=945\rho ,{\mu }_{11,1}=10395\rho .\hfill \\ {\mu }_{22}=1+2{\rho }^2,{\mu }_{42}/3=1+4{\rho }^2,{\mu }_{62}/15=1+6{\rho }^2,{\mu }_{82}/105=1+8{\rho }^2,\hfill \\ {\mu }_{10,2}/945=1+10{\rho }^2,{\mu }_{13}=3\rho ,{\mu }_{33}/3=3\rho +2{\rho }^3,{\mu }_{53}/15=3\rho +4{\rho }^3,\hfill \\ {\mu }_{73}/105=3\rho +6{\rho }^3,{\mu }_{93}/945=3\rho +8{\rho }^3.\hfill \\ {\mu }_{04}=3,{\mu }_{24}=3+12{\rho }^2,{\mu }_{44}/3=3+24{\rho }^2+8{\rho }^4,\hfill \\ {\mu }_{64}/15=3(1+12{\rho }^2+8{\rho }^4),{\mu }_{84}/105=3(1+16{\rho }^2+16{\rho }^4).\hfill \end{array}$$

By swapping r and s, this includes all bivariate moments up to order 10. (1) is not a symmetric rule. Setting $Y_1=X_2$ and $Y_2=X_1$ we see that (6) $\Rightarrow {\mu }_{rs}={\mu }_{sr}$ giving its dual

$$\begin{array}{cc}{\mu }_{rs}\hfill & =\sum _{0\le k\le r/2}{m}_{srk}{(1-{\rho }^2)}^k{\rho }^{r-2k}.\hfill \end{array}$$

(6)

This has about $r/2$ terms while (1) has about $s/2$ terms if $s>r$. The same is true if we expand ${(1-{\rho }^2)}^k$ to write (1) as

$$\begin{array}{cc}{\mu }_{rs}\hfill & =\sum _{-1\le n\le s/2}{M}_{rsn}{\rho }^{s-2n}\mathrm{where}\hfill \\ M_{rsn}\hfill & =\sum _{0\le i\le s/2}{\left[m_{rsk}{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k+1}{i+1}}\right)\right]}_{k=n+i+1}.\hfill \end{array}$$

(7)

The method becomes increasingly cumbersome as s increases. In §4 we give a new and simpler formula for the bivariate moment ${\mu }_{rs}$.

4. Successive Generalisation Using Quasi-Differential Operators

An alternative to deriving particular cases from the general moment, is to derive bivariate moments from univariate moments, then trivariate moments from bivariate moments, and so on, by the method of 3.29 of Stuart and Ord (1987), given there to obtain multivariate moments in terms of multivariate cumulants. But equally well it applies to relations among moments. To increase consistency with the notation there, we set

$$\begin{array}{c}\mu \left(ij\right)=V_{ij},\mu (1^{n_1}\cdots p^{n_p})=E{X}_1^{n_1}\cdots X_p^{n_p}={\mu }_n\mathrm{of}\left(2\right).\hfill \end{array}$$

(1)

We do not assume that $V_{jj}\equiv 1$. Define the operator $\partial (1.2)$ by

$$\begin{array}{c}\partial \left(1.2\right)\mu (1^r{2}^s{3}^t\cdots )=r\mu (1^{r-1}{2}^{s+1}{3}^t\cdots ).\hfill \\ So\partial {\left(1.2\right)}^k\mu (1^r{2}^s\cdots )={\left[r\right]}_k\mu (1^{r-k}{2}^{s+k}\cdots ).\hfill \end{array}$$

(2)

We show how to obtain ${\mu }_{n_1\cdots n_{p+1}}$ by applying $\partial {(j.p+1)}^m$ to ${\mu }_{n_1\cdots n_p}$ for any $j=1,\cdots ,p$. Applying $\partial (1.2)$ to

$$\mu \left(1^{2r}\right)=E{X}_1^{2r}={\nu }_{2r}\mu {\left(1^2\right)}^r,$$

gives

$$\begin{array}{c}2r\mu \left(1^{2r-1}2\right)={\nu }_{2r}r\mu {\left(1^2\right)}^{r-1}2\mu \left(12\right),\hfill \end{array}$$

(3)

or dividing by $2r$,

$$\mu \left(1^{2r-1}2\right)={\nu }_{2r}\mu {\left(1^2\right)}^{r-1}\mu \left(12\right),thatis,E{X}_1^{2r-1}{X}_2={\nu }_{2r}{V}_{11}^{r-1}{V}_{12},$$

proving (2). Applying this operator a 2nd, 3rd and 4th time gives (3)–(5). We can apply the operator k times using Faa di Bruno’s rule for differentiating $h\left(x\right)=f\left(g\right(x\left)\right)$, [4c] p137 of Comtet (1974): for $k\ge 0$, its kth derivative is

$$h_{.k}\left(x\right)=\sum _{j=0}^k{B}_{kj}{f}_{.j}\left(g\left(x\right)\right)$$

where $B_{kj}=B_{kj}\left(g\right)$ is the partial exponential Bell polynomial in $g=(g_1,g_2,\cdots )$ and $g_k=g_{.k}\left(x\right)$. Comtet tables them on p307–308. $B_{kj}\left(g\right)$ is defined by

$$\begin{array}{c}{(\sum _{k=1}^{\infty }{g}_k{t}^k/k!)}^j/j!=\sum _{k=j}^{\infty }{B}_{kj}{t}^k/k!.\mathrm{So}{B}_{k0}=I(k=0).\hfill \end{array}$$

(4)

We apply this with

$$\begin{array}{c}d/dx=\partial \left(1.2\right),g\left(x\right)=\mu \left(1^2\right),f\left(g\right)=g^r,h\left(x\right)=\mu {\left(1^2\right)}^r.\hfill \\ \mathrm{So}{f}_{.j}\left(g\left(x\right)\right)={\left[r\right]}_j\mu {\left(1^2\right)}^{r-j},g_1=2\mu \left(12\right),g_2=2\mu \left(2^2\right),g_k=0\mathrm{for}k\ge 3,\hfill \end{array}$$

and $B_{kj}/k!$ is the coefficient of $t^k$ in ${[2\mu \left(12\right)t+\mu \left(2^2\right)t^2]}^j/j!$:

$$\begin{array}{c}{B}_{kj}={\left[k\right]}_j{\left[2\mu \left(12\right)\right]}^{2j-k}\mu {\left(2^2\right)}^{k-j}/(2j-k)!,k/2\le j\le k.\hfill \end{array}$$

So by (2) we obtain for $k\ge 1$,

$$\begin{array}{c}\mu \left(1^{2r-k}{2}^k\right)2^{r}r!/(2r-k)!={\left[2r\right]}_k\mu \left(1^{2r-k}{2}^k\right)/{\nu }_{2r}\hfill \end{array}$$

$$\begin{array}{c}=\partial {\left(1.2\right)}^k\mu \left(1^{2r}\right)/{\nu }_{2r}=\partial {\left(1.2\right)}^k\mu {\left(1^2\right)}^r=\sum _{j=1}^k{B}_{kj}{\left[r\right]}_j\mu {\left(1^2\right)}^{r-j}\hfill \end{array}$$

(5)

$$\begin{array}{c}=\sum _{k/2\le j\le min(k,r)}{\left[r\right]}_j{\left[k\right]}_j\mu {\left(1^2\right)}^{r-j}{\left[2\mu \left(12\right)\right]}^{2j-k}\mu {\left(2^2\right)}^{k-j}/(2j-k)!=H_{rk}^{12},\hfill \end{array}$$

(6)

say. Dividing by $r!k!$ gives

Theorem 1.

At $s=2r-k$, ${\mu }_{sk}=\mu \left(1^s{2}^k\right)=E{X}_1^s{X}_2^k$ is given by

$$\begin{array}{c}{\mu }_{sk}{2}^r/s!k!=\sum _{k/2\le j\le min(k,r)}{B}_{r-j}{C}_{2j-k}{D}_{k-j}\mathrm{at}2r=s+kwhere\hfill \end{array}$$

(7)

$$\begin{array}{c}{B}_r=\mu {\left(1^2\right)}^r/r!,C_r=C_r^{12}={\left[2\mu \left(12\right)\right]}^r/r!,D_r=\mu {\left(2^2\right)}^r/r!,\hfill \\ \mu \left(j^2\right)=V_{jj}={\sigma }_j^{2}say,\mu \left(12\right)=V_{12}={\rho }_{12}{\sigma }_1{\sigma }_2.\hfill \end{array}$$

(8)

This is our 1st new formula for the bivariate normal moments.

Corollary 1.

At $s=2r-k$, if $V_{jj}\equiv 1$, then ${\mu }_{sk}/{\nu }_{2r}=H_{rk}^{12}/{\left[2r\right]}_k$

$$\begin{array}{c}where{H}_{rk}^{12}=\sum _{k/2\le j\le min(k,r)}{\left[k\right]}_j{\left[r\right]}_j{\left(2{\rho }_{12}\right)}^{2j-k}/(2j-k)!.\hfill \end{array}$$

(9)

$$\begin{array}{c}So,{\mu }_{sk}{2}^r/s!=\sum _{k/2\le j\le min(k,r)}{\left[k\right]}_j{\left(2{\rho }_{12}\right)}^{2j-k}/(2j-k)!(r-j)!.\hfill \end{array}$$

(10)

$$\begin{array}{c}So,H_{r0}^{12}=1,H_{r1}^{12}=2r{\rho }_{12},H_{r2}^{12}=2r+4{\left[r\right]}_2{\rho }_{12}^2,\hfill \\ H_{r3}^{12}=12{\left[r\right]}_2{\rho }_{12}+8{\left[r\right]}_3{\rho }_{12}^3,H_{r4}^{12}/4=3{\left[r\right]}_2+12{\left[r\right]}_3{\rho }_{12}^2+4{\left[r\right]}_4{\rho }_{12}^4.\hfill \end{array}$$

(11)

Swapping s and k in (7) and (10) give equivalent formulas. (10) is simpler than Soper and Isserlis’s (1) as that is a polynomial in both ${\rho }_{12}$ and $1-{\rho }_{12}^2$.

Putting ${\rho }_{12}=1$ in Corollary 4.1 gives the new identity ${\left[H_{rk}^{12}\right]}_{{\rho }_{12}=1}=$

$$\begin{array}{c}\sum _{k/2\le j\le min(k,r)}{\left[k\right]}_j{\left[r\right]}_j{2}^{2j-k}/(2j-k)!={\left[2r\right]}_k\mathrm{for}0\le k\le 2r.\hfill \end{array}$$

(12)

5. The Multinomial Method

We now obtain (2)–(5) by another new method. Given

$$\begin{array}{c}t\in R^p,t^{\prime }X\sim {\mathcal{N}}_1(0,v_t)\mathrm{where}{v}_t=t^{\prime }Vt,\mathrm{so}E{\left(t^{\prime }X\right)}^{2r}={\nu }_{2r}{v}_t^r.\hfill \end{array}$$

(1)

By the multinomial theorem for $\left|n\right|$ and $n!$ of (1),

$$\begin{array}{c}{\left(t^{\prime }X\right)}^{2r}=\sum _{\left|n\right|=2r}\left({\displaystyle \genfrac{}{}{0pt}{}{2r}{n}}\right)t^n{X}^n\mathrm{where}\left({\displaystyle \genfrac{}{}{0pt}{}{2r}{n}}\right)=\left(2r\right)!/n!.\hfill \\ So,{\nu }_{2r}{v}_t^r=E{\left(t^{\prime }X\right)}^{2r}=\sum _{\left|n\right|=2r}\left({\displaystyle \genfrac{}{}{0pt}{}{2r}{n}}\right)t^n{\mu }_n,\mathrm{and}{\mu }_n=E_n{F}_n\mathrm{where}\hfill \end{array}$$

$$\begin{array}{c}{E}_n={\nu }_{2r}/\left({\displaystyle \genfrac{}{}{0pt}{}{2r}{n}}\right)=n!/2^{r}r!,\mathrm{and}{F}_n=coeff\left(t^n\right)\mathrm{in}{v}_t^r,\left|n\right|=2r,\hfill \end{array}$$

(2)

$F_n$ can be obtained from a 2nd application of the multinomial theorem. Suppose that $V_{jj}\equiv 1$. So

$$\begin{array}{c}{\rho }_{jk}=E{X}_j{X}_k\mathrm{and}{v}_t=2\sum _{1\le j<k\le p}{\rho }_{jk}{t}_j{t}_k+\sum _{j=1}^p{t}_j^2.\hfill \end{array}$$

(3)

For $c=\left(c_{jk}\right)$ a symmetric $p\times p$ matrix of integers in N and $a\in N^p$, set

$${\rho }^c={\Pi }_{1\le j<k\le p}{\rho }_{jk}^{c_{jk}},c_j=\sum _{k\ne j}{c}_{jk}.$$

By (3), for $a,c\in N^p$, the coefficient of $t^n$ in $v_t^r$ is

$$\begin{array}{c}{F}_n=\sum \{\left({\displaystyle \genfrac{}{}{0pt}{}{r}{ac}}\right)2^{\left|c\right|}{\rho }^c:2a_j+c_j=n_k,k=1,\cdots ,p\},\left|n\right|=2r\hfill \end{array}$$

(4)

where $\left({\displaystyle \genfrac{}{}{0pt}{}{r}{ac}}\right)=r!/a!c!,a!={\Pi }_{j=1}^p{a}_j!$ and $c!={\Pi }_{1\le j<k\le p}{c}_{jk}!$.

The multinomial method for bivariate moments.

Set $\rho ={\rho }_{12},c=c_{12}$. So $v_t=2\rho t_1{t}_2+t_1^2+t_2^2$ and the coefficient of $t^n$ in $v_t^r$ needed for (2) is

$$\begin{array}{cc}\hfill & {\mu }_n/E_n=2^{r}r!{\mu }_n/n!=F_n\hfill \\ \hfill & =\sum \{\left({\displaystyle \genfrac{}{}{0pt}{}{r}{c{a}_1{a}_2}}\right){\left(2\rho \right)}^c:c+2a_1=n_1,c+2a_2=n_2\}\mathrm{where}\hfill \\ \hfill & 2r=\left|n\right|,\mathrm{and}\left({\displaystyle \genfrac{}{}{0pt}{}{r}{c{a}_1{a}_2}}\right)=r!/(c!a_1!a_2!).\hfill \end{array}$$

(5)

Taking $n=(2r-i,i)$ for $i\le 5$ gives (2)–(5).

Example 1.

Take $n=(2r-5,5).$ Then $(c,a_2)=(1,2)$, $(3,1)$ or $(5,0)$. So $F_n$ sums over $(c,a_1,a_2)=(1,r-3,2),(3,r-4,1),(5,r-5,0)$ giving

$$\begin{array}{cc}\hfill & F_n=\sum _{j=3}^5{\left[r\right]}_j{\alpha }_{j}where{\alpha }_3=\rho ,{\alpha }_4=4{\rho }^3/3,{\alpha }_5=4{\rho }^5/15.\hfill \end{array}$$

So since $E_n=(2r-5)!5!/2^{r}r!$, by (5), $r=3,4,5,6$ give

$$\begin{array}{c}{\mu }_{15}=15\rho ,{\mu }_{35}=8(3\rho +4{\rho }^3),{\mu }_{55}=15(15\rho +40{\rho }^3+8{\rho }^5),\hfill \\ {\mu }_{75}=105(5\rho +20{\rho }^3+8{\rho }^5).\hfill \end{array}$$

Example 2.

Take $n=(2r-6,6).$ Then $(c,a_2)=(0,3),(2,2),(4,1)$ or $(6,0)$. So $F_n$ sums over $(c,a_1,a_2)=(c,a_2)=(0,r-3,3),(2,r-4,2),(4,r-5,1),(6,r-6,0)$:

$$\begin{array}{cc}\hfill & F_n=\sum _{j=3}^6{\left[r\right]}_j{\alpha }_{j}where{\alpha }_3=1/6,{\alpha }_4={\rho }^2,{\alpha }_5={\rho }^4/3,{\alpha }_6=4{\rho }^6/45.\hfill \end{array}$$

By (5), $E_n=(2r-6)!6!/2^{r}r!$, so $r=3,4,5,6,7$ give ${\mu }_{06},{\mu }_{26},{\mu }_{46}$ above, and

$$\begin{array}{c}{\mu }_{66}=45(5+90{\rho }^2+120{\rho }^4+16{\rho }^6),{\mu }_{86}=5.7.9(5+120{\rho }^2+240{\rho }^4+64{\rho }^6).\hfill \end{array}$$

Moments with $p>2$ can be dealt with similarly, but the operator method of §6 is easier.

6. Trivariate Moments

Here we give ${\mu }_{rst}$ for arbitrary $r,s,t$. Define $B_r,C_r,D_r$ as in (8). Assuming $V_{jj}\equiv 1$, Isserlis (1918) gave (12) and the following 2 trivariate moments.

$q_I$ is his notation.

$$\begin{array}{c}{\mu }_{2r,11}=E{X}_1^{2r}{X}_2{X}_3=q_{1^{2r}23}={\nu }_{2r}({\rho }_{23}+2r{\rho }_{12}{\rho }_{13}),\hfill \end{array}$$

(1)

$$\begin{array}{c}{\mu }_{2r-1,21}=q_{1^{2r-1}{2}^{2}3}={\nu }_{2r}[{\rho }_{13}+2{\rho }_{12}{\rho }_{23}+2(r-1){\rho }_{12}^2{\rho }_{13}]\hfill \\ ={\rho }_{13}{a}_{2r-1,2}+{\rho }_{23}{b}_{2r-1,2}\mathrm{where}{a}_{2r-1,2}={\nu }_{2r}[1+2(r-1){\rho }_{12}^2],b_{2r-1,2}=2{\nu }_{2r}{\rho }_{12}.\hfill \end{array}$$

(2)

So ${\mu }_{211},{\mu }_{411},{\mu }_{611},{\mu }_{321},{\mu }_{521}$ are given by (7), (5), (15), (10), (16).

Trivariate moments using the differential operator method.

We now extend the operator $\partial (1.2)$ of (2) to $\partial (i.j)$. So

$$\partial \left(1.3\right)\mu (1^r{2}^s{3}^t{4}^u\cdots )=r\mu (1^{r-1}{2}^s{3}^{t+1}{4}^u\cdots ).$$

That is, $\partial (1.3){\mu }_{rstu\cdots }=r{\mu }_{r-1,s,t+1,u\cdots }$. By (2), for $B_r,C_r,D_r$ of (8),

$$\begin{array}{c}\partial \left(1.3\right)B_r=2B_{r-1}\mu \left(13\right),\partial \left(1.3\right)C_r=2C_{r-1}\mu \left(23\right).\hfill \\ So,\partial \left(1.3\right)\mu {\left(1^2\right)}^r=2r\mu {\left(1^2\right)}^{r-1}\mu \left(13\right),\hfill \end{array}$$

(3)

$$\begin{array}{c}\partial \left(1.3\right)\mu {\left(12\right)}^r=r\mu {\left(12\right)}^{r-1}\mu \left(23\right),\partial {\left(1.3\right)}^e\mu {\left(12\right)}^r={\left[r\right]}_e\mu {\left(12\right)}^{r-e}\mu {\left(23\right)}^e.\hfill \end{array}$$

(4)

Applying $\partial (1.3)$ to (7) gives

Theorem 2.

For $2r=s+k+1,{\mu }_{sk1}=\mu \left(1^s{2}^{k}3\right)$ is given by

$$\begin{array}{cc}\hfill & {\mu }_{sk1}{2}^{r-1}/s!k!=\sum {[B_{r-j-1}{C}_J\mu \left(13\right)+B_{r-j}{C}_{J-1}\mu \left(23\right)]}_{J=2j-k}{D}_{k-j}\hfill \end{array}$$

(5)

summed over $k/2\le j\le min(k,r)$.

For example (5) with $k=0$ and (11) with $k=1$ reduce to (2) with r changed to $2r-1$:

$${\mu }_{2r-1,1}=\mu \left(1^{2r-1}2\right)={\nu }_{2r}\mu {\left(1^2\right)}^{r-1}\mu \left(12\right).$$

Corollary 2.

When $V_{jj}\equiv 1$, for $2r=s+k+1,$

$$\begin{array}{cc}\hfill & {\mu }_{sk1}{2}^{r-1}/s!=\sum _j{\left[k\right]}_j{2}^J[R{\rho }_{13}{\rho }_{12}^J+J{\rho }_{23}{\rho }_{12}^{J-1}/2]/J!R!\hfill \end{array}$$

(6)

summed over $k/2\le j\le min(k,r)$ at $J=2j-k,R=r-j$.

For $k=1,2$ (6) gives Isserlis’s (1), (2) but otherwise (6) is new.

Corollary 3.

$$\begin{array}{c}{\mu }_{sk1}={\rho }_{13}{a}_{sk}+{\rho }_{23}{b}_{sk}where\hfill \end{array}$$

(7)

$$\begin{array}{c}{a}_{2r,3}/{\nu }_{2r}=3(1+2r{\rho }^2),b_{2r,3}/{\nu }_{2r}=2r[3\rho +2(r-1){\rho }^3].\hfill \\ Sofor{\mu }_{211},a_{21}=2\rho ,b_{21}=1;for{\mu }_{321},a_{32}/3=1+2{\rho }^2,b_{32}/3=2\rho ;\hfill \end{array}$$

(8)

$$\begin{array}{c}for{\mu }_{431},a_{43}/3=4(3\rho +2{\rho }^3),b_{43}/3=1+4{\rho }^2;\hfill \end{array}$$

(9)

$$\begin{array}{c}for{\mu }_{541},a_{54}/15=3+24{\rho }^2+8{\rho }^4,b_{54}/15=4(3\rho +4{\rho }^3);\hfill \\ for{\mu }_{631},a_{63}/15=3(1+6{\rho }^2),b_{63}/15=6(3\rho +4{\rho }^3);\hfill \\ for{\mu }_{651},a_{65}=90(15\rho +40{\rho }^3+8{\rho }^5),b_{65}=225(1+12{\rho }^2+8{\rho }^4);\hfill \\ for{\mu }_{741},a_{74}/105=3(1+12{\rho }^2+8{\rho }^4),b_{74}/105=12(\rho +2{\rho }^3);\hfill \\ for{\mu }_{831},a_{83}/{\nu }_8=3(1+8{\rho }^2),b_{83}/{\nu }_8=24(\rho +2{\rho }^3).\hfill \end{array}$$

By (2),

$$\begin{array}{c}\partial \left(2.3\right)\mu \left(1^r{2}^s\right)=s\mu \left(1^r{2}^{s-1}3\right),\mathrm{that}\mathrm{is},\partial \left(2.3\right){\mu }_{rs}=s{\mu }_{r,s-1,1},\hfill \\ \mathrm{so}\partial \left(2.3\right)C_r=2C_{r-1}\mu \left(13\right),\partial \left(2.3\right)D_r=2D_{r-1}\mu \left(23\right).\hfill \end{array}$$

(10)

Applying $\partial (2.3)$ to (7) gives an alternative to Theorem 6.1:

Theorem 3.

For $2r=s+k,{\mu }_{s,k-1,1}=\mu \left(1^s{2}^{k-1}3\right)$ is given by

$$\begin{array}{cc}\hfill & {\mu }_{s,k-1,1}{2}^{r-1}/s!(k-1)!\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\sum _{k/2\le j\le min(k,r)}{B}_{r-j}[C_{J-1}{D}_K\mu \left(13\right)+C_J{D}_{K-1}\mu \left(23\right)]\hfill \end{array}$$

(11)

$$\begin{array}{c}atJ=2j-k,K=k-j.\hfill \end{array}$$

(12)

Corollary 4.

When $V_{jj}\equiv 1$, for $2r=s+k$, and $J,K$ of (12),

$$\begin{array}{c}{\mu }_{s,k-1,1}{2}^{r-1}/s!(k-1)!\hfill \\ =\sum _{k/2\le j\le min(k,r)}[J{\rho }_{13}{\left(2{\rho }_{12}\right)}^{J-1}+K{\rho }_{23}{\left(2{\rho }_{12}\right)}^J]/J!K!(r-j)!.\hfill \end{array}$$

By (10), applying $\partial (2.3)$ to (11) gives

Theorem 4.

For $2r=s+k$, and $J,K$ of (12),

$$\begin{array}{c}{\mu }_{s,k-2,2}{2}^{r-1}/s!(k-2)!=\sum _{k/2\le j\le min(k,r)}{B}_{r-j}[2C_{J-2}{D}_K\mu {\left(13\right)}^2\hfill \\ +C_{J-1}{D}_{K-1}4\mu \left(13\right)\mu \left(23\right)+C_J{D}_{K-2}2\mu {\left(23\right)}^2+C_J{D}_{K-1}\mu \left(33\right)].\hfill \end{array}$$

(13)

Corollary 5.

When $V_{jj}\equiv 1$, for $2r=s+k$ and $J,K$ of (12),

$$\begin{array}{c}{\mu }_{s,k-2,2}{2}^{r-1}/s!(k-2)!=\sum _{k/2\le j\le k}[2{\left(2{\rho }_{12}\right)}^{J-2}{\rho }_{13}^2/(J-2)!K!\hfill \\ +4{\rho }_{13}{\rho }_{23}{\left(2{\rho }_{12}\right)}^{J-1}/(J-1)!(K-1)!\hfill \\ +\{1/(K-1)!+2{\rho }_{23}^2/(K-2)!\}{\left(2{\rho }_{12}\right)}^J/J!]/(r-j)!.\hfill \end{array}$$

(14)

Here are some examples. $k=2$ gives the 1st equation in (3) with r changed to $2r-2$ and $X_2$ replaced by $X_3$. (13) with $k=3$ gives (2) with $\lambda$ changed to $2r-3$ and $X_2$ replaced by $X_3$. (14) with $k=4$ gives for $2r=s+4$,

$$\begin{array}{c}{\mu }_{2r-4,22}={\nu }_{2r-4}{A}_r\mathrm{where}{A}_r=a+(2r-4)b+4(2r-4)(2r-6)c,\hfill \end{array}$$

(15)

$$\begin{array}{c}a=1+2{\rho }_{23}^2,b={\rho }_{12}^2+{\rho }_{13}^2+4{\rho }_{12}{\rho }_{13}{\rho }_{23},c={\rho }_{12}^2{\rho }_{13}^2.\hfill \\ \mathrm{So},{\mu }_{222}=a+2b\mathrm{as}\mathrm{in}\left(12\right),{\mu }_{422}=3(a+4b+8c),{\mu }_{622}=15(a+6b+24c),\hfill \\ {\mu }_{822}=105(a+8b+48c),{\mu }_{10,22}=945(a+10b+80c),{\mu }_{12,22}={\nu }_{12}(a+12b+120c).\hfill \end{array}$$

(14) with $k=5$ gives for $2r=s+5$,

$$\begin{array}{c}{\mu }_{s32}{2}^{r-1}/s!3!=\sum _{j=3}^5{a}_j/(r-j)!\mathrm{where}{a}_3=4{\rho }_{13}{\rho }_{23}+2(1+2{\rho }_{23}^2){\rho }_{12},\hfill \end{array}$$

(16)

$$\begin{array}{c}{a}_4=4{\rho }_{12}{\rho }_{13}^2+8{\rho }_{13}{\rho }_{23}{\rho }_{12}^2,a_5=8{\rho }_{12}^2{\rho }_{13}^2.\hfill \\ \mathrm{So},2{\mu }_{332}/9=a_3+a_4,{\mu }_{532}/45=a_3/2+a_4+a_5.\hfill \end{array}$$

(14) with $k=6$ gives $2r=s+6$,

$$\begin{array}{c}{\mu }_{2r-6,42}{2}^{r-1}/(2r-6)!4!=\sum _{j=3}^6{a}_j/(r-j)!\mathrm{where}\hfill \end{array}$$

(17)

$$\begin{array}{c}{a}_3=1/2+2{\rho }_{23}^2,a_4={\rho }_{13}^2+8{\rho }_{12}{\rho }_{13}{\rho }_{23}+2{\rho }_{12}^2+2{\rho }_{23}^2,\hfill \\ a_5=4{\rho }_{12}^2{\rho }_{13}^2+16{\rho }_{12}^3{\rho }_{13}{\rho }_{23}/3+1+2{\rho }_{12}^4/3,a_6=4{\rho }_{12}^4{\rho }_{13}^2/3.\hfill \\ \mathrm{So},{\mu }_{442}/36=a_3/2+a_4+a_5,{\mu }_{642}/540=a_3/6+a_4/2+a_5+a_6.\hfill \end{array}$$

(14) with $k=6$ gives $2r=s+7$,

$$\begin{array}{c}{\mu }_{2r-7,52}{2}^{r-1}/(2r-7)!5!=\sum _{j=4}^7{a}_j/(r-j)!\mathrm{where}\hfill \\ a_4=2{\rho }_{13}{\rho }_{23}+(1+4{\rho }_{23}^2){\rho }_{12},\hfill \end{array}$$

(18)

$$\begin{array}{c}{a}_5=2{\rho }_{12}{\rho }_{13}^2+8{\rho }_{12}^2{\rho }_{13}{\rho }_{23}+4{\rho }_{12}^3(1+2{\rho }_{23}^2)/3,\hfill \\ a_6=8{\rho }_{12}^3{\rho }_{13}({\rho }_{13}+2{\rho }_{23})/3+4{\rho }_{12}^5/15,a_7=8{\rho }_{12}^5{\rho }_{13}^2/15.\hfill \\ \mathrm{So},8{\mu }_{552}/15=\sum _{j=4}^6{a}_j/(6-j)!,8{\mu }_{752}/315=\sum _{j=4}^7{a}_j/(7-j)!.\hfill \end{array}$$

Now apply $\partial {(1.3)}^t$ to (7) with $u=s-t$ using Faa di Bruno’s rule of (5) with 2 changed to 3, and Leibniz’ rule for the tth derivative of a product. So,

Theorem 5.

For $B_r,C_r,D_r$ of (8) and $2r=u+k+t,$

$$\begin{array}{c}{\mu }_{ukt}{2}^r/u!k!=\sum _{k/2\le j\le min(k,r)}{D}_{k-j}\partial {\left(1.3\right)}^t\left[B_{r-j}{C}_{2j-k}\right],\hfill \end{array}$$

(19)

$$\begin{array}{c}where\partial {\left(1.3\right)}^t\left[B_h{C}_J\right]=\sum _{d+e=t}\left({\displaystyle \genfrac{}{}{0pt}{}{t}{d}}\right)B_{h.3d}{C}_{J.3e}\hfill \end{array}$$

(20)

$$\begin{array}{c}{B}_{h.3d}=\partial {\left(1.3\right)}^d{B}_h=H_{hd}^{13}/h!,C_{J.3e}=\partial {\left(1.3\right)}^e{C}_J=C_{J-e}{\left[2\mu \left(23\right)\right]}^e,\hfill \end{array}$$

(21)

and $H_{hd}^{13}$ is $H_{rk}^{12}$ of (6) with $2,r,k$ replaced by $3,h,d$. That is,

$$\begin{array}{c}{\mu }_{ukt}{2}^r/u!=\sum _{k/2\le j\le min(k,r)}{\left[k\right]}_j\mu {\left(2^2\right)}^{k-j}{I}_{r-j,2j-k,t}^{123}{2}^{2j-k}/(r-j)!(2j-k)!,\hfill \\ I_{hJt}^{123}=\partial {\left(1.3\right)}^t\mu {\left(1^2\right)}^h\mu {\left(12\right)}^J=\sum _{d+e=t}\left({\displaystyle \genfrac{}{}{0pt}{}{t}{d}}\right){\left[J\right]}_e{H}_{hd}^{13}\mu {\left(12\right)}^{J-e}\mu {\left(23\right)}^e.\hfill \end{array}$$

(22)

(21) follows from (4). (22) follows from

$$\partial {\left(1.3\right)}^e\mu {\left(12\right)}^J={\left[J\right]}_e\mu {\left(12\right)}^{J-e}.$$

This is our 1st formula for the general trivariate moment. Example 7.1 will need

$$\begin{array}{c}{I}_{h0t}^{123}=H_{ht}^{13},I_{h1t}^{123}=H_{ht}^{13}\mu \left(12\right)+t{H}_{h,t-1}^{13}\mu \left(23\right),\hfill \\ I_{h2t}^{123}=H_{ht}^{13}\mu {\left(12\right)}^2+2t{H}_{h,t-1}^{13}\mu \left(12\right)\mu \left(23\right)+{\left[t\right]}_2{H}_{h,t-2}^{13}\mu {\left(23\right)}^2.\hfill \\ \mathrm{Set}{L}_{hJt}=h!RHS\left(20\right)=\sum _{d+e=t}\left({\displaystyle \genfrac{}{}{0pt}{}{t}{d}}\right)H_{hd}^{13}{\left(2{\rho }_{12}\right)}^g{\left(2{\rho }_{23}\right)}^e/g!atg=J-e.\hfill \end{array}$$

(23)

Corollary 6.

When $V_{jj}\equiv 1$, for $2r=u+k+t$ and $H_{rk}^{12}$ of (9),

$$\begin{array}{c}{\mu }_{ukt}{2}^r/u!=\sum _{k/2\le j\le min(k,r)}{\left[k\right]}_j{L}_{r-j,2j-k,t}/(r-j)!\hfill \end{array}$$

(24)

$$\begin{array}{c}where{L}_{hJt}=2^J\sum _{e=0}^J{\left[t\right]}_e{\rho }_{12}^g{\rho }_{23}^e{H}_{h,t-e}^{13}/g!e!\mathrm{at}g=J-e.\hfill \end{array}$$

(25)

$$\begin{array}{c}\mathrm{So}{L}_{h0t}=H_{ht}^{13},L_{h1t}=2{\rho }_{12}{H}_{ht}^{13}+2t{\rho }_{23}{H}_{h,t-1}^{13},\hfill \\ L_{h2t}=2{\rho }_{12}^2{H}_{ht}^{13}+4t{\rho }_{12}{\rho }_{23}{H}_{h,t-1}^{13}+2{\left[t\right]}_2{\rho }_{23}^2{H}_{h,t-2}^{13},\hfill \\ L_{h3t}=4{\rho }_{12}^3{H}_{ht}^{13}/3+4t{\rho }_{12}^2{\rho }_{23}{H}_{h,t-1}^{13}+4{\left[t\right]}_2{\rho }_{12}{\rho }_{23}^2{H}_{h,t-2}^{13}+4{\left[t\right]}_3{\rho }_{23}^3{H}_{h,t-3}^{13}/3,\hfill \\ L_{hJ1}=2^J[2h{\rho }_{12}^J{\rho }_{13}/J!+{\rho }_{12}^{J-1}{\rho }_{23}/(J-1)!.\hfill \end{array}$$

Putting ${\tilde{L}}_{hJt}=L_{hJt}$ when ${\rho }_{ij}\equiv 1$, gives the new identity

$$\begin{array}{c}{\left[2r\right]}_{k+t}=\sum _{k/2\le j\le min(k,r)}{\left[k\right]}_j{\left[r\right]}_j{\tilde{L}}_{r-j,2j-k,t}/(r-j)!\hfill \\ \mathrm{where}{\tilde{L}}_{hJt}=2^J\sum _{e=0}^{min(J,t)}{\left[t\right]}_e{\left[2h\right]}_{t-e}/(J-e)!e!.\hfill \end{array}$$

Some examples: $k=3$ gives for $2r-3=u+t$,

$$\begin{array}{c}{\mu }_{u3t}(r-2)!2^r/u!3!=L_{r-2,1t}+(r-2)L_{r-3,3t}.\hfill \\ t=3,u=2r-6,h=r-2\Rightarrow {\mu }_{u33}(r-2)!2^r/u!3!=L_{h13}+h{L}_{h-1,33}.\hfill \end{array}$$

$$\begin{array}{c}\mathrm{Set}a={\rho }_{12}{\rho }_{13},\mathrm{and}b={\rho }_{12}^2+{\rho }_{13}^2.\hfill \end{array}$$

$$\begin{array}{c}\mathrm{For}h=r-2,{\mu }_{2r-6,33}h!2^r/(2r-6)!3!=\sum _{j=0}^2{c}_j{\rho }_{23}^j\mathrm{where}\hfill \end{array}$$

(26)

$$\begin{array}{c}{c}_0/16=3{\left[h\right]}_{2}a/2+{\left[h\right]}_{3}ab+2{\left[h\right]}_4{a}^3/3,\hfill \\ c_1/12=h+2{\left[h\right]}_{2}b+4{\left[h\right]}_3{a}^2,c_2=48{\left[h\right]}_{2}a,c_3=8h.\hfill \\ \mathrm{So},r=5,u=4,h=3\mathrm{gives}4{\mu }_{433}/3=\sum _{j=0}^2{c}_j{\rho }_{23}^j\mathrm{where}\hfill \\ c_0=48(3a+2ab),c_1=36(1+4b+8a^2),c_2=288a,c_3=24.\hfill \\ r=u=6,h=4\mathrm{gives}16{\mu }_{633}/45=\sum _{j=0}^2{c}_j{\rho }_{23}^j\mathrm{where}\hfill \\ c_0/32=9a+12ab+8a^3,c_1/48=1+6b+24a^2,c_2=576a,c_3=32.\hfill \\ r=7,u=8,h=5\mathrm{gives}4{\mu }_{833}/63=\sum _{j=0}^2{c}_j{\rho }_{23}^j\mathrm{where}\hfill \\ c_0=160(3a+6ab+8a^3),c_1=60(1+8b+48a^2),c_2=960a,c_3=40.\hfill \end{array}$$

By Corollary 6.5 with $k=4$,

$$\begin{array}{c}\mathrm{for}2r-4=u+t,{\mu }_{u4t}{2}^r/u!4!=\sum _{j=2}^4{L}_{r-j,2j-4,t}/(r-j)!(4-j)!.\hfill \end{array}$$

(27)

$$\begin{array}{c}\mathrm{So},{\mu }_{242}/3=1+2{\rho }_{13}^2+4{\rho }_{12}^2+4{\rho }_{23}^2+16{\rho }_{12}{\rho }_{13}{\rho }_{23}+8{\rho }_{12}^2{\rho }_{23}^2,\hfill \end{array}$$

(28)

$$\begin{array}{c}{\mu }_{341}/9={\rho }_{13}+4{\rho }_{12}({\rho }_{23}+{\rho }_{12}{\rho }_{13})+8{\rho }_{12}^3{\rho }_{23}/3,\hfill \\ L_{143}=16({\rho }_{12}{\rho }_{23}^3+{\rho }_{12}^3{\rho }_{23}+3{\rho }_{12}^3{\rho }_{23}^2{\rho }_{13},\hfill \end{array}$$

(29)

$$\begin{array}{c}{\mu }_{343}/9=3{\rho }_{13}+2{\rho }_{13}^3+12[{\rho }_{12}^2{\rho }_{13}+{\rho }_{12}{\rho }_{23}(1+2{\rho }_{13}^2)+{\rho }_{13}{\rho }_{23}^2]\hfill \\ +8[{\rho }_{12}{\rho }_{23}^3+{\rho }_{12}^3{\rho }_{23}+3{\rho }_{12}^2{\rho }_{23}^2{\rho }_{13}],\hfill \\ {\mu }_{444}/9=3+24b+72a+8c+96d+128ab+192a^2\mathrm{where}\hfill \\ a={\Pi }_{123}^3{\rho }_{12},b=\sum _{123}^3{\rho }_{12}^2,c=\sum _{123}^3{\rho }_{12}^4,d=\sum _{123}^3{\rho }_{12}^2{\rho }_{13}^2=(b^2-c)/2.\hfill \end{array}$$

An alternative to Theorem 6.4 is given by applying $\partial {(2.3)}^t$ to (7) with $u=k-t$.

Trivariate moments by the multinomial method.

We now apply (4) with $p=3$. Set $c=(c_{12},c_{13},c_{23})$. Then

$${\rho }^c={\rho }_{12}^{c_{12}}{\rho }_{13}^{c_{13}}{\rho }_{23}^{c_{23}},c_1=c_{12}+c_{13},c_2=c_{12}+c_{23},c_3=c_{13}+c_{23}.$$

Example 3.

Take $n=(2r-2,1,1).$ Then $a_2=a_3=0$ and either $c=(0,0,1)$ or $(1,1,0)$, so that $C_n$ sums over either $a=(r-1,0,0),c=(0,0,1)$ or $a=(r-2,0,0),c=(1,1,0)$ giving

$$\begin{array}{c}{C}_n=2r{\rho }_{23}+4{\left[r\right]}_2{\rho }_{12}{\rho }_{13},B_n={\nu }_{2r}/{\left[2r\right]}_2={\nu }_{2r-2}/\left(2r\right),\hfill \\ {\mu }_n/{\nu }_{2r-2}={\rho }_{23}+2(r-1)\alpha where\alpha ={\rho }_{12}{\rho }_{13}:\hfill \\ {\mu }_{211}={\rho }_{23}+2\alpha ,{\mu }_{411}/3={\rho }_{23}+4\alpha ,{\mu }_{611}/15={\rho }_{23}+6\alpha ,\hfill \\ {\mu }_{811}/105={\rho }_{23}+8\alpha ,{\mu }_{10,11}/945={\rho }_{23}+10\alpha .\hfill \end{array}$$

This was given by Isserlis with $\lambda =2r-2$.

Example 4.

Take $n=(2r-3,2,1).$

Then $C_n$ sums over either $c=(0,1,0),a=(r-2,1,0)$ or

$c=(1,0,1),a=(r-2,0,0)$ or $c=(2,1,0),a=(r-3,0,0)$ giving

$$\begin{array}{c}{C}_n/2{\left[r\right]}_2={\alpha }_0+(r-2){\alpha }_{1}where{\alpha }_0={\rho }_{13}+2{\rho }_{12}{\rho }_{23},{\alpha }_1=2{\rho }_{12}^2{\rho }_{23},\hfill \\ B_n={\nu }_{2r}/DwhereD=\left({\displaystyle \genfrac{}{}{0pt}{}{2r}{2r-3,2,1}}\right)={\left[2r\right]}_3/2=2(2r-1){\left[r\right]}_2\Rightarrow \hfill \\ B_n={\nu }_{2r-2}/2{\left[r\right]}_2,{\mu }_n/{\nu }_{2r-2}={\alpha }_0+(r-2){\alpha }_1:\hfill \\ {\mu }_{121}={\alpha }_{0}asin\left(7\right),{\mu }_{321}/3={\alpha }_0+{\alpha }_{1}asin\left(10\right),\hfill \\ {\mu }_{521}={\alpha }_0+2{\alpha }_{1}asin\left(16\right),{\mu }_{721}={\alpha }_0+3{\alpha }_1,{\mu }_{921}={\alpha }_0+4{\alpha }_1.\hfill \end{array}$$

This agrees with (1) with $\lambda =2r-3$.

Example 5.

Take $n=(2r-4,3,1).$ $2a_3+c_{13}+c_{23}=1$ so $a_3=0$ and $(c_{13},c_{23})=\left(01\right)$ (Case 1) or $\left(10\right)$ (Case 2).

Case 1: $2a_2+c_{12}+c_{23}=3$ so $a_2=0$ and $(c_{12},c_{23})=\left(21\right)$ (I) say, or $\left(30\right)$, (II) say. If (I) then $\left({\displaystyle \genfrac{}{}{0pt}{}{r}{ac}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{r}{r-3,00201}}\right)={\left[r\right]}_3/2,{\left(2\rho \right)}^c=2^3{\rho }_{12}^2{\rho }_{23}$.

If (II) then $\left({\displaystyle \genfrac{}{}{0pt}{}{r}{ac}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{r}{r-2,10001}}\right)={\left[r\right]}_2,{\left(2\rho \right)}^c=2{\rho }_{23}$.

Case 2: $2a_2+c_{12}+c_{23}=3$ so $a_2=0$ and $(c_{12},c_{23})=\left(30\right)$ (I) say, or $\left(10\right)$, (II) say. If (I) then $\left({\displaystyle \genfrac{}{}{0pt}{}{r}{ac}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{r}{r-4,00310}}\right)={\left[r\right]}_4/6,{\left(2\rho \right)}^c=2^4{\rho }_{12}^3{\rho }_{13}$.

If (II) then $\left({\displaystyle \genfrac{}{}{0pt}{}{r}{ac}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{r}{r-3,10110}}\right)={\left[r\right]}_3,{\left(2\rho \right)}^c=4{\rho }_{12}{\rho }_{13}$.

Also $B_n={\nu }_{2r}/\left({\displaystyle \genfrac{}{}{0pt}{}{2r}{2r-4,31}}\right)=3{\nu }_{2r-4}/2{\left[r\right]}_2$. So finally,

$$\begin{array}{cc}\hfill & {\mu }_n/{\nu }_{2r-4}=\sum _{j=0}^2{[r-2]}_j{\alpha }_{j}for{\alpha }_0=3{\rho }_{23},{\alpha }_1=6{\rho }_{12}({\rho }_{13}+{\rho }_{12}{\rho }_{23}),{\alpha }_2=4{\rho }_{12}^3{\rho }_{13}.\hfill \end{array}$$

For example $r=2,\cdots ,6$ give (2), (10), (9),

$$\begin{array}{c}{\mu }_{631}/15={\alpha }_0+3{\alpha }_1+6{\alpha }_2,{\mu }_{831}/105={\alpha }_0+4{\alpha }_1+12{\alpha }_2.\hfill \end{array}$$

Example 6.

Take $n=(2r-4,2,2).$

Then $C_n$ sums over $a=(r-2,0,0),c=(0,0,2)$, $a=(r-3,0,0),c=(1,1,1)$, $a=(r-4,0,0),c=(2,2,0)$, $a=(r-3,0,1),c=(2,0,0)$,

$a=(r-3,1,0),c=(0,2,0)$, $a=(r-2,1,1),c=(0,0,0)$, giving

$$\begin{array}{c}{C}_n/{\left[r\right]}_2=\sum _{j=0}^2{[r-2]}_j{\alpha }_{j}where\hfill \\ {\alpha }_0=1+2{\rho }_{23}^2,{\alpha }_1=2\sum _{j=1}^2{\rho }_{1j}^2+8{\rho }_{12}{\rho }_{13}{\rho }_{23},{\alpha }_2=4{\rho }_{12}^2{\rho }_{13},\hfill \\ B_n={\nu }_{2r-4}/{\left[r\right]}_2\Rightarrow {\mu }_n/{\nu }_{2r-4}=\sum _{j=0}^2{[r-2]}_j{\alpha }_j.\hfill \\ So,{\mu }_{022}={\alpha }_0,{\mu }_{222}={\alpha }_0+{\alpha }_1,{\mu }_{422}/3={\alpha }_0+2{\alpha }_1+4{\alpha }_2,\hfill \\ {\mu }_{622}/15={\alpha }_0+3{\alpha }_1+6{\alpha }_2,{\mu }_{822}/105={\alpha }_0+4{\alpha }_1+12{\alpha }_2.\hfill \end{array}$$

Moments with $p>3$ can be obtained by either of the last 2 methods.

7. Moments of Dimension 4

We use $H_{rk}^{12}$ of (6) and (9). Recall that by (21), for $C_J={\left[2\mu \left(12\right)\right]}^J/J!,\partial {(1.3)}^e{C}_J=C_{J-e}{\left[2\mu \left(23\right)\right]}^e.$ That is,

$$\partial {\left(1.3\right)}^e\mu {\left(12\right)}^J={\left[J\right]}_e\mu {\left(12\right)}^{J-e}\mu {\left(23\right)}^e.$$

By (2), applying $\partial {(1.4)}^v$ to (19) and setting $w=u-v$ gives

Theorem 6.

For $B_r,C_r^{12}=C_r,D_r$ of (8) and $I_{hJt}^{123}$ of (22),

$$\begin{array}{c}{\mu }_{wktv}{2}^r/w!k!=\sum _{k/2\le j\le min(k,r)}{D}_{k-j}{E}_{r-j,2j-k,tv}/(r-j)!at2r=w+k+t+v,\hfill \end{array}$$

$$\begin{array}{c}{E}_{hJtv}=h!\partial {\left(1.4\right)}^{v}RHS\left(20\right)=\sum _{0\le e_1\le min(J,t)}\left({\displaystyle \genfrac{}{}{0pt}{}{t}{e_1}}\right)F_{hJ,t-e_1,e_{1}v},\hfill \end{array}$$

(1)

$$\begin{array}{c}{F}_{hJ{d}_1{e}_{1}v}=h!\partial {\left(1.4\right)}^v{B}_{h.3d_1}{C}_{J.3e_1}=\sum _{0\le e_2\le min(J-e_1,v)}\left({\displaystyle \genfrac{}{}{0pt}{}{v}{e_2}}\right)S_{h{d}_1,v-e_2}{T}_{J{e}_1{e}_2},\hfill \end{array}$$

(2)

$$\begin{array}{c}{S}_{h{d}_1{d}_2}=h!\partial {\left(1.4\right)}^{d_2}{B}_{h.3d_1}=\partial {\left(1.4\right)}^{d_2}{H}_{h{d}_1}^{13}\hfill \\ =\sum _{d_1/2\le j\le min(d_1,h)}{\left[h\right]}_j{\left[d_1\right]}_j{I}_{h-j,2j-d_1,d_2}^{134}\mu {\left(3^2\right)}^{d_1-j}{2}^{2j-d_1}/(2j-d_1)!,\hfill \\ T_{J{e}_1{e}_2}=\partial {\left(1.4\right)}^{e_2}{C}_{J.3e_1}=T^{\prime }{\left[2\mu \left(23\right)\right]}^{e_1},T^{\prime }=\partial {\left(1.4\right)}^{e_2}{C}_{J-e_1}={\left[2\mu \left(24\right)\right]}^{e_2}{C}_{J-e_1-e_2}.\hfill \end{array}$$

Corollary 7.

For $2r=w+k+t+v$, $V_{jj}\equiv 1$, $E_{hJtv}$ of (1) and (2), and $I_{hJt}^{123}$ of (22),

$$\begin{array}{c}{\mu }_{wktv}{2}^r/w!=\sum _{k/2\le j\le min(k,r)}{\left[k\right]}_j{E}_{r-j,2j-k,tv}/(r-j)!where\hfill \end{array}$$

$$\begin{array}{c}{S}_{h{d}_1{d}_2}=\sum _{d_1/2\le j\le min(d_1,h)}{\left[h\right]}_j{\left[d_1\right]}_j{I}_{h-j,2j-d_1,d_2}^{134}{2}^{2j-d_1}/(2j-d_1)!,\hfill \end{array}$$

(3)

$$\begin{array}{c}{T}_{J{e}_1{e}_2}={\left(2{\rho }_{23}\right)}^{e_1}{\left(2{\rho }_{24}\right)}^{e_2}{C}_{J-e_1-e_2}=2^J{\rho }_{23}^{e_1}{\rho }_{24}^{e_2}{\rho }_{12}^{J-e_1-e_2}/(J-e_1-e_2)!.\hfill \end{array}$$

(4)

For Examples 7.1 and 7.2, and $H_{rk}^{12}$ of (6), we need

$$\begin{array}{c}{S}_{h0d_2}=I_{h0d_2}^{134}=H_{h{d}_2}^{14},S_{h1d_2}/2h=I_{h-1,1d_2}^{134}=H_{h-1,d_2}^{14}{\rho }_{13}+d_2{H}_{h-1,d_2-1}^{14}{\rho }_{34},\hfill \end{array}$$

$$\begin{array}{c}{S}_{h2d_2}/2h=H_{h-1,d_2}^{14}+2(h-1)I_{h-2,2d_2}^{134}.\hfill \end{array}$$

(5)

$$\begin{array}{c}\mathrm{So}{S}_{h00}=1,S_{h01}=2h{\rho }_{14},S_{h02}=H_{h2}^{14}=2h+4{\left[h\right]}_2{\rho }_{14}^2,S_{h10}=2h{\rho }_{13},\hfill \\ S_{h11}=2h{\rho }_{34}+4{\left[h\right]}_2{\rho }_{13}{\rho }_{14},S_{h12}=4{\left[h\right]}_2({\rho }_{13}+2{\rho }_{14}{\rho }_{34})+8{\left[h\right]}_3{\rho }_{13}{\rho }_{14}^2.\hfill \end{array}$$

$$\begin{array}{c}{S}_{h20}=2h+4{\left[h\right]}_2{\rho }_{13}^2,S_{h21}=4{\left[h\right]}_2{\rho }_{14.3}^{\left(2\right)}+8{\left[h\right]}_3{\rho }_{13}^2{\rho }_{14},\hfill \end{array}$$

(6)

$$\begin{array}{c}{S}_{h22}/4={\left[h\right]}_2(1+2{\rho }_{34}^2)+2{\left[h\right]}_3({\rho }_{13}^2+{\rho }_{14}^2+4{\rho }_{13}{\rho }_{14}{\rho }_{34})+4{\left[h\right]}_4{\rho }_{13}^2{\rho }_{14}^2,\hfill \end{array}$$

(7)

$$\begin{array}{c}{S}_{h23}/8=3{\left[h\right]}_3({\rho }_{14}+2{\rho }_{13}{\rho }_{34}+2{\rho }_{14}{\rho }_{34}^2)+2{\left[h\right]}_4({\rho }_{14}^3+3{\rho }_{13}^2{\rho }_{14}+6{\rho }_{13}{\rho }_{14}^2{\rho }_{34})\hfill \\ +4{\left[h\right]}_5{\rho }_{13}^2{\rho }_{14}^3,\hfill \\ T_{J{e}_{1}0}={\left(2{\rho }_{23}\right)}^{e_1}{C}_{J-e_1},T_{J{e}_{1}1}={\left(2{\rho }_{23}\right)}^{e_1}\left(2{\rho }_{24}\right)C_{J-e_1-1},\hfill \\ T_{J00}=C_J,T_{J01}=2{\rho }_{24}{C}_{J-1},T_{J10}=2{\rho }_{23}{C}_{J-1},T_{J11}=4{\rho }_{23}{\rho }_{24}{C}_{J-2},\hfill \\ T_{J20}=2^J{\rho }_{23}^2{\rho }_{12}^{J-2}/(J-2)!,T_{J21}=2^J{\rho }_{23}^2{\rho }_{24}{\rho }_{12}^{J-3}/(J-3)!.\hfill \\ I_{h23}^{134}=H_{h3}^{14}{\rho }_{13}^2+6H_{h2}^{14}{\rho }_{13}{\rho }_{34}+6H_{h1}^{14}{\rho }_{34}^2.\hfill \end{array}$$

Example 7.

Take $v=1$, $2r-1=w+k+t$, $S_{h{d}_1{d}_2}$ of (3), and $T_{J{e}_1{e}_2}$ of (4). If $V_{jj}\equiv 1$, then

$$\begin{array}{c}{\mu }_{wkt1}{2}^r/w!=\sum _{k/2\le j\le min(k,r)}{\left[k\right]}_j{E}_{r-j,2j-k,t1}/(r-j)!where\hfill \\ E_{hJt1}=\sum _{0\le e_1\le min(J,t)}\left({\displaystyle \genfrac{}{}{0pt}{}{t}{e_1}}\right)F_{hJ,t-e_1,e_{1}1},F_{hJ{d}_1{e}_{1}1}=S_{h{d}_{1}0}{T}_{J{e}_{1}1}+S_{h{d}_{1}1}{T}_{J{e}_{1}0}.\hfill \end{array}$$

(8)

Now take $t=1,2r-2=w+k$. Then by (11) and (5),

$$\begin{array}{c}{\mu }_{wk11}{2}^r/w!=\sum _{k/2\le j\le min(k,r)}{\left[k\right]}_j{E}_{r-j,2j-k,11}/(r-j)!where\hfill \end{array}$$

(9)

$$\begin{array}{c}{E}_{hJ11}=F_{hJ011}+F_{hJ101},\hfill \\ F_{hJ011}=S_{h00}{T}_{J11}+S_{h01}{T}_{J10},F_{hJ101}=S_{h10}{T}_{J01}+S_{h11}{T}_{J00},\hfill \\ So{E}_{hJ11}=4{\rho }_{23}{\rho }_{24}{C}_{J-2}+4hc{C}_{J-1}+(2h{\rho }_{34}+4{\left[h\right]}_2{\rho }_{13}{\rho }_{14})C_J\hfill \end{array}$$

$$\begin{array}{c}wherec={\rho }_{13}{\rho }_{24}+{\rho }_{14}{\rho }_{23}.\hfill \end{array}$$

(10)

$$\begin{array}{c}Sofork=1,w=2r-3,{\mu }_{w111}{2}^r/w!=E_{r-1,111}/(r-1)!where\hfill \end{array}$$

(11)

$$\begin{array}{c}{E}_{h111}=4hb+8{\left[h\right]}_{2}a,a={\rho }_{12}{\rho }_{13}{\rho }_{14},b={\rho }_{12}{\rho }_{34}+{\rho }_{13}{\rho }_{24}+{\rho }_{14}{\rho }_{23}:\hfill \end{array}$$

(12)

$$\begin{array}{c}{\mu }_{1111}=b\Rightarrow \left(8\right),{\mu }_{3111}=3(b+2a)\Rightarrow \left(10\right),{\mu }_{5111}=15(b+4a),\hfill \end{array}$$

(13)

$$\begin{array}{c}{\mu }_{7111}=105(b+6a),{\mu }_{2r+1,111}={\nu }_{2r+2}(b+2ra).\hfill \end{array}$$

$$\begin{array}{c}Fork=2,w=2r-4,{\mu }_{w211}{2}^{r-1}/w!=\sum _{j=1}^2{E}_{r-j,2j-2,11}/(r-j)!,\hfill \end{array}$$

(14)

$$\begin{array}{c}where{E}_{h011}=2h{\rho }_{34}+4{\left[h\right]}_2{\rho }_{13}{\rho }_{24},\hfill \\ E_{h211}=4{\rho }_{23}{\rho }_{24}+8h{\rho }_{12}c+4{\rho }_{12}^2(h{\rho }_{34}+2{\left[h\right]}_2{\rho }_{13}{\rho }_{14}):\hfill \\ {\mu }_{0211}={\rho }_{34.2}^{\left(2\right)}\Rightarrow \left(7\right),\hfill \\ {\mu }_{2211}={\rho }_{34}+2{\rho }_{13}{\rho }_{14}+2{\rho }_{23}{\rho }_{24}+4{\rho }_{12}c+2{\rho }_{12}^2{\rho }_{34}\Rightarrow \left(13\right),\hfill \end{array}$$

$$\begin{array}{c}{\mu }_{4211}/3={\rho }_{34}+4{\rho }_{13}{\rho }_{14}+2{\rho }_{23}{\rho }_{24}+8{\rho }_{12}c+4{\rho }_{12}^2{\rho }_{34.1}^{\left(2\right)},\hfill \end{array}$$

(15)

$$\begin{array}{c}{\mu }_{6211}/15={\rho }_{34}+6{\rho }_{13}{\rho }_{14}+2{\rho }_{23}{\rho }_{24}+12{\rho }_{12}c+6{\rho }_{12}^2{\rho }_{34.1}^{\left(4\right)},\hfill \\ where{\rho }_{ij.k}^{\left(r\right)}={\rho }_{ij}+r{\rho }_{ik}{\rho }_{jk}.\hfill \end{array}$$

$$\begin{array}{c}Fork=3,w=2r-5,{\mu }_{w311}{2}^{r-1}/3w!=\sum _{j=2}^3{E}_{r-j,2j-3,11}/(r-j)!\hfill \end{array}$$

(16)

$$\begin{array}{c}where{E}_{h311}=8{\rho }_{12}{\rho }_{23}{\rho }_{24}+8h{\rho }_{12}^{2}c+8{\rho }_{12}^3(h{\rho }_{34}+2{\left[h\right]}_2{\rho }_{13}{\rho }_{14})/3:\hfill \\ r=3\Rightarrow {\mu }_{1311}=3b+6{\rho }_{21}{\rho }_{23}{\rho }_{24}\Rightarrow \left(11\right),\hfill \end{array}$$

$$\begin{array}{c}r=4\Rightarrow {\mu }_{3311}/3=3b+6{\rho }_{12}({\rho }_{13}{\rho }_{14}+{\rho }_{23}{\rho }_{24})+6{\rho }_{12}^{2}c+2{\rho }_{12}^3{\rho }_{34},\hfill \end{array}$$

(17)

$$\begin{array}{c}r=5\Rightarrow {\mu }_{5311}/15=3b+12a+6{\rho }_{12}{\rho }_{23}{\rho }_{24}+12{\rho }_{12}^{2}c+4{\rho }_{12}^3{\rho }_{34.1}^{\left(2\right)},\hfill \\ r=6\Rightarrow {\mu }_{7311}/315=b+6a+2{\rho }_{21}{\rho }_{23}{\rho }_{24}+6{\rho }_{12}^{2}c+2{\rho }_{12}^3{\rho }_{34.1}^{\left(4\right)}.\hfill \\ r=7\Rightarrow {\mu }_{9311}/945=3(b+6a)+6{\rho }_{12}{\rho }_{23}{\rho }_{24}+12{\rho }_{12}^{2}c+8{\rho }_{12}^3{\rho }_{34.1}^{\left(6\right)}.\hfill \\ r=8\Rightarrow 4{\mu }_{11,311}/10395=12(b+6a)+24{\rho }_{12}{\rho }_{23}{\rho }_{24}+120{\rho }_{12}^{2}c+120{\rho }_{12}^3{\rho }_{34.1}^{\left(8\right)}.\hfill \end{array}$$

Now take $t=2,2r-3=w+k$. Then by (11) and (5),

$$\begin{array}{c}{\mu }_{wk21}{2}^r/w!=\sum _{k/2\le j\le min(k,r)}{\left[k\right]}_j{E}_{r-j,2j-k,21}/(r-j)!where\hfill \end{array}$$

(18)

$$\begin{array}{c}{E}_{hJ21}=F_{hJ201}+2F_{hJ111}+F_{hJ021},F_{hJ201}=S_{h20}{T}_{J01}+S_{h21}{T}_{J00},\hfill \\ F_{hJ111}=S_{h10}{T}_{J11}+S_{h11}{T}_{J10},F_{hJ021}=S_{h00}{T}_{J21}+S_{h01}{T}_{J20}.\hfill \\ k=2,w=2r-5\Rightarrow {\mu }_{w221}{2}^{r-1}/w!=E_{r-1,021}/(r-1)!+E_{r-2,221}/(r-2)!\hfill \\ where{E}_{h021}=S_{h21}of\left(6\right),E_{h221}=F_{h2201}+2F_{h2111}+F_{h2021},\hfill \\ F_{h2201}=S_{h20}{T}_{201}+S_{h21}{T}_{200},F_{h2111}=S_{h10}{T}_{211}+S_{h11}{T}_{210},F_{h2021}=S_{h01}{T}_{220}.\hfill \end{array}$$

$$\begin{array}{c}r=3\Rightarrow {\mu }_{1221}={\rho }_{14}(1+2{\rho }_{23}^2)+2{\rho }_{12}{\rho }_{24}+2{\rho }_{13}{\rho }_{14}\hfill \\ +4{\rho }_{23}({\rho }_{12}{\rho }_{34}+2{\rho }_{13}{\rho }_{24})\Rightarrow \left(13\right).\hfill \end{array}$$

(19)

$$\begin{array}{c}r=4\Rightarrow {\mu }_{3221}=\sum _{i=1}^4{a}_{i}where{a}_1=3{\rho }_{14},a_2=6{\rho }_{12}{\rho }_{24}+6{\rho }_{13}{\rho }_{34},\hfill \end{array}$$

(20)

$$\begin{array}{c}{a}_3=6{\rho }_{14}({\rho }_{12}^2+{\rho }_{13}^2+{\rho }_{23}^2)+12{\rho }_{23}({\rho }_{13}{\rho }_{24}+{\rho }_{12}{\rho }_{34}),\hfill \\ a_4=12{\rho }_{12}^2{\rho }_{13}{\rho }_{34}+12{\rho }_{12}{\rho }_{13}^2{\rho }_{24}+24{\rho }_{12}{\rho }_{13}{\rho }_{14}{\rho }_{23}.\hfill \\ r=5\Rightarrow 2{\mu }_{5221}/5=E_{4021}/8+E_{3221}/2,E_{4021}=S_{421}=48({\rho }_{14.3}^{\left(2\right)}+4{\rho }_{13}^2{\rho }_{14}),\hfill \\ by\left(6\right),E_{3221}=S_{320}{T}_{201}+S_{321}{T}_{200}+2(S_{310}{T}_{211}+S_{311}{T}_{210})+S_{301}{T}_{220},\hfill \\ \Rightarrow {\mu }_{5221}/15=\sum _{i=1}^5{a}_{i}where{a}_1={\rho }_{14},a_2=2{\rho }_{12}{\rho }_{24}+2{\rho }_{13}{\rho }_{34},,\hfill \\ a_3=4{\rho }_{12}{\rho }_{23}{\rho }_{34}+4({\rho }_{12}^2+{\rho }_{13}^2){\rho }_{14}+4{\rho }_{13}{\rho }_{23}{\rho }_{24}+2{\rho }_{14}{\rho }_{23}^2,\hfill \\ a_4=16{\rho }_{12}{\rho }_{13}{\rho }_{14}{\rho }_{23}+8\sum _{23}^2{\rho }_{12}{\rho }_{13}^2{\rho }_{24},a_5=8{\rho }_{12}^2{\rho }_{13}^2{\rho }_{14}.\hfill \end{array}$$

Now take $t=3,2r-7=w+k.$ By (8),

$$\begin{array}{c}{\mu }_{wk31}{2}^r/w!=\sum _{k/2\le j\le min(k,r)}{\left[k\right]}_j{E}_{r-j,2j-k,31}/(r-j)!,\hfill \\ w=2r-8\Rightarrow {\mu }_{w131}{2}^r/w!=E_{r-1,131}/(r-1)!,\hfill \\ w=2r-9\Rightarrow {\mu }_{w231}{2}^{r-1}/w!=E_{r-1,031}/(r-1)!+E_{r-2,231}/(r-2)!,\hfill \\ w=2r-10\Rightarrow {\mu }_{w331}{2}^{r-1}/3w!=E_{r-2,131}/(r-2)!+E_{r-3,331}/(r-3)!,\hfill \\ w=2r-11\Rightarrow {\mu }_{w431}{2}^{r-3}/3w!=E_{r-2,031}/2(r-2)!+E_{r-3,231}/(r-3)!\hfill \\ +E_{r-4,431}/(r-4)!.\hfill \end{array}$$

The reader can now easily work out special cases.

Example 8.

Take $v=2$, $2r-2=w+k+t$. If $V_{jj}\equiv 1$, then

$$\begin{array}{c}{\mu }_{wkt2}{2}^r/w!=\sum _{k/2\le j\le min(k,r)}{\left[k\right]}_j{E}_{r-j,2j-k,t2}/(r-j)!where\hfill \end{array}$$

(21)

$$\begin{array}{c}{E}_{hJt2}=\sum _{0\le e_1\le min(J,t)}\left({\displaystyle \genfrac{}{}{0pt}{}{t}{e_1}}\right)F_{hJ,t-e_1,e_{1}2},\hfill \\ F_{hJ{d}_1{e}_{1}2}=S_{h{d}_{1}0}{T}_{J{e}_{1}2}+2S_{h{d}_{1}1}{T}_{J{e}_{1}1}+S_{h{d}_{1}2}{T}_{J{e}_{1}0}.\hfill \end{array}$$

Now take $t=2,2r-4=w+k$. Then

$$\begin{array}{c}{\mu }_{wk22}{2}^r/w!=\sum _{k/2\le j\le min(k,r)}{\left[k\right]}_j{E}_{r-j,2j-k,22}/(r-j)!\hfill \end{array}$$

(22)

$$\begin{array}{c}where{E}_{hJ22}=F_{hJ022}+2F_{hJ112}+F_{hJ202},\hfill \\ F_{hJ022}=S_{h00}{T}_{J22}+2S_{h01}{T}_{J21}+S_{h02}{T}_{J20},F_{hJ112}=S_{h10}{T}_{J12}+2S_{h11}{T}_{J11}\hfill \\ +S_{h12}{T}_{J10},F_{hJ202}=S_{h20}{T}_{J02}+2S_{h21}{T}_{J01}+S_{h22}{T}_{J00}.\hfill \\ Sofork=2,w=2r-6\Rightarrow \hfill \end{array}$$

$$\begin{array}{c}{\mu }_{w222}{2}^{r-1}/w!=E_{r-1,022}/(r-1)!+E_{r-2,222}/(r-2)!where\hfill \end{array}$$

(23)

$$\begin{array}{c}{E}_{h022}=S_{h22}of\left(7\right),E_{h222}=8\sum _{j=1}^4{\left[h\right]}_j{a}_j,\hfill \\ a_1=a_{12}+a_{13},a_{12}={\rho }_{23}^2+{\rho }_{24}^2,a_{13}=4{\rho }_{23}{\rho }_{24}{\rho }_{34},\hfill \\ a_2={\rho }_{12}^2+\sum _{i=3}^4{a}_{2i},a_{23}=4{\rho }_{12}{\rho }_{13}{\rho }_{23}+4{\rho }_{12}{\rho }_{14}{\rho }_{24},\hfill \\ a_{24}=2{\rho }_{12}^2{\rho }_{34}^2+8{\rho }_{12}{\rho }_{13}{\rho }_{14}{\rho }_{34}+8{\rho }_{12}{\rho }_{32}{\rho }_{24}{\rho }_{34}+8{\rho }_{13}{\rho }_{14}{\rho }_{23}{\rho }_{24},\hfill \\ a_3=\sum _{i=4}^5{a}_{3i},a_{34}=2{\rho }_{12}^2({\rho }_{13}^2+{\rho }_{14}^2),\hfill \end{array}$$

$$\begin{array}{c}{a}_{35}=8{\rho }_{12}{\rho }_{13}{\rho }_{14}({\rho }_{12}{\rho }_{34}+{\rho }_{13}{\rho }_{24}+{\rho }_{14}{\rho }_{23}),\hfill \end{array}$$

(24)

$$\begin{array}{c}{a}_4=4{\rho }_{12}^2{\rho }_{13}^2{\rho }_{14}^2.\hfill \end{array}$$

$$\begin{array}{c}r=3\Rightarrow {\mu }_{0222}=1+2{\rho }_{14}^2+2a_1\Rightarrow \left(12\right),\hfill \end{array}$$

(25)

$$\begin{array}{c}r=4\Rightarrow {\mu }_{2^4}=1+2\sum _{i=2}^4{c}_{i}where{c}_2=\sum ({\rho }_{ij}^2:1\le i<j\le 4),\hfill \end{array}$$

(26)

$$\begin{array}{c}{c}_3/4={\rho }_{12}{\rho }_{13}{\rho }_{23}+{\rho }_{12}{\rho }_{14}{\rho }_{24}+{\rho }_{13}{\rho }_{14}{\rho }_{34}+{\rho }_{23}{\rho }_{24}{\rho }_{34},\hfill \\ c_4=2\sum _{1234}^3({\rho }_{12}^2{\rho }_{34}^2+4{\rho }_{12}{\rho }_{13}{\rho }_{23}{\rho }_{34}).\hfill \\ r=5,w=4\Rightarrow {\mu }_{4222}=S_{422}/16+E_{3222}/4=3+2\sum _{i=2}^5{d}_5,\hfill \\ d_2=2{\rho }_{12}^2+2{\rho }_{13}^2+2{\rho }_{14}^2+{\rho }_{23}^2+{\rho }_{24}^2+{\rho }_{34}^2,\hfill \\ d_3/4=2{\rho }_{12}({\rho }_{13}{\rho }_{23}+{\rho }_{14}{\rho }_{24})+(2{\rho }_{13}{\rho }_{14}+{\rho }_{23}{\rho }_{24}){\rho }_{34},\hfill \\ d_4/4={\rho }_{12}^2{\rho }_{13}^2+a_{24}/2+a_{34}/2={\rho }_{12}^2({\rho }_{13}^2+{\rho }_{14}^2+{\rho }_{34}^2)+{\rho }_{13}^2{\rho }_{14}^2+{\rho }_{14}^2{\rho }_{23}^2\hfill \\ +4{\rho }_{12}({\rho }_{14}{\rho }_{23}+{\rho }_{13}{\rho }_{24}){\rho }_{34}+4{\rho }_{13}{\rho }_{14}{\rho }_{23}{\rho }_{24},d_5=2a_{35}of\left(24\right).\hfill \\ r=w=6\Rightarrow {\mu }_{6222}/15=1+2\sum _{i=2}^5{e}_{i}where\hfill \\ e_2=3{\rho }_{12}^2+3{\rho }_{13}^2+3{\rho }_{14}^2+{\rho }_{23}^2+{\rho }_{24}^2+{\rho }_{34}^2,\hfill \\ e_3/4=3{\rho }_{12}{\rho }_{13}{\rho }_{23}+3{\rho }_{12}{\rho }_{14}{\rho }_{24}+3{\rho }_{13}{\rho }_{14}{\rho }_{34}+{\rho }_{23}{\rho }_{24}{\rho }_{34},\hfill \\ e_4={\rho }_{12}^2({\rho }_{13}^2+{\rho }_{14}^2+{\rho }_{34}^2)+a_{24}/2,e_5=6a_{35}.\hfill \end{array}$$

$$\begin{array}{c}Fork=3,w=2r-7,{\mu }_{w322}{2}^{r-1}/3w!=\sum _{j=2}^3{E}_{r-j,2j-3,22}/(r-j)!\hfill \end{array}$$

(27)

$$\begin{array}{c}where{E}_{h122}=2S_{h12}{T}_{110}+2S_{h21}{T}_{101}+S_{h23}{T}_{100},\hfill \\ E_{h322}=F_{h3022}+2F_{h3112}+F_{h3202},F_{h3022}=2S_{h01}{T}_{321}+S_{h02}{T}_{320},\hfill \\ F_{h3112}=S_{h10}{T}_{312}+2S_{h11}{T}_{311}+S_{h12}{T}_{310},\hfill \\ F_{h3202}=S_{h20}{T}_{302}+2S_{h21}{T}_{301}+S_{h23}{T}_{300}.\hfill \\ Forexample,w=3,r=5\Rightarrow {\mu }_{3322}/3=E_{3122}/16+E_{4322}/64.\hfill \end{array}$$

For ${\mu }_{wkt3}$, put $v=3$ in Corollary 7.1. Of course the method can be continued for higher moments. For example for moments of dimension 5, one route is to apply ${\partial }_{1.5}^e$ to Theorem 7.1.

8. Moments of $Z\sim {\mathcal{N}}_p(\mu ,V)$

So far we have given the moments of $X\sim {\mathcal{N}}_p(0,V)$. It is worth while adding some examples of the non-central moments of $Z=X+\mu$. Consider the case $p=2$. Then

$$m_{rs}=E{Z}_1^r{Z}_2^s=\sum _{j=0}^r\left({\displaystyle \genfrac{}{}{0pt}{}{r}{j}}\right){\mu }_1^{r-j}\sum _{k=0}^s\left({\displaystyle \genfrac{}{}{0pt}{}{s}{k}}\right){\mu }_2^{s-k}{\mu }_{jk}$$

for ${\mu }_{jk}$ of (1). For example

$$\begin{array}{c}{m}_{r2}=\sum _{jeven}\left({\displaystyle \genfrac{}{}{0pt}{}{r}{j}}\right){\mu }_1^{r-j}{\nu }_j({\mu }_2^2+1+j{\rho }^2)+2\rho {\mu }_2\sum _{jodd}\left({\displaystyle \genfrac{}{}{0pt}{}{r}{j}}\right){\mu }_1^{r-j}{\nu }_{j+1}:\hfill \\ m_{22}={\mu }_1^2{\mu }_2^2+4\rho {\mu }_1{\mu }_2+{\mu }_2^2+1+2{\rho }^2,\hfill \\ m_{32}={\mu }_1^3({\mu }_2^2+1)+3{\mu }_1({\mu }_2^2+1+2{\rho }^2)+6\rho ({\mu }_1^2+1){\mu }_2,\hfill \\ m_{42}={\mu }_1^4({\mu }_2^2+1)+6{\mu }_1^2({\mu }_2^2+1+2{\rho }^2)+3({\mu }_2^2+1+4{\rho }^2)+8\rho ({\mu }_1^3+{\mu }_1){\mu }_2.\hfill \end{array}$$