Go to:Table of Contents

Composition Relations

We observe the measures $^\alpha{\ensuremath{\boldsymbol{\mathscr{I}}}}^s_r(P_1,P_2,...,P_M)$ ($\alpha=1,2$ and $3$) $^\alpha{\ensuremath{\boldsymbol{\mathscr{J}}}}^s_r(P_1,P_2,...,P_M)$ ($\alpha=1,2$ and $3$) and$^\alpha{\ensuremath{\boldsymbol{\mathscr{T}}}}^s_r(P_1,P_2,...,P_M)$ ($\alpha=1,2$ and $3$) are continuous with respect to the parameters $r$ and $s$. This allows us to write them in the following simplified forms:

$$^\alpha{\ensuremath{\boldsymbol{\mathscr{I}}}}^s_r(P_1,P_2,...,P_......Big\{\,^\alphaI^s_r(P_1,P_2,...,P_M)\left\vert r\neq 1, s\neq 1\Big\}\right.,$$

$$^\alpha{\ensuremath{\boldsymbol{\mathscr{J}}}}^s_r(P_1,P_2,...,P_......Big\{\, ^\alphaJ^s_r(P_1,P_2,...,P_M)\left\vert r\neq 1,s\neq 1\Big\}\right.,$$ and $$^\alpha{\ensuremath{\boldsymbol{\mathscr{T}}}}^s_r(P_1,P_2,...,P_......\Big\{\,^\alphaT^s_r(P_1,P_2,...,P_M)\left\vert r\neq 1,s\neq 1\Big\}\right.,$$ $\alpha=1,2$ and $3$, for all $r,s \in (-\infty,\infty)$, where "CE" stands for "continuous extension" with respect to $r$ and $s$.

Also we can write

$$^1{\ensuremath{\boldsymbol{\mathscr{I}}}}^s_r(P_1,P_2,...,P_M)={......}}_s\Big(\ ^1{\ensuremath{\boldsymbol{\mathscr{J}}}}^1_r(P_1,P_2,...,P_M)\Big),$$

$$^2{\ensuremath{\boldsymbol{\mathscr{I}}}}^s_r(P_1,P_2,...,P_M)=\......boldsymbol{\mathscr{D}}}}^1_r(P_j\vert\vert \sum_{k=1}^M{\lambda_k P_k})\Big)},$$

$$^3{\ensuremath{\boldsymbol{\mathscr{I}}}}^s_r(P_1,P_2,...,P_M)={......ol{\mathscr{M}}}}_s\Big({\ensuremath{\boldsymbol{\mathscr{H}}}}^1_r(P_j)\Big)},$$

$$^\alpha{\ensuremath{\boldsymbol{\mathscr{J}}}}^s_r(P_1,P_2,...,P......\ensuremath{\boldsymbol{\mathscr{J}}}}^1_r(P_1,P_2,...,P_M)\Big),\ (\alpha =1\$$   and$\3)$

$$^2{\ensuremath{\boldsymbol{\mathscr{J}}}}^s_r(P_1,P_2,...,P_M)=\......}}}_s\Big({\ensuremath{\boldsymbol{\mathscr{D}}}}^1_r(P_j\vert\vert P_k)\Big)},$$

$$^\alpha{\ensuremath{\boldsymbol{\mathscr{T}}}}^s_r(P_1,P_2,...,P......\ensuremath{\boldsymbol{\mathscr{T}}}}^1_r(P_1,P_2,...,P_M)\Big),\ (\alpha =1\$$   and$\3)$

and

$$^2{\ensuremath{\boldsymbol{\mathscr{T}}}}^s_r(P_1,P_2,...,P_M)=\......\boldsymbol{\mathscr{D}}}}^1_r(\sum_{j=1}^M{\lambda_jP_j}\vert\vert P_k)\Big)},$$

where ${\ensuremath{\boldsymbol{\mathscr{M}}}}_s(x)$ and ${\ensuremath{\boldsymbol{\mathscr{N}}}}_s(x)$ are as given by (3.10) and (4.4) respectively.