I asked suggestion for good book here. After reading the suggested book and others I want to ensure what I have learned is correct and want to clear some of my doubt.
Whole life Annuity-due
The formula is,
\begin{align} \require{enclose} \ddot a_{x} &= 1+{}_1p_xv+{}_2p_xv^2+\cdots\\ &=1+v\frac{l_{x+1}}{l_x}+v^2\frac{l_{x+2}}{l_x}+\cdots \\ &= 1+\frac{v^{x+1}}{v^x}\frac{l_{x+1}}{l_x}+\frac{v^{x+2}}{v^x}\frac{l_{x+2}}{l_x}+\cdots \\ &= \frac{\sum_{t=0}^\infty D_{x+t}}{D_x}\qquad\textrm{Where }D_x=l_xv^x \textrm{ is a commutation function}\\ &= \frac{N_x}{D_x} \end{align}
Interpretation of ${}_tp_xv^t$: Multiplied by ${}_tp_x$ to reflect the probability that a payment is made at time $t$ (${}_tp_x$ is the probability that a person aged $x$ survives to age $x+t$) and discount at interest back to time zero by multiplying $v^t$.
But What $N_x$ or $\sum_{t=0}^\infty D_{x+t}$ and $\sum_{t=0}^\infty N_{x+t}=S_{x}$ mean? (I mean the interpretation of these)
Whole life Assurance
The value of a whole life assurance of $1$ to a person aged $x$ is,
\begin{align} A_x &= v\frac{d_x}{l_x}+v^2\frac{l_{x+1}}{l_x}\frac{d_{x+1}}{l_x}+v^3\frac{l_{x+1}}{l_x}\frac{l_{x+2}}{l_{x+1}}\frac{d_{x+2}}{l_{x+1}}+\cdots \\ &\vdots \\ &= \frac{\sum_{t=0}^\infty C_{x+t}}{D_x}\qquad\textrm{Where }C_x=v^{x+1}d_x\textrm{ is a c.f.}\\ &= \frac{M_x}{D_x} \end{align}
But I couldn't understand the interpretation of $v^{x+1}d_x$ here.
Could someone help me to understand it like I did in whole life Annuity-due above? Or Could someone redirect/suggest/provide any link/book/article where I can get the proper definition/explanation of the commutation functions?
Updated
OK, it seems commutation functions are introduced for calculation purpose only. But still suspect there should exist a meaning of $\frac{d_x}{l_x}$ in $A_x$ (Whole life Assurance).
The commutation functions are/were used for making manual calculations easier (when things were done by hand on paper). They don't really have an interpretation other than an abstract way of moving things forwards or backwards. The actuarial methodologies have changed away from using them in practical sense. This other book may be helpful: Statutory Valuation of Life Insurance Liabilities