I am reading Euler's Introduction to analysis of the infinite and get stuck here. I wish to ask two questions:
1) The part I don't understand is circled in red box. If I understand correctly, he is trying expand the binomial series for binomial, trinomial and quadrinomial and finally polynomial expression. But I don't understand how did he derive the recurrent relationship between the first and the second coefficients, the second and the third, and so forth.
I have also numbered the red box for easy reference.
For example, for red box number 2, I don't understand where does $\dfrac{2m-n}{n}\beta\cdot L$ come from?
2) I also don't understand the yellow part. Is this the consequence of expanding a trinomial into trinomial series?
For the first red box, they're getting an expression for the $(m+1)$th (or $n$th) coefficient (defined as $N$) in terms of the $m$th coefficient (defined as $M$).
Note the last two terms in the equations preceding red box 1 and note that in this case $m=2$ and $n=m+1=3$:
$$M = \alpha^2\frac{(m-1)(m-2)}{1\cdot 2}; N = \alpha^3\frac{(m-1)(m-2)(m-3)}{1\cdot 2 \cdot 3},$$
so
$$N=\alpha \frac{m-3}{3}M.$$
The other red boxes are similar redefinition of coefficients to get recurrence formulas.
For the yellow part, these terms come from multiplying out $(\alpha z + \beta z z)^2$:
$$(\alpha z + \beta z z)^2 = \alpha^2z^2+2\alpha\beta z^3+\beta^2z^4.$$
(Only the first two terms are written out explicitly. The last term is a higher-order term covered by the "etc.")
To verify these formulas, it really is just a matter of multiplying the things out and doing all of the bookkeeping, and looking at the patterns. If you pick a particular value of $m$, this will determine a particular value of $n$, which is the highest power of $z$ in the overall expansion. Then just call the other coefficients $A, B, C,$ and so forth, and you'll find that the relations in the red boxes hold for the particular values of $m$ and $n$.
You can start simply with $m=2$. Then there's nothing even to multiply out and you can verify things immediately.
Then move to $m=3$. This will square the polynomial on the left sides of the equations. In section 75, $n=4$ because your highest power is $z^4$. In section 76, $n=6$ because your highest power is $z^6$.
Then try $m=4$, and if you're not tired yet, try $m=5$.
Hope this helps.