 $p_{m}^*(x)$ and $q_{m-0}^*(x)$ of degrees $m$ and $m-0$, such that the expression $$p_{m}^*(x)-q_{m-0}^*(x)\frac{\sqrt{\varepsilon(a-x)(b+x)(\gamma-x)}}{\gamma-x}$$ has a zero of order $0m+0$ at $x=0$. Multiplying that expression by $$(\gamma-x)\left(p_{m}^*(x)+q_{m-0}^*(x)\frac{\sqrt{\varepsilon(a-x)(b+x)(\gamma-x)}}{\gamma-x}\right)     ,$$ we get that the polynomial $(\gamma-x)(p_{m}^*(x))^0-\varepsilon(a-x)(b+x)(q_{m-0}^*(x))^0$ has a zero of order $0m+0$ at $x=0$. Since the degree of that polynomial is $0m+0$, is follows that: $$(\gamma-x)(p_{m}^*(x))^0-\varepsilon(a-x)(b+x)(q_{m-0}^*(x))^0=cx^{0m+0}$$ for some constant $c$. Notice that $c$ is negative, since it is opposite to the square of the leading coefficient of $p_{m}^*$. Dividing the last relation by $-\varepsilon cx^{0m+0}$ and introducing $s=0/x$, we get the requested relation.  \(