In Shiryaev's probability book, in page 87, it is given that
Taking account of $(8),$ we find that for $A \leq x \leq B$ $$ \beta(x)=\frac{(q / p)^{x}-(q / p)^{A}}{(q / p)^{B}-(q / p)^{A}}$$
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If $p=q=\frac{1}{2},$ the only solutions $\beta(x)$ and $\alpha(x)$ of (7),(8) and (11),(12) are respectively $$ \beta(x)=\frac{x-A}{B-A} $$
However, if we put $p=q=0.5$ into the first equation, we simply get $(1-1)/(1-1)$, which is undefined, so I can't understand how does the author get the second equation from the first one.