Maximizing weighted sums of binomial coefficients using generalized continued fractions

Let $m,r\in \mathbb{Z}$ and $\omega \in \mathbb{R}$ satisfy $0⩽r⩽m$ and $\omega ⩾1$. Our main result is a generalized continued fraction for an expression involving the partial binomial sum $s_m(r)={\sum }_{i=0}^r\left(\genfrac{}{}{0px}{}{m}{i}\right)$. We apply this to create new upper and lower bounds for $s_m(r)$ and thus for $g_{\omega ,m}(r)={\omega }^{-r}{s}_m(r)$. We also bound an integer $r_0\in \{0,1,\dots ,m\}$ such that $g_{\omega ,m}(0)<\cdots <g_{\omega ,m}(r_0-1)⩽g_{\omega ,m}(r_0)$ and $g_{\omega ,m}(r_0)>\cdots >g_{\omega ,m}(m)$. For real $\omega ⩾\sqrt{3}$ we prove that $r_0\in \{\lfloor \frac{m+2}{\omega +1}\rfloor ,\lfloor \frac{m+2}{\omega +1}\rfloor +1\}$, and also $r_0=\lfloor \frac{m+2}{\omega +1}\rfloor$ for $\omega \in \{3,4,\dots \}$ or $\omega =2$ and $3\nmid m$.