Maximizing weighted sums of binomial coefficients using generalized continued fractions

Abstract

Let $m,r\in\mathbb{Z}$ and $\omega\in\mathbb{R}$ satisfy $0\leqslant r\leqslant m$ and $\omega\geqslant1$. Our main result is a generalized continued fraction for an expression involving the partial binomial sum $s_m(r) = \sum_{i=0}^r\binom{m}{i}$. 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 $g_{\omega,m}(0)<\cdots<g_{\omega,m}(r_0-1)\leqslant g_{\omega,m}(r_0)$ and $g_{\omega,m}(r_0)>\cdots>g_{\omega,m}(m)$. For real $\omega\geqslant\sqrt3$ 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 $3\nmid m$.