Maximizing weighted sums of binomial coefficients using generalized continued fractions

Abstract

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

14 pages, 1 figure, 1 table. Version 2 adds Remarks 4.9 and 5.4. To appear in Proc. Royal Soc. Edinburgh