Maximizing weighted sums of binomial coefficients using generalized continued fractions

Let m,\,r∈ Z and ω ∈ R satisfy 0 r m and ω 1 . Our main result is a generalized continued fraction for an expression involving the partial binomial sum sₘ(r) = ∑ ᵢ₌₀ʳmi . We apply this to create new upper and lower bounds for sₘ(r) and thus for gω,m(r)=ω ⁻ʳsₘ(r) . We also bound an integer r₀ ∈ \0,\,1,\,…,\,m\ such that gω,m(0)<⋯ < gω,m(r₀-1) gω,m(r₀) and gω,m(r₀)>⋯ >gω,m(m) . For real ω √ 3 we prove that r₀∈ \ m+2/ω +1,\, m+2/ω +1 +1\ , and also r₀ = m+2/ω +1 for ω ∈ \3,\,4,\,… \ or ω =2 and 3 m .