Let $m,\,r\in {\mathbb {Z}}$ and $\omega \in {\mathbb {R}}$ satisfy $0\leqslant r\leqslant m$ and $\omega \geqslant 1$. 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 $r_0 \in \{0,\,1,\,\ldots,\,m\}$ 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 $\omega \geqslant \sqrt 3$ 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 $\omega \in \{3,\,4,\,\ldots \}$ or $\omega =2$ and $3\nmid m$.

In this paper, motivated by Catalan numbers and higher-order Catalan numbers, we study factors of products of at most two binomial coefficients.

There are two infinitesimal (i.e., additive) versions of the K-theory of a field F: one introduced by Cathelineau, which is an F-module, and the other introduced by Bloch-Esnault, which is an F*-module. Both versions are equipped with a regulator map, when F is the field of complex numbers.

We will introduce an extended version of Cathelineau's group, and a complex-valued regulator map given by the entropy. We will also give a comparison map between our extended version and Cathelineau's group.

Our results were motivated by two unrelated sources: Neumann's work on the extended Bloch group (which is isomorphic to indecomposable K3 of the complex numbers), and the study of singularities of generating series of hypergeometric multisums.

A basic result in the theory of total positivity is that the convolution of any two Pólya frequency sequences is again a Pólya frequency sequence. The like result for binomial convolution, associated with exponential generating functions, is proved. This and similar results are used to obtain an upper bound on the probability that the union of independent random subsets of a finite set N is all of N. Parallels from the theory of reliability involving sums of random variables with increasing failure rates are noted.