Digit sums of binomial sums
The authors show that the sum of digits in an integer base \(b \geq 2\) of a binomial sum of the form \[ S(n) := \sum_{k=0}^n \binom{n}{k}^{r_0} \binom{n+k}{k}^{r_1} \cdots \binom{n+km}{k}^{r_m} \] with nonnegative integers \(r_0, r_1, \dots, r_m\) is at least \(c_0 \log n / \log \log n\) for a set of positive integers \(n\) of asymptotic density 1 (except when \(m = 0\), \(r_0 = 1\), and \(b\) is a power of 2), where \(c_0\) is a positive constant depending on \(b, r_0, r_1, \dots r_m\). Note that for \(m = 0\), \(r_0 = 2\), \(S(n) = \binom{2n}{n}\) is the middle binomial coefficient, and for \(m = 1\), \(r_0 = r_1 = 2\), \(S(n)\) is the \(n\)th Apéry number. The same conclusion holds for Catalan numbers. This improves and generalizes results from [\textit{F. Luca} and \textit{I. E. Shparlinski}, Ann. Comb. 14, No. 4, 507--524 (2010; Zbl 1233.05020); Rocky Mt. J. Math. 41, No. 4, 1291--1301 (2011; Zbl 1221.11020)]. The proof uses a result from [\textit{R. J. McIntosh}, J. Number Theory 58, No. 1, 158--172 (1996; Zbl 0858.05003)] on the asymptotic expansions of such binomial sums as well as Baker's theorem on lower bounds for linear forms in logarithms of algebraic numbers. Furthermore, the authors conjecture that (when \(m \geq 1\) or \(r_0 \geq 2\)) the sum of digits of \(S(n)\) is always larger than \(c_0 n\) for a constant \(c_0\) and that the Cesàro means converge to a constant \(c_1\), which is given explicitly.
scientific article; zbMATH DE number 5990334