On the \(g\)-ary expansions of middle binomial coefficients and Catalan numbers (Q635279)

The \(q\)-ary nonzero digits of the central binomial coefficients and Catalan numbers are investigated. As a first approximation, the authors prove that these numbers have at least three nonzero digits in their \(q\)-ary expansions (for all but finitely many indices). An approximative but much more informative result is also proven: Let the base \(g>1\) be fixed and let \(\varepsilon(n)\) be a function tending to zero. Then \[ w_q(b_n)\gg\varepsilon(n)(\log(n))^{1/2},\quad w_q(C_n)\gg\varepsilon(n)(\log(n))^{1/2} \] hold for all \(n\leq X\) with at most \(o(X)\) exceptions as \(X\to\infty\), where \(b_n=\binom{2n}{n}\), \(C_n\) is the \(n\)th Catalan number and \(w_g\) is the number of nonzero digits of the argument in the \(q\)-ary expansion.