On a function related of multinomial coefficients. I. (Q1565979)

Let \(p\) be a prime number, \(t\) and \(n\) be positive integers, and \(F_{p,t}(n)\) denote the number of coefficients of \((x_1+\dots+x_t)^{j}\) for \(0\leq j\leq n-1\) which are not divisible by \(p\). Set \(G_{p,t}(n)=F_{p,t}(n)/n^{\theta}\), where \(\theta=\log({p+t-1\choose t})/\log p\). By prior work of the authors, it is known that the \(\liminf_{n\rightarrow \infty} G_{p,t}(n)=\beta(p,t)\) is a positive real number. In this paper, the authors show that both \(F_{p,t}\) and \(G_{p,t}\) can be extended to continuous functions on \({\mathbb R}^+\), denoted by \(F\) and \(G\), respectively, which further satisfy various conditions such as \(F(px)=p^{\theta}F(x)\) and \(G(px)=G(x)\). Moreover, \(F\) is strictly increasing on \({\mathbb R}^+\), \(G\) is nowhere monotonic on \({\mathbb R}^+\) and both sets of differential and nondifferential points of \(G(x)\) are dense in \({\mathbb R}^+\). The function \(F(x)\) is explicitly given in the sense that given \(x\in {\mathbb R}^+\), the authors give a formula for \(F(x)\) in terms of the base \(p\) expansion of \(x\). Thus, in order to prove all their results, it just remains to check that \(F\) (and \(G\)) have all the announced properties, a task which is achieved in 14 pages of cumbersome algebraic manipulations with power series.