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Fix a prime number $p$. If $n$ is a positive integer, then denote $$\text{$\omega_{p,k}(n):=\#$ of $k$'s in the $p$-ary expansion of $n$}$$ and the total sum of all its $p$-ary digits by $$\Omega_p(n):=\sum_{k=0}^{p-1}k\cdot\omega_{p,k}(n).$$

Question. Given a prime $p$ and for each $n\in\Bbb{N}$, is this true? $$\prod_{k=0}^{p-1}\left(\frac{X^{k+1}-Y^{k+1}}{X-Y}\right)^{\omega_{p,k}(n)} =\sum_{\Omega_p(j)+\Omega_p(n-j)=\Omega_p(n)}^{0\leq j\leq n} X^{\Omega_p(j)}\cdot\,\,Y^{\Omega_p(n-j)}.$$

Example. If $p=2$ and $s_2(n)=\#$ the sum of binary digits of $n$, then $$(X+Y)^{s_2(n)}=\sum_{s_2(j)+s_2(n-j)=s_2(n)}^{0\leq j\leq n} X^{s_2(j)}\cdot\,\,Y^{s_2(n-j)}.$$

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First we notice that the equality $\Omega_p(a)+\Omega_p(b)=\Omega_p(a+b)$ happens if and only if there are no carries when adding $a+b$ in base $p$. Indeed the number of carries is equal to $$\frac{\Omega_p(a)+\Omega_p(b)-\Omega_p(a+b)}{p-1}$$ which is also equal to $\nu_p\left(\binom{a+b}{a}\right)$ by Kummer's theorem. This is equivalent to saying that the base $p$ expansions of $a,b$ $$a=a_0+a_1p+a_2p^2+\cdots , b=b_0+b_1p+b_2p^2+\cdots$$ satisfy $a_i+b_i\le p-1$ for all $i$.

Consider the generating function $$F(X,Y,t)=\sum_{n\geq 0}\left(\sum_{\Omega_p(j)+\Omega_p(n-j)=\Omega_p(n)}^{0\leq j\leq n} X^{\Omega_p(j)}\cdot\,\,Y^{\Omega_p(n-j)}\right)t^n$$ where the coefficient of $t^n$ is the right hand side of your identity. By the observation above this can also be written as $$F(X,Y,t)=\sum_{\Omega_p(a)+\Omega_p(b)=\Omega_p(a+b)}t^{a+b}X^{\Omega(a)}Y^{\Omega(b)}$$ $$=\sum_{a_i+b_i\le p-1}t^{a+b}X^{\sum a_i}Y^{\sum b_i}=\prod_{i=1}^{\infty} \left(\sum_{a_i+b_i\le p-1} X^{a_i}Y^{b_i}t^{(a_i+b_i)p^i}\right)$$ $$=\prod_{i=1}^{\infty} \left(\sum_{k=0}^{p-1}\left(\frac{X^{k+1}-Y^{k+1}}{X-Y}\right) t^{kp^i}\right)$$ from this last expression we see that the coefficient of $t^n$ is equal to $\prod_{k=0}^{p-1}\left(\frac{X^{k+1}-Y^{k+1}}{X-Y}\right)^{\omega_{p,k}(n)}$ which is the left hand side of our identity, as desired.

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  • $\begingroup$ Really nice, Gjergji. $\endgroup$ – T. Amdeberhan Sep 28 '18 at 17:31

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