AI 中文总结
研究多项式\\(g_{p,r}(n)=\binom{pn}{p+r}\\)在移位二项式基下的展开，通过生成函数等得到系数公式，刻画其回文性及相关性质，证明\\(p\\)整除系数，还发现与多项式三角形选定行一致，揭示了二项式系数展开的一些规律。
AI 中文摘要
我们研究多项式\\(g_{p,r}(n)=\binom{pn}{p+r}\\)（其中整数\\(p\geq2\\)且\\(r\geq1\\)）在移位二项式基\\(\{\binom{n + k - 1}{p + r}\}\\)下的展开。利用生成函数和有限差分，得到展开系数\\(B_{p,r,k}\\)的封闭形式公式。接着刻画系数序列何时为回文，表明当且仅当\\(r\equiv1\pmod{p}\\)时在其支撑上具有反射对称性。证明结合了对序列支撑的分析和\\(\binom{pX}{p + r}\\)的根结构。在相同同余条件下，表明\\(p\\)整除每个系数。对于\\(r = 1\\)，首项系数简化为\\(pC_p\\)，其中\\(C_p\\)是第\\(p\\)个卡特兰数。最后，对小的\\(p\\)和\\(r\\)值的计算表明，所得系数序列与\\(p\\)抽取的多项式三角形的选定行一致。
英文摘要
For integers $m\ge2$ and $r\ge1$, put $d=m+r-1$ and define the shifted-binomial coordinates $B_{m,r,k}$ by \[ \binom{mn}{d} = \sum_{k=1}^{d+1}B_{m,r,k}\binom{n+k-1}{d}. \] The main purpose of this paper is to identify these coordinates with a specific residue-class decimation of a multinomial coefficient sequence. We prove \[ B_{m,r,k} = [x^{mk-1}](1+x+\cdots+x^{m-1})^{m+r}. \] This identity converts the alternating finite-difference formula for the coordinates into a positive coefficient formula, and yields their exact support. We then prove that the nonzero coefficient vector is palindromic if and only if $r\equiv2\pmod m$. This symmetry criterion is derived here directly within the multinomial-decimation framework. The principal arithmetic results are an unconditional divisibility law and an exact formula for the greatest common divisor of each nonzero row. We prove \[ \frac{m}{\gcd(m,r-1)}\mid B_{m,r,k}, \] and, more precisely, determine \[ \gcd_{1\le k\le K_{m,r}} B_{m,r,k} \] as an explicit product of prime powers determined by the $p$-adic valuations of $m$ and $m+r-1$. Finally, a roots-of-unity filter gives the exact row sum \[ \sum_k B_{m,r,k}=m^{m+r-1}. \] Thus the shifted-binomial coordinates form a positive, arithmetically structured decimation of an $m$-nomial coefficient sequence.
Comments14 pages
