论文标题
在Chaundy和Bullard的组合身份
On a combinatorial identity of Chaundy and Bullard
论文作者
论文摘要
我们给出了两个新证明，证明了chaundy-bullard公式$$（1-x）^{n+1} \ sum_ {k = 0}^m {n+k \ select k} x^k+x^k+x^{m+1} \ sum_} \ sum_ {m+sum_ {k = 0} \ frac {（1-x）^{（n+1）}}} {（n+1）！} \ sum_ {k = 0}^m \ frac {n+1} {n+k+1} \ frac {x^{x^{（k）}+ \ frac {x^{（M+1）}}} {（M+1）！} \ sum_ {k = 0}^n \ frac {m+1} {m+k+1} \ frac {（1-x）此外，我们提出涉及不完整beta函数和一定组合总和的身份。
We give two new proofs of the Chaundy-Bullard formula $$ (1-x)^{n+1} \sum_{k=0}^m {n+k\choose k} x^k +x^{m+1}\sum_{k=0}^n {m+k\choose k} (1-x)^k=1 $$ and we prove the "twin formula" $$ \frac{ (1-x)^{(n+1)}}{(n+1)!} \sum_{k=0}^m \frac{n+1}{n+k+1} \frac{ x^{(k)}}{k!} + \frac{ x^{(m+1)}}{(m+1)!} \sum_{k=0}^n \frac{m+1}{m+k+1} \frac{ (1-x)^{(k)}}{k!}=1, $$ where $z^{(n)}$ denotes the rising factorial. Moreover, we present identities involving the incomplete beta function and a certain combinatorial sum.
