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Changed names of the sum indices
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In a recent calculation I obtain a result involving the following expression depending on two integers $L^+,L^-\geq 0$$n,m\geq 0$: $$S(L^+,L^-):=\frac{(L^++L^-+1)!}{L^+!L^-!}\sum_{l=0}^{L^++L^-}\frac{1}{L^++L^--l+1}\sum_{\substack{l^++l^-=l\\ 0\leq l^\pm\leq L^\pm}}(-1)^{l^+}{L^+ \choose {l^+}}{L^- \choose {l^-}}.$$$$S(n,m):=\frac{(n+m+1)!}{n!m!}\sum_{l=0}^{n+m}\frac{1}{n+m-l+1}\sum_{\substack{j+k=l\\ 0\leq j\leq n\\0\leq k \leq m}}(-1)^{j}{n \choose j}{m \choose k}.$$ Numerical evidence suggests that one has $$ S(L^+,L^-)=\begin{cases} \sum_{k=0}^{L^-/2}{L^++L^-+2 \choose 2k}, \quad & L^-\in \{0,2,4,\ldots\},\\ -\sum_{k=0}^{(L^-+1)/2-1}{L^++L^-+2 \choose 2k+1}, & L^-\in \{1,3,5,\ldots\}.\end{cases} $$$$ S(n,m)=\begin{cases} \sum_{k=0}^{m/2}{n+m+2 \choose 2k}, \quad & m\in \{0,2,4,\ldots\},\\ -\sum_{k=0}^{(m+1)/2-1}{n+m+2 \choose 2k+1}, & m\in \{1,3,5,\ldots\}.\end{cases} $$ Besides being much simpler, this formula has the great advantage that it makes obvious that $S(L^+,L^-)$$S(n,m)$ is an integer and non-zero.

How would I go about proving the claimed identity?

I guess it must be quite simple compared to many other binomial coefficient identities discussed here on MO. My hope is that experts in this business immediately see an "obvious" simplification step that I am not aware of.

In a recent calculation I obtain a result involving the following expression depending on two integers $L^+,L^-\geq 0$: $$S(L^+,L^-):=\frac{(L^++L^-+1)!}{L^+!L^-!}\sum_{l=0}^{L^++L^-}\frac{1}{L^++L^--l+1}\sum_{\substack{l^++l^-=l\\ 0\leq l^\pm\leq L^\pm}}(-1)^{l^+}{L^+ \choose {l^+}}{L^- \choose {l^-}}.$$ Numerical evidence suggests that one has $$ S(L^+,L^-)=\begin{cases} \sum_{k=0}^{L^-/2}{L^++L^-+2 \choose 2k}, \quad & L^-\in \{0,2,4,\ldots\},\\ -\sum_{k=0}^{(L^-+1)/2-1}{L^++L^-+2 \choose 2k+1}, & L^-\in \{1,3,5,\ldots\}.\end{cases} $$ Besides being much simpler, this formula has the great advantage that it makes obvious that $S(L^+,L^-)$ is an integer and non-zero.

How would I go about proving the claimed identity?

I guess it must be quite simple compared to many other binomial coefficient identities discussed here on MO. My hope is that experts in this business immediately see an "obvious" simplification step that I am not aware of.

In a recent calculation I obtain a result involving the following expression depending on two integers $n,m\geq 0$: $$S(n,m):=\frac{(n+m+1)!}{n!m!}\sum_{l=0}^{n+m}\frac{1}{n+m-l+1}\sum_{\substack{j+k=l\\ 0\leq j\leq n\\0\leq k \leq m}}(-1)^{j}{n \choose j}{m \choose k}.$$ Numerical evidence suggests that one has $$ S(n,m)=\begin{cases} \sum_{k=0}^{m/2}{n+m+2 \choose 2k}, \quad & m\in \{0,2,4,\ldots\},\\ -\sum_{k=0}^{(m+1)/2-1}{n+m+2 \choose 2k+1}, & m\in \{1,3,5,\ldots\}.\end{cases} $$ Besides being much simpler, this formula has the great advantage that it makes obvious that $S(n,m)$ is an integer and non-zero.

How would I go about proving the claimed identity?

I guess it must be quite simple compared to many other binomial coefficient identities discussed here on MO. My hope is that experts in this business immediately see an "obvious" simplification step that I am not aware of.

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B K
B K
  • 1.9k
  • 14
  • 18

A binomial coefficient identity involving two parameters

In a recent calculation I obtain a result involving the following expression depending on two integers $L^+,L^-\geq 0$: $$S(L^+,L^-):=\frac{(L^++L^-+1)!}{L^+!L^-!}\sum_{l=0}^{L^++L^-}\frac{1}{L^++L^--l+1}\sum_{\substack{l^++l^-=l\\ 0\leq l^\pm\leq L^\pm}}(-1)^{l^+}{L^+ \choose {l^+}}{L^- \choose {l^-}}.$$ Numerical evidence suggests that one has $$ S(L^+,L^-)=\begin{cases} \sum_{k=0}^{L^-/2}{L^++L^-+2 \choose 2k}, \quad & L^-\in \{0,2,4,\ldots\},\\ -\sum_{k=0}^{(L^-+1)/2-1}{L^++L^-+2 \choose 2k+1}, & L^-\in \{1,3,5,\ldots\}.\end{cases} $$ Besides being much simpler, this formula has the great advantage that it makes obvious that $S(L^+,L^-)$ is an integer and non-zero.

How would I go about proving the claimed identity?

I guess it must be quite simple compared to many other binomial coefficient identities discussed here on MO. My hope is that experts in this business immediately see an "obvious" simplification step that I am not aware of.