Checking an identity involving binomial coefficients I need some help to check the following identity:
for every $0\leq i\leq l\leq r$ $$\sum_{j=0}^i\binom{r-l+i-j}{i-j}\binom{l-i+j}{j}=\binom{r+1}{i}.$$
Is this true ? 
Answering to John, this identity come from a geometric problem. I was computing the top Chern class of a vector bundle of the form $S\otimes Q$, $S$ is a line bundle, and $Q$ a rank $r$ vector bundle whose Chern classes have a particularly easy form. I have a guess for what should be the answer and imposing equality between the two led me to this identity. I am pretty sure my guess is true and this identity indeed holds; it's interesting that it does not depend on $l$.
I think it should be possible to prove this with somee induction argument.
 A: Yes, that is true, it is the "double convolution" formula.
To put the formula in more general terms, define the binomial in the extended way:
$$
\left( \matrix{
  x \cr 
  m \cr}  \right) = \left\{ \matrix{
  {{x^{\,\underline {\,m\,} } } \over {m!}}\quad \left| {\;0 \le {\rm integer}\;m} \right. \hfill \cr 
  0\quad \quad \left| {\;{\rm otherwise}} \right. \hfill \cr}  \right.
$$
Then the Upper Negation rule tells that
$$
\left( \matrix{
  x \cr 
  m \cr}  \right) = \left( { - 1} \right)^m \left( \matrix{
  m - x - 1 \cr 
  m \cr}  \right)
$$
and therefore, applying it twice
$$
\eqalign{
  & \sum\limits_{\left( {0\, \le } \right)\,j\,\left( { \le \,\,i} \right)} {\left( \matrix{
  r - l + i - j \cr 
  i - j \cr}  \right)\left( \matrix{
  l - i + j \cr 
  j \cr}  \right)}  =   \cr 
  &  = \sum\limits_{\left( {0\, \le } \right)\,j\,\left( { \le \,\,i} \right)} {\left( { - 1} \right)^i \left( \matrix{
   - r + l - 1 \cr 
  i - j \cr}  \right)\left( \matrix{
   - l + i - 1 \cr 
  j \cr}  \right)}  =  \cr} 
$$
and by the "simple" convolution
$$
 = \left( { - 1} \right)^i \left( \matrix{
   - r + i - 2 \cr 
  i \cr}  \right) = \left( \matrix{
  r + 1 \cr 
  i \cr}  \right)\quad \left| {\;\left\{ \matrix{
  {\rm integer}\;i \hfill \cr 
  {\rm real}\;r,l \hfill \cr}  \right.} \right.
$$
A: $\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
 \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\half}{{1 \over 2}}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\iff}{\Leftrightarrow}
 \newcommand{\imp}{\Longrightarrow}
 \newcommand{\ol}[1]{\overline{#1}}
 \newcommand{\pars}[1]{\left(\, #1 \,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\, #2 \,}\,}
 \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$

The Question:
  $\ds{\sum_{j=0}^{k}{r - \ell + k - j \choose k - j}{\ell - k + j \choose j} =
     {r + 1 \choose k}}$


\begin{align}
&\color{#f00}{\sum_{j=0}^{k}{r - \ell + k - j \choose k - j}
{\ell - k + j \choose j}}
\\[3mm] = &\
\sum_{j=0}^{\infty}{-r + \ell - k + j + k - j - 1\choose k - j}\pars{-1}^{k - j}
{-\ell + k - j + j - 1 \choose j}\pars{-1}^{j}
\\[3mm] = &\
\pars{-1}^{k}\sum_{j=0}^{\infty}{\ell - r - 1\choose k - j}
{k - \ell - 1 \choose j} =
\pars{-1}^{k}\sum_{j=0}^{\infty}{\ell - r - 1\choose k - j}
{k - \ell - 1 \choose k - \ell -1 - j}
\\[3mm] = &\
\pars{-1}^{k}\sum_{j=0}^{\infty}\bracks{\oint_{\verts{z} = 1^{-}}
{\pars{1 + z}^{\ell - r - 1} \over z^{k - j + 1}}\,{\dd z \over 2\pi\ic}}
\bracks{\oint_{\verts{w} = 1^{-}}
{\pars{1 + w}^{k - \ell - 1} \over w^{k - \ell - j}}
\,{\dd w \over 2\pi\ic}}
\\[3mm] = &\
\pars{-1}^{k}\oint_{\verts{z} = 1^{-}}
{\pars{1 + z}^{\ell - r - 1} \over z^{k + 1}}
\oint_{\verts{w} = 1^{-}}
{\pars{1 + w}^{k - \ell - 1} \over w^{k - \ell}}\sum_{j = 0}^{\infty}\pars{zw}^{j}
\,{\dd w \over 2\pi\ic}\,{\dd z \over 2\pi\ic}
\\[3mm] = &\
\pars{-1}^{k}\oint_{\verts{z} = 1^{-}}
{\pars{1 + z}^{\ell - r - 1} \over z^{k + 1}}
\oint_{\verts{w} = 1^{-}}
{\pars{1 + w}^{k - \ell - 1} \over w^{k - \ell}\pars{1 - zw}}
\,{\dd w \over 2\pi\ic}\,{\dd z \over 2\pi\ic}
\\[3mm] \stackrel{w\ \to\ 1/w}{=}\ &\
\pars{-1}^{k}\oint_{\verts{z} = 1^{-}}
{\pars{1 + z}^{\ell - r - 1} \over z^{k + 1}}\
\overbrace{\oint_{\verts{w} = 1^{+}}
{\pars{1 + w}^{k - \ell - 1} \over w - z}\,{\dd w \over 2\pi\ic}}
^{\ds{\pars{1 + z}^{k - \ell - 1}}}\ \,{\dd z \over 2\pi\ic}
\\[3mm] = &\
\pars{-1}^{k}\oint_{\verts{z} = 1^{-}}
{\pars{1 + z}^{k - r - 2} \over z^{k + 1}}\,{\dd z \over 2\pi\ic} =
\pars{-1}^{k}{k - r - 2 \choose k}
\\[3mm] = &\
\pars{-1}^{k}{-k + r + 2 + k - 1 \choose k}\pars{-1}^{k} =
\color{#f00}{{r + 1 \choose k}}
\end{align}
