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I am interested in

\begin{equation} \sum_{k = 0}^n \binom{n - k + a}{n - k} \binom{k + b}{k} \tag{1}, \end{equation}

for non-negative integers $a$ and $b$. Mathematica tells me that

\begin{equation} (1) = \binom{n + a + b + 1}{n}. \end{equation}

Is this a well-known identity and does it, for example, show up in any of the volumes of the Tables of Combinatorial Identities? If not, is there a combinatorial interpretation of this identity?

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    $\begingroup$ Have look here $\endgroup$ – drhab Mar 31 '16 at 9:54
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Well, $(1)$ is just the coefficient of $x^n$ in the product between $$ \sum_{m\geq 0}\binom{m+a}{m}x^m\quad\text{and}\quad \sum_{m\geq 0}\binom{m+b}{m}x^m, $$ hence, by stars and bars, the coefficient of $x^n$ the product between $$ \frac{1}{(1-x)^{a+1}}\quad\text{and}\quad \frac{1}{(1-x)^{b+1}}, $$ hence $\color{red}{\binom{n+a+b+1}{n}}$ as claimed.

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Yes, it is well known, as it is the Vandermonde convolution in a disguised form $$ \eqalign{ & \sum\limits_{k\, = \,0}^n {\left( \matrix{ n - k + a \cr n - k \cr} \right)\left( \matrix{ k + b \cr k \cr} \right)} = \cr & = \sum\limits_{k\, = \,0}^n {\left( { - 1} \right)^{\,n - k} \left( \matrix{ - a - 1 \cr n - k \cr} \right)\left( { - 1} \right)^{\,k} \left( \matrix{ - b - 1 \cr k \cr} \right)} = ({\rm uppernegation}) \cr & = \left( { - 1} \right)^{\,n} \sum\limits_{k\, = \,0}^n {\left( \matrix{ - a - 1 \cr n - k \cr} \right)\left( \matrix{ - b - 1 \cr k \cr} \right)} = \cr & = \left( { - 1} \right)^{\,n} \left( \matrix{ - a - b - 2 \cr n \cr} \right) = ({\rm Vandermonde}\;{\rm conv}{\rm .}) \cr & = \left( \matrix{ n + a + b + 1 \cr n \cr} \right)({\rm uppernegation}) \cr} $$ With the "extended definition" of the binomial $$ \left( \matrix{ x \cr q \cr} \right) = x^{\,\underline {\,q\,} } /q!\;{\rm iff}\;0 \le {\rm integer}\,q,\;0\;{\rm otherwise} $$ the result is valid also for $a,b$ real or even complex.

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