<s>[INST] Proving a binomial identity Prove that ($k\le m$)
$$\sum_{j=k}^m(_{2m+1}^{2j+1})(_k^j)=\frac{2^{2(m-k)}(2m-k)!}{(2m-2k)!k!} , (k\le m)$$
Please help me with this identity, I've spent a lot of time on it but didn't solve the problem.
 [/INST] According to OPs expression we show:

The following is valid
\begin{align*}
\sum_{j=k}^{m}\binom{2m+1}{2j+1}\binom{j}{k}=4^{m-k}\binom{2m-k}{k}\qquad\qquad 0\leq k \leq m
\end{align*} 

In the following we use the coefficient of operator $[x^k]$ to denote the coefficient of $x^{k}$ in a series $\sum_{j=0}^{\infty}a_jx^j$.

We start with the left hand side and transform the sum into a coefficient of $x^k$ of a polynomial in $x$.
\begin{align*}
\sum_{j=k}^{m}\binom{2m+1}{2j+1}\binom{j}{k}&=\sum_{j=0}^{m}\binom{2m+1}{2j+1}\binom{j}{k}\tag{1}\\
&=[x^k]\sum_{j=0}^{m}\binom{2m+1}{2j+1}(1+x)^j\tag{2}
\end{align*}

Comment:


*

*In (1) we start the index $j$ from $0$. Note that $\binom{j}{k}=0$ if $j<k$. We also write $\binom{j}{k}=[x^k](1+x)^j$.

*In (2) we use the linearity of the coefficient of operator.
Note, that (2) contains binomial coefficients with odd $2j+1$. We consider $(1\pm y)^{2m+1}$ and split it into odd and even parts.
\begin{align*}
(1\pm y)^{2m+1}&=\sum_{j=0}^{m}\binom{2m+1}{2j}y^{2j}\pm \sum_{j=0}^{m}\binom{2m+1}{2j+1}y^{2j+1}\\
\end{align*}

We get
  \begin{align*}
\sum_{j=0}^{m}\binom{2m+1}{2j+1}y^{2j}=\frac{(1+y)^{2m+1}-(1-y)^{2m+1}}{2y}
\end{align*}
Substituting: $y\rightarrow\sqrt{1+x}$ we obtain from (2)
\begin{align*}
\sum_{j=k}^{m}\binom{2m+1}{2j+1}\binom{j}{k}
=[x^k]\frac{(1+\sqrt{1+x})^{2m+1}-(1-\sqrt{1+x})^{2m+1}}{2\sqrt{1+x}}\tag{3}
\end{align*}
Note, the expression on the RHS is a polynomial, since the factor $\sqrt{1+x}$ cancels out.

Next we transform the RHS of OPs expression. We consider it as coefficient of the polynomial

\begin{align*}
4^{m-k}\binom{2m-k}{k}&=4^{m-k}[x^k]\sum_{j=0}^{m}\binom{2m-k}{k}x^{j}\\
&=4^m[x^k]\sum_{j=0}^{m}\binom{2m-k}{k}\left(\frac{x}{4}\right)^{j}\tag{4}
\end{align*}

In order to prove (4) we refer to a useful identity stated as formula (2.1) in H.W.Goulds Combinatorial Identities, Vol. 4. With $n=2m+1$ we get the following expression
\begin{align*}
\sum_{j=0}^{m}(-1)^j\binom{2m-j}{j}(t_1t_2)^j(t_1+t_2)^{2m-2j}=\frac{t_1^{2m+1}-t_2^{2m+1}}{t_1-t_2}
\end{align*}

Setting $t_1=1+\sqrt{1+x}$ and $t_2=1-\sqrt{1+x}$ we get
\begin{align*}
t_1t_2&=-x\\
t_1+t_2&=2\\
t_1-t_2&=2\sqrt{1+x}
\end{align*}
It follows
  \begin{align*}
\sum_{j=0}^{m}\binom{2m-j}{j}x^j2^{2m-2j}
=\frac{(1+\sqrt{1+x})^{2m+1}-(1-\sqrt{1+x})^{2m+1}}{2\sqrt{1+x}}
\end{align*}
  and we obtain
  \begin{align*}
4^{m-k}\binom{2m-k}{k}=[x^k]\frac{(1+\sqrt{1+x})^{2m+1}-(1-\sqrt{1+x})^{2m+1}}{2\sqrt{1+x}}\tag{5}
\end{align*}

The RHS of (3) and (5) coincide and OPs identity follows.
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