# Rewriting a binomial coefficient in terms of Pochhammer symbols

I am working with the equation $$\sum^{2n+1}_{k=0} \binom{2n+1}{k}(x^k -(-x)^k), \ n = 0,1,2,..$$ and want to rewrite it in terms of rising Pochhammer symbols.

I am aware of the relation $$\frac{(x)_n}{n!} = \binom{x+n-1}{n}.$$ But how could I manipulate my binomial coefficient to get it in to this form?

Any help is greatly appreciated!

Edit: Just to clarify, here we denote $(x)_n$ as the rising factorial i.e $x(x+1)(x+2)\dots(x+n-1)$

• Since $$\frac{(x)_k}{k!} = \binom{x+k-1}{k}$$ then just solve for $x$ in $x+k-1=2n+1$ to get the argument of your Pochhammer symbol. – Tito Piezas III Jan 31 '18 at 15:42