Closed expression for hypergeometric sum

Question

I am trying to simplify an expression and find a closed form for $$\sum_{m=0}^l \binom{s-m}{s-l} \binom{s-1+m}{s-1}x^m$$

How could I get rid of this summation?

Answer 1

You may argue as GH from MO from your other post.

• the coefficient of $y^m$ in $(1-xy)^{-s}$ equals $\binom{s+m-1}{s-1}x^m$;
• the coefficient of $y^{\ell-m}$ in $(1-y)^{\ell-s-1}$ equals $\binom{s-m}{s-\ell}$.

Therefore, the sum on your LHS equals to

• the coefficient of $y^{\ell}$ in $(1-xy)^{-s}(1-y)^{\ell-s-1}$.

Unfortunately, this has no closed form. How can we be sure? To this end, denote your sum by $$f(\ell):=\sum_{m=0}^\ell\binom{s+m-1}{s-1}\binom{s-m}{s-\ell}x^m.$$ As I explained the WZ-method in the other post, the procedure generates a recurrence. However, this time it is a three-term relation $$(\ell+2)f(\ell+2)+(-sx-\ell x-s+2\ell-x+2)f(\ell+1)+(x-1)(s-\ell)f(\ell)=0$$ which reveals that $f(s)$ can not have a closed form.

If you're not interested in the sum, then formulate this as a contour integral. Let $\gamma$ be a closed path (oriented positive) around $z=0$, and apply Cauchy's Integral Formula: $$f(\ell)=\frac1{2\pi i}\int_{\gamma}\frac{dz} {z^{\ell+1}(1-xz)^s(1-z)^{s+1-\ell}}.$$ On a positive note, we can derive a generating function for the sequence $f(\ell)$: $$\sum_{\ell=0}^{\infty}f(\ell)y^{\ell}=\left(\frac{(1+y)^2}{1+y-xy}\right)^s.$$ To see this, start by interchanging summations to proceed as follows: \begin{align} \sum_{\ell\geq0}f(\ell)y^{\ell} &=\sum_{m\geq0}\binom{s+m-1}mx^m\sum_{\ell=m}^s\binom{s-m}{\ell-m}y^{\ell} \\ &=\sum_{m\geq0}\binom{s+m-1}mx^my^m(1+y)^{s-m} \\ &=(1+y)^s\sum_{m=0}^{\infty}\binom{s+m-1}m\left(\frac{yx}{1+y}\right)^m \\ &=(1+y)^s\left(1-\frac{yx}{1+y}\right)^{-s} \\ &=\left(\frac{(1+y)^2}{1+y-xy}\right)^s. \end{align}

Answer 2

Mathematica says: $$\binom{s}{s-l} \, _2F_1(-l,s;-s;x).$$

(without hypergeometrics for special values of $x,$ like $x=1:$

$$\frac{(-1)^l (l-2 s-1)!}{l! (-2 s-1)!}$$

Answer 3

The Mathematica formula mentioned by Rivin is derived via \begin{equation} \sum_{m=0}^l \binom{s-m}{s-l}\binom{s-1+m}{s-1}x^m = \sum_{m=0}^l \frac{\Gamma(s-m+1)\Gamma(s+m)}{\Gamma(1+s-l)\Gamma(l+1-m)\Gamma(s)}\frac{x^m}{m!} \end{equation} \begin{equation} = \frac{1}{\Gamma(1+s-l)}\sum_{m=0}^l \frac{\Gamma(s-m+1)(s)_m}{\Gamma(l+1-m)}\frac{x^m}{m!} \end{equation} \begin{equation} = \frac{1}{\Gamma(1+s-l)}\sum_{m=0}^l (-)^m \Gamma(s+1)\frac{(s)_m}{(-s)_m\Gamma(l+1-m)}\frac{x^m}{m!} \end{equation} \begin{equation} = \frac{1}{\Gamma(1+s-l)}\sum_{m=0}^l (-)^m \Gamma(s+1)\frac{(s)_m(-l)_m}{(-s)_m(-)^m\Gamma(l+1)}\frac{x^m}{m!} \end{equation} \begin{equation} = \frac{1}{\Gamma(1+s-l)}\sum_{m=0}^l \Gamma(s+1)\frac{(s)_m(-l)_m}{(-s)_m\Gamma(l+1)}\frac{x^m}{m!} \end{equation} \begin{equation} = \frac{\Gamma(s+1)}{\Gamma(1+s-l)\Gamma(l+1)}\sum_{m=0}^l \frac{(s)_m(-l)_m}{(-s)_m}\frac{x^m}{m!} \end{equation} \begin{equation} = \frac{\Gamma(s+1)}{\Gamma(1+s-l)\Gamma(l+1)}{}_2F_1(s,-l;-s;x) \end{equation} \begin{equation} = \frac{s!}{(s-l)!l!}{}_2F_1(s,-l;-s;x) \end{equation} \begin{equation} = \binom{s}{l}{}_2F_1(s,-l;-s;x) \end{equation}