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Is there an easy way to see that no polynomial $P(X)$ of degree $2n$ satisfies the polynomial equation $$(X-n)^2P(X+1)-X^2P(X)-1=0,$$ where $n\geq 1$ is some integer?

I am trying to show that the hypergeometric term $\binom{n}{k}^2$ is not Gosper-summable in $k$, and the equation above is Gosper's corresponding equation.

Zuy
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  • You reference $p(X)$ but I do not see $p$. Did you instead mean $p$ where you have written $s$? – FShrike Dec 15 '21 at 19:15
  • @FShrike Thank you! – Zuy Dec 15 '21 at 19:16
  • By "satisfies", this could mean the identity holds for all $X$ ( for all $X$ in ?) or just that it holds for at least one $X$ ($X$ in ?), $\Bbb C,\Bbb R,\Bbb N$? – FShrike Dec 15 '21 at 19:20
  • @FShrike This is an equation between polynomials; the RHS is the zero polynomial. – Zuy Dec 15 '21 at 19:21
  • We then require satisfaction for all $X$ – FShrike Dec 15 '21 at 19:22
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    We can easily see that the left-hand side is a polynomial of degree $2n$. Since its $(n+1)$ coefficients should be 0, we obtain a linear system of the kind $M x = b$, where $x$ is the vector stacking all coefficients of $P$, $b=(0,...,0, 1)'$ and $M$ is a $(2n+1)\times (2n+1)$ matrix. Then one needs to show that $b$ is not in the range of $M$. This is easy to see when $n=1$ but otherwise I don't know... (I have not computed $M$ in its full generality). – bdx77 Dec 16 '21 at 05:20
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    Unfortunately, this isn't really discussed in papers in great generality because , by virtue of this being an algorithm, the discussion centers around the fact that expressing this as a matrix equation involving the non-determined coefficients and then using methods to verify whether the linear system has a solution or not , is the optimal way of solving this problem. I've not seen anybody focus on the solvability of the equation from a classical standpoint. – Sarvesh Ravichandran Iyer Dec 16 '21 at 10:05

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