hyper

Usage

hyper L

Parameter	Type	Description
L	${\mathbb{Q}}[n,E]$	A difference operator

Returns

Returns $[]$ if $L y = 0$ has no hypergeometric solution over ${\mathbb{Q}}(n)$, or $f(n)$ such that any solution of $y(n+1) = f(n) y(n)$ is also a solution of $L y = 0$.

Remarks

hyper returns only the solutions that are hypergeometric over ${\mathbb{Q}}(n)$, i.e. the solutions $y$ such that $y(n+1)/y(n) \in {\mathbb{Q}}(n)$. It does not return any eventual solution that is hypergeometric over $\overline{{\mathbb{Q}}}(n)$ but not over ${\mathbb{Q}}(n)$. In addition, hyper does not attempt to return several linearly independent hypergeometric solutions, so if it finds one, there could be some others.

Example

The equation

$$L = (n-1)y(n+2) - (n^2+3n-2)y(n+1) + 2n(n+1)y(n) = 0$$

has hypergeometric solutions:

1 --> L := (n-1)*E^2 - (n^2+3*n-2)*E + 2*n*(n+1);
2 --> tex(hyper(L));

$$2$$

This shows that $2^n$ is a solution, but there are others: we can verify that $E - n - 1$ divides $L$ on the right, which proves that $n!$ is another hypergeometric solution:

3 --> R := E - n - 1; 
4 --> Q := L / R;
5 --> tex(L - Q*R);

$$0$$