Algebraic and modular properties of hypergeometric functions

[xcaruso]

We implement the class HypergeometricAlgebraic which provides methods for dealing with algebraic properties of hypergeometric functions, including algebraicity, global boundedness and stuffs about reductions modulo primes (p-curvature, annihilating polynomial, Lucas property, etc.)

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Dependences

#41205

[xcaruso]

@mantepse: we (Florian and I) point out this PR that you might be interested in reviewing (when finished) :-)

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[mantepse]

Is the new framework suitable to express whether a P-recursive sequence can be written as a linear combination of hypergeometric terms (using https://en.wikipedia.org/wiki/Petkov%C5%A1ek%27s_algorithm)?

(I'm not asking to implement the algorithm, it is only whether the framework would allow to express spaces of hypergeometric functions)

[xcaruso]

I'm not entirely sure to understand your question but we have parents for hypergeometric functions over any base ring. Nonetheless, our hypergeometric functions are not that general than the ones of the Wikipedia page you linked since we require that the numerators and denominators of the rational function $r(n)$ are split.

However, I usually consider Petkovšek algorithm as an algorithm for finding linear factors of a Ore polynomial in $K(n)[\text{shift}]$. So maybe, the right place to implement it is in (the suitable subclass of) OrePolynomial.

[xcaruso]

I'm not entirely sure to understand your question but we have parents for hypergeometric functions over any base ring. Nonetheless, our hypergeometric functions are not that general than the ones of the Wikipedia page you linked since we require that the numerators and denominators of the rational function $r(n)$ are split.

Well, they are actually also more general in another direction since they include the case of positive characteristic (with parameters in $\mathbb Q$).