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Let $n$ be a positive integer and $f: \mathbb{C} \rightarrow \mathbb{C}$ be a multivalued function, analytic everywhere except for branch points at $0$, $1$ and $\infty$. Around one of those singularity $p$, $f$ admits a development of the form:

$$ f(z) = \sum_{i=1}^n (z-p)^{\alpha_{p,i}} h_i(z) $$

With $h_i$ holomorphic around $p$. $\alpha_{p,i} \in \mathbb{C}$ are called the exponents of $f$ ( and their sum $\sum_{i,p} \alpha_{p,i}$ should be $0$ ).

Given a set of generic exponents around the three singularities $\{\alpha_{p,i}\}$, I want to know everything there is to know about $f$.

This is a little too broad, so to be more specific:

  • For $n=1$, for exponents $(a)$, $(b)$ and $(-a-b)$ there is an unique solution in the form of $z \rightarrow z^a (1-z)^b$.

  • A more interesting example is $n=2$, the hypergeometric function $_2F_1(a,b; c ; \cdot) $, with generic parameters ($a$, $b$ and $c$ not separated by an integer) verifies the hypothesis and has exponents $(0,0)$, $(0,c-a-b)$ and $(a,b)$ respectively. So the vector space formed by the two functions $z \rightarrow z^e (1-z)^d \ _2F_1(a,b; c ; z)$ and $z \rightarrow z^{e+1-c} (1-z)^{d} \ _2F_1(b-c+1,a-c+1; 2-c ; z)$ is solution.

But for $n=3$, the obvious candidate, the generalized hypergeometric function $_3 F_2(a_1, a_2, a_3 ; b_1, b_2 ; z)$ don't give all the possible exponents : For the singularity in $1$, the exponents are always $(0,1,\gamma)$.

For $n=3$ and for a given set of exponent (generic, in the sense that no two exponents are separated by an integer), is there a function $f$ ? Is it unique and is there an efficient way to compute it ?


Remarks :

  • By multiplying by the $n=1$ solution and playing with the monodromy it's always possible to put a function $f$ in the canonical form $(0,0,0)$, $(0,a,b)$, $(c,d,e)$.

  • It seems natural to expect a vector space of solutions of dimension 3, if there is one.

  • $z \rightarrow \ _2 F_1(a, b ; c ; z) \ _2 F_1(d, b + d - a, c + 2d - 2a)$ is another valid example for $n=3$. But again it is easy to see that the exponents are not completely generic.

  • Contrary to the case of $n=2$ it seems not possible to have a generic differential equation associated with this problem.

Any insight is appreciated, thanks in advance,

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