ENH: add parameter finder for degrees or freedom for students_t distribution

[dschmitz89]

Towards #1305

This adds a parameter finder for the student t distribution with respect to the degrees or freedom.

Disclaimer: this PR is heavily LLM supported as C++ is not (yet?) my forte.

I verified the math with a simple Python program before. I also ran a simple sanity script to see that the function actually executes. I am not so confident about the tests though as I cannot really decipher the output of b2.

Approach: We use bracket_and_solve_root with an initial guess from a second order Edgeworth expansion of the CDF. The initial guess is very good for $df > 10$ but useless for low degrees of freedom as it returns NaN. In this case or if it gives a relative error of the CDF worse than 10%, we fall back to an initial guess of 0.01.

Detailed approach for finding the initial guess

Given a quantile $x$ and a CDF value $p = P(T \leq x)$, we want to recover the degrees of freedom $\nu$.

Step 1 — Edgeworth warm start.

We use the 2nd-order Edgeworth expansion of the $t$ CDF in powers of $1/\nu$:

$$F(x;\nu) \approx \Phi(x) - \frac{\varphi(x)(x + x^3)}{4\nu} + \frac{\varphi(x)(3x + 5x^3 + 7x^5 - 3x^7)}{96\nu^2}$$
where $\Phi(x)$ is the standard normal CDF and $\phi(x)$ the standard normal PDF.
Setting $u = 1/\nu$ and $F(x;\nu) = p$ yields the quadratic

$$B u^2 - A u + (\Phi(x) - p) = 0$$

where

$$A = \frac{\varphi(x)(x + x^3)}{4}, \qquad B = \frac{\varphi(x)(3x + 5x^3 + 7x^5 - 3x^7)}{96}$$

The physically meaningful root (smallest positive $u$, i.e. largest $\nu$) gives a closed-form starting estimate $\hat\nu = 1/u$.

Step 2 — Validation.

We plug $\hat\nu$ into the exact $t$ CDF. If the relative residual $|F(x;\hat\nu) - p|/|p|$ exceeds 10%, we fall back to a safe low starting value $\nu_0 = 10^{-2}$.