# Hypergeometric and Theta Functions Solution to the Quintic (in python!)

On github, in my math diary (which is unorganized at the moment), there is code which implements (essentially, it’s only a special case, but is easily generalized0 both the hypergeometric (Glasser) and the theta functions solution of the quintic. The Glasser method quoted on the wikipedia page about the Bring Radical contains sign errors but is essentially correct. The theta functions solution on the mathworld page about the quintic also appears to be essentially correct, although the orthography on that page contains some spurious mistakes — such as the “$q^{5/8}(q^{5})^{-1/8}$
coefficient, and is painful to write, but is elegant.

The polynomial in question is $w(z) = z^5 -z + \frac{4}{5^{5/4}}$, and this one was chosen in particular because the constant interacts nontrivially with both of these solutions: For the theta function solution it corresponds to elliptic modulus $\sqrt{2}-1$ and thus to nome $e^{-\pi\sqrt{2}}$, and in the hypergeometric case, the fourth power of this means that the $z$-slot in the hypergeometric functions is simply 1 — which is a little bit of a problem for mpmath, and it’s noted in the documentation that there might be problems when $z$ is around one. I am looking forward to reimplementing both of these in arb

 owen@orrery ~/atal » ./hypg.verify.py (0.66880437318301406386 - 0.006218779022452095584j) (0.66879191050396671825 + 0.0062116365382047596402j) (-1.1038422688861613827 + 8.7204858053738321163e-19j) (-0.11681917053334134398 + 1.03445357140566252j) (-0.11681917053334134475 - 1.0344535714056625207j) 

 owen@orrery ~/atal » ./theta.verify.py (-1.103842268886161584 + 8.6656680999461563393e-17j) (0.66874030497642225453 + 4.459681438480688112e-9j) (-0.11681917053334137041 - 1.0344535714056627406j) (-0.11681917053334137041 + 1.0344535714056627406j) (0.66874030497642225453 - 4.459681438480688112e-9j) 

More generally different root finding methods must give the same results, and I suspect there are nifty things to find by equating the Umemura/theta functions approach with the Glasser/differential resolvents approach. I did ask, and will update relationship between solution of quintic in terms of 4F3 hypergeometric function and theta functions on m.se, but given how scant public knowledge of these methods are and the errors in their transcriptions, I rather doubt there has been detailed study so far.