Tyma Gaidash has recently posted solutions to some quintics in terms of Inverse Beta Regularized function. He also found the closed form for the equation $\cos x=x$ using the same Inverse Beta Regularized function.
So, the question is: can a general quintic be solved in terms of this function?
Best Answer
Yes, it seems, it can. Any general quintic can be reduced to the Bring-Jerrard form $x^5+ax+b=0$.
Then Tyma Gaidash found a solution for the Bring-Jerrard form via Inverse Beta Regularized function:
$x=\frac{5b}{4a\left(\text I^{-1}_{\frac{3125b^4}{256a^5}+1}(2,4)-1\right)}$.
In computer algebra systems and spreadsheets like Excel the function is defined only for the subscript argument in the range $[0,1]$. But there is nothing that prevents to define this function on a bigger domain either by functional equations or analytic continuation.
Tyma Gaidash gives only one solution, but other roots can be obtained by factoring and solving general quartic.
Moreover, in his recent finds he found solutions for equations of other orders, including non-integer ones:
$x^r+ax+b=0\implies x=\frac{br}{a(1-r)}\text I^{-1}_\frac{ b^{r-1}(r-1)}{a^r\left(\frac1r-1\right)^r}(-r,2)$
and
$x^r+ax+b=0\implies x=\frac{b r}{a(1-r)\text I^{-1}_\frac{b^{r-1}(r-1)}{a^r\left(\frac1r-1\right)^r}(r-1,2)}$
So, yes. A solution of general quintic and some equations of higher orders can be expressed in terms of inverse beta regularized function granted the function is defined on a greater range than its usual range of definition.
As a side note: Tyma Gaidash managed to express using inverse beta regularized function besides polynomial roots also such things as Dottie number, elliptic and trigonometric and hyperbolic functions, logarithms and exponents, Lambert W-function, inverse functions of $\text{erf}(x)$ and $\text{erfc}(x)$, logarithmic and exponential integrals $\text{Ei}(x)$ and $\text{li}(x)$.