# Eikonal Blog

## 2010.01.04

### Trinomial equation related to Mellin transform

Filed under: mathematics — Tags: , , , — sandokan65 @ 20:08

The algebraic equation $y^n + x y^p -1 = 0$ for ($n>p$) has following series solutions:

$y_0(x) = \frac1{n} \sum_{r=0}^\infty \frac{(-)^r}{r!} \frac{\Gamma(\frac{1+pr}{n}) x^r}{\Gamma(\frac{1+pr}{n}+1-r)}.$

Other $n-1$ solutions are given by the complex rotations as $y_k(x) = \omega^k y_0(\omega^{pk} x)$ where $\omega:\equiv e^{i\frac{2\pi}{n}}$, $k=\overline{0,n-1}$.

The solutions satisfy following ODE:

$(-d_x)^n y(x) = p (-x d_x -n) y(x)$.

Source: “The Functions of Mathematical Physics” by Harry Hochstadt