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

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