On a Functional Differential Equation of Determinantal Type

Braden, H. W.; Byatt-Smith, J. G. B.

On a Functional Differential Equation of Determinantal Type

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We solve the functional equations $$ \begin{vmatrix} 1 & 1 & 1 f(x) & f(y) & f(z) f\sp{\prime}(x)& f\sp{\prime}(y)& f\sp{\prime}(z) \end{vmatrix} =0,\quad\quad \begin{vmatrix} 1 & 1 & 1 f(x) & g(y) & h(z) \\ f\sp{\prime}(x)& g\sp{\prime}(y)& h\sp{\prime}(z) \end{vmatrix} =0, for suitable functions $f$, $g$ and $h$ subject to $x+y+z=0$. These equations essentially characterise the Weierstrass $\wp$-function and its degenerations. %\quad\quad x+y+z=0. $$
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