The General Analytic Solution of a Functional Equation of Addition Type

Details The General Analytic Solution of a Functional Equation of Addition Type The General Analytic Solution of a Functional Equation of Addition Type

1995-08-28

arxiv.org/abs/funct-an/9508002v2

Mathematics

Operator Algebras

26 pages latex2e. A further example is included

Scientific paper

The general analytic solution to the functional equation $$ \phi_1(x+y)= { { \biggl|\matrix{\phi_2(x)&\phi_2(y)\cr\phi_3(x)&\phi_3(y)\cr}\biggr|} \over { \biggl|\matrix{\phi_4(x)&\phi_4(y)\cr\phi_5(x)&\phi_5(y)\cr}\biggr|} } $$ is characterised. Up to the action of the symmetry group, this is described in terms of Weierstrass elliptic functions. We illustrate our theory by applying it to the classical addition theorems of the Jacobi elliptic functions and the functional equations $$ \phi_1(x+y)=\phi_4(x)\phi_5(y)+\phi_4(y)\phi_5(x) $$ and \[ \Psi _1(x+y)=\Psi _2(x+y) \phi_2(x)\phi_3(y) +\Psi_3(x+y) \phi_4(x)\phi_5(y). \]

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