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The Jordan-Kronecker function is defined by the infinite sum

$$ F(x, y) = \sum_{n=-\infty}^\infty \frac{y^n}{1 - x q^n}, \quad |q|<|y|<1 $$

and, obviously, restrictions on $x$ to avoid poles. Surprisingly, $$ F(x, y) = F(y, x) = -F(-1/x,1/y). $$