A law of mortality, proposed by Heligman and Pollard in 1980, and refining an empirical law observed by Gompertz in 1825, implies that beyond the age of 50, the odds of dying between age x and age x + 1 increase exponentially with x. Two natural solutions of a functional equation arising from the Heligman-Pollard law are obtained. Among other results, it is shown that while the ratio of these solutions is an entire function with infinitely many zeros, on the real axis this ratio may be considered constant for actuarial purposes.