, Volume 113, Issue 3, pp 307-317

Sums of exponential functions having only real zeros

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Let H n (z) be the function of a complex variable z defined by where the summation is over all 2 n possible plus and minus sign combinations, the same sign combination being used in both the argument of G and in the exponent. The numbers ${{a_1,a_2,a_3,\ldots}}$ and ${{b_1,b_2,b_3,\ldots}}$ are assumed to be positive, and G is an entire function of genus 0 or 1 that is real on the real axis and has only real zeros. Then the function H n (z) has only real zeros.