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In this paper, we discuss the conditions for the function $F_a(z) =\sum_{k=0}^\infty \frac{z^k}{(a+1)(a^2+1) \ldots (a^k+1)}, a>1,$ to belong tothe Laguerre-P\'olya class, or to have only real zeros.

On the conditions for a special entire function relative to the partial theta-function and the Euler function to belong to the Laguerre-P\'olya class