Geodesic
Analytical Properties of a Generating Function for a Number of Renewals
Teoriâ veroâtnostej i ee primeneniâ, Tome 8 (1963) no. 1, pp. 108-112

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Let $\{t_i\}$ be a renewal process, $N_t=\max(n:t_n$. Some analytic properties, such as analyticy within the circle $|z|{1/r}$, of the function $\Pi _t (z)=\sum _{k=0}^\infty z^k P\{N_t=k\}$ and others are proved. 
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     title = {Analytical {Properties} of a {Generating} {Function} for a {Number} of {Renewals}},
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Yu. K. Belyaev; V. M. Maksimov. Analytical Properties of a Generating Function for a Number of Renewals. Teoriâ veroâtnostej i ee primeneniâ, Tome 8 (1963) no. 1, pp. 108-112. http://geodesic.mathdoc.fr/item/TVP_1963_8_1_a9/