A renewal-process definition of the discrete-time fractional Poisson process using discrete Mittag-Leffler waiting times yields distributions that are not mathematically equivalent to the Sibuya-subordinated construction.
Nagai, Discrete Mittag-Leffler function and its applications, RIMS Kokyuroku,1302(2003), 1–20
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An alternative formulation of the discrete-time fractional Poisson process
A renewal-process definition of the discrete-time fractional Poisson process using discrete Mittag-Leffler waiting times yields distributions that are not mathematically equivalent to the Sibuya-subordinated construction.