Sufficient conditions of unimodality

The following definition is applied the Py jeNo distribution is unimodal, if in the series - Pq, P2 P P P2, there is precisely one change of sign. 

For the generating function Fof the stationary distribution we can write the following ordinary differential equation  

A theorem of Medgyessy (1977) Let us assume that the generating function F of the Pq 0) discrete distribution satisfies the differential equation 

Equations (5.139) and (5.140) hold precisely if r exists such that P(r) = 0 (and then a(r) = 1). From the chemical point of view it means that the reaction (5.136) contains an elementary reaction 0 (first-order decay or outflow). 

Based on another theorem we may state that a birth and death-type chemical reaction with one internal component leads to unimodal stationary distribution. More precisely, let us assume that a chemical reaction can be identified with a birth and death-type process with the birth / and death p rate  

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