Convolution of Dirichlet Distributions

I consider a random time-varying probability vector alpha_t.
A markovian model is made on this probability vector such as:

p(alpha_t|alpha_ {t-1})=Dirichlet(beta*alpha_{t-1}) with beta a known
constant.

If alpha_t-1 is distributed from a Dirichlet distribution of parameter
gamma, I would like to find an analytic expression of the marginal pdf
p(alpha_t)=\int(p(alpha_t|alpha_ {t-1})p(alpha_t-1) d(alpha_t-1) )
i.e., to find the convolution of two dirichlet distributions.
Is p(alpha_t) still a Dirichlet, and, if so, with which parameters ?
Thank you in advance

.

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