A spectral decomposition for the block counting process and the fixation line of the beta(3,1)-coalescent

Abstract

A spectral decomposition for the generator of the block counting process of the $\beta (3,1)$-coalescent is provided. This decomposition is strongly related to Riordan matrices and particular Fuss–Catalan numbers. The result is applied to obtain formulas for the distribution function and the moments of the absorption time of the $\beta (3,1)$-coalescent restricted to a sample of size $n$. We also provide the analog spectral decomposition for the fixation line of the $\beta (3,1)$-coalescent. The main tools in the proofs are generating functions and Siegmund duality. Generalizations to the $\beta (a,1)$-coalescent with parameter $a\in (0,\infty )$ are discussed leading to fractional differential or integral equations.