2025/02/19 by Hitczenko, Pawel
#05A16 #60C05 #60F05 #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2502.13382
Graham, Knuth and Patashnik in their book Concrete Mathematics called for development of a general theory of the solutions of recurrences defined by | n\atop k|=(αn+βk+γ)|n-1\atop k|+(α' n+β' k+γ')|n-1\atop k-1|+In=k=0 for 0≤ k≤ n and six parameters α,β,γ,α'β',γ'. Since then, a number of authors investigated various properties of the solutions of these recurrences. In this note we consider a probabilistic aspect, namely we consider the limiting distributions of sequences of integer valued random variables naturally associated with the solutions of such recurrences. We will give a complete description of the limiting behavior when α'=0 and the remaining five parameters are non--negative.