2024/05/03 by Agarwal, Archit, Bhoria, Subhash Chand, Eyyunni, Pramod +1
#33D15 #60F99 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary 11P84 #Probability (math.PR) #Secondary 05C80
paper · doi:10.48550/arxiv.2405.01877
In I981, Uchimura studied a divisor generating q-series that has applications in probability theory and in the analysis of data structures, called heaps. Mainly, he proved the following identity. For |q|<1, ∑n=1^∞ n qn (qn+1)_∞ =∑n=1∞ \frac(-1)n-1 q(n(n+1))/(2) (1-qn) ( q)n = ∑n=1∞ ( qn )/(1-qn). Over the years, this identity has been generalized by many mathematicians in different directions. Uchimura himself in 1987, Dilcher (1995), Andrews-Crippa-Simon (1997), and recently Gupta-Kumar (2021) found a generalization of the aforementioned identity. Any generalization of the right most expression of the above identity, we name as divisor-type sum, whereas a generalization of the middle expression we say Ramanujan-type sum, and any generalization of the left most expression we refer it as Uchimura-type sum. Quite surprisingly, Simon, Crippa and Collenberg (1993) showed that the same divisor generating function has a connection with random acyclic digraphs. One of the main themes of this paper is to study these different generalizations and present a unified theory. We also discuss applications of these generalized identities in probability theory for the analysis of heaps and random acyclic digraphs.