Generalized Eulerian distributions from the wreath product $$\pmb {{\mathcal {C}}}_{\varvec{r}} \pmb {\wr } {\varvec{S}}_{\varvec{n}}$$ C r ≀ S n

Bingran Cui (), Lily Li Liu () and Haiyan Yuan ()
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Bingran Cui: Qufu Normal University
Lily Li Liu: Qufu Normal University
Haiyan Yuan: Rizhao No.1 Middle School of Shandong

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 1, 125-139

Abstract: Abstract In this paper, we consider two sequences of polynomials $$(P_n(q;a,b,c,d,k,s,t))_{n\ge 0}$$ ( P n ( q ; a , b , c , d , k , s , t ) ) n ≥ 0 and $$(Q_n(q;a,b,c,d))_{n\ge 0}$$ ( Q n ( q ; a , b , c , d ) ) n ≥ 0 , where $$P_n(q;a,b,c,d,k,s,t)$$ P n ( q ; a , b , c , d , k , s , t ) generalizes the cyclic Eulerian polynomials, the cyclic derangement polynomials, the binomial Eulerian polynomials for the wreath product $${\mathcal {C}}_r\wr S_n$$ C r ≀ S n and the binomial Eulerian polynomials of type B; and $$Q_n(q;a,b,c,d)$$ Q n ( q ; a , b , c , d ) generalizes the polynomials of involutions in $${\mathcal {C}}_r\wr S_n$$ C r ≀ S n . Using two different methods, we show the Jacobi continued fraction expressions of $$\sum P_n(q;a,b,c,d,k,s,t)x^n$$ ∑ P n ( q ; a , b , c , d , k , s , t ) x n and $$\sum Q_n(q;a,b,c,d)x^n$$ ∑ Q n ( q ; a , b , c , d ) x n . Then we discuss the strong q-log-convexity and Hankel determinants of these two sequences of polynomials.

Keywords: Continued fraction; q-log-convexity; Eulerian polynomial; Wreath product; Involution; 05A20; 11A55; 11B83 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-023-00466-3

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