3 research outputs found
On certain combinatorial expansions of the Legendre-Stirling numbers
The Legendre-Stirling numbers of the second kind were introduced by Everitt
et al. in the spectral theory of powers of the Legendre differential
expressions. In this paper, we provide a combinatorial code for
Legendre-Stirling set partitions. As an application, we obtain combinatorial
expansions of the Legendre-Stirling numbers of both kinds. Moreover, we present
grammatical descriptions of the Jacobi-Stirling numbers of both kinds.Comment: 11 page
On the unimodality of the Taylor expansion coefficients of Jacobian elliptic functions
The Jacobian elliptic functions are standard forms of elliptic functions, and
they were independently introduced by C.G.J. Jacobi and N.H. Abel. In this
paper, we study the unimodality of Taylor expansion coefficients of the
Jacobian elliptic functions sn(u,k) and cn(u,k). By using the theory of
gamma-positivity, we obtain that the Taylor expansion coefficients of sn(u,k)
are symmetric and unimodal, and that of cn(u,k) are unimodal and alternatingly
increasing.Comment: 14 page
Gamma-positivity and partial gamma-positivity of descent-type polynomials
In this paper, we study gamma-positivity of descent-type polynomials by
introducing the change of context-free grammars method. We first present
grammatical proofs of the gamma-positivity of the Eulerian polynomials, type B
Eulerian polynomials, derangement polynomials, Narayana polynomials and type B
Narayana polynomials. We then provide partial gamma-positive expansions for
several multivariate polynomials associated to Stirling permutations,
Legendre-Stirling permutations, Jacobi-Stirling permutations and type B
derangements, and the recurrences for the partial gamma-coefficients of these
expansions are also obtained. Moreover, we define variants of the Foata-Strehl
group action which are used to give combinatorial interpretations for the
coefficients of most of these partial gamma-positive expansions.Comment: 31 page