Abstract
In this paper we introduce different families of numerical and polynomial sequences by using Riordan pseudo involutions and Sheffer polynomial sequences. Many examples are given including dual of Hermite numbers and polynomials, dual of Bell numbers and polynomials, among other. The coefficients of some of these polynomials are related to the counting of different families of set partitions and permutations. We also studied the dual of Catalan numbers and dual of Fuss-Catalan numbers, giving several combinatorial identities.
In this paper, we perform a further investigation for the Catalan numbers. By making use of the method of derivatives and some properties of the Bell polynomials, we establish two new expressions for sums of products of arbitrary number of the Catalan numbers. The results presented here can be regarded as the development of some known formulas.
In the paper, motivated by the generating function of the Catalan
numbers in combinatorial number theory and with the aid of Cauchy’s
integral formula in complex analysis, the authors generalize the Catalan
numbers and its generating function, establish an explicit formula and
an integral representation for the generalization of the Catalan numbers
and corresponding generating function, and derive several integral
formulas and combinatorial identities.
In 2021, Mork and Ulness studied the Mandelbrot and Julia sets for a generalization of the well-explored function ηλ(z)=z2+λ. Their generalization was based on the composition of ηλ with the Möbius transformation μ(z)=1z at each iteration step. Furthermore, they posed a conjecture providing a relation between the coefficients of (each order) iterated series of μ(ηλ(z)) (at z=0) and the Catalan numbers. In this paper, in particular, we prove this conjecture in a more precise (quantitative) formulation.
Variations of the Catalan numbers from some nonassociative binary operations
