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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.

This note discussed the computation of Fermat's pseudoprimes (Fermat's Psp). [ABSTRACT FROM AUTHOR]

Inspired by the Fibonacci tree shown in the recent book, Catalan Numbers, by Richard Stanley, we present a combinatorial way to construct double Riordan arrays by using ECO technique. The sequence characterizations of double Riordan arrays and subgroups of the double Riordan group are found. As an extension of Pascal–Fibonacci triangle, the compression forms of double Riordan arrays called double quasi-Riordan arrays are defined. The connections of lower triangular matrices and double Riordan arrays as well as their compressions are given. Those results are also extended to the case of high order Riordan arrays, which are defined in the paper. The pairs of Sheffer polynomials and pairs of summation formulas associated with double Riordan arrays are defined and discussed. [ABSTRACT FROM AUTHOR]

In this paper, we present the Riordan arrays called Fuss–Catalan matrices which are constructed by the convolutions of the generating functions of the Fuss–Catalan numbers. We also discuss weighted sums of the Fuss–Catalan matrices, using such matrices as transformations of recursive sequences, and their connection with stabilizer subgroups of the Riordan group. [ABSTRACT FROM AUTHOR]

Here we use row sum generating functions and alternating sum generating functions to characterize Riordan arrays and subgroups of the Riordan group. Numerous applications and examples are presented which include the construction of Girard–Waring type identities. We also show the extensions to weighted sum (generating) functions, called the expected value (generating) functions of Riordan arrays. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a linear operator T defined in Riordan group R by using the upper shift matrix U and lower shift matrix U T , namely for each R ∈ R , T : R ↦ U R U T . Some isomorphic properties of the operator T and the structures of its range sets for different domains are studied. By using the operator T and the properties of Bell subgroup of R , the Riordan type Chu–Vandermonde identities and the Riordan equivalent identities of Format Last Theorem and Beal Conjecture are shown. The applications of the shift operators to the complementary Riordan arrays and to the Riordan involutions and Riordan pseudo-involutions are also presented. [ABSTRACT FROM AUTHOR]