The dual of number sequences, Riordan polynomials, and Sheffer polynomials

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.

Factorizations of some lower triangular matrices and related combinatorial identities

In this study, we investigate the connection between second order recurrence matrix and several combinatorial matrices such as generalized r-eliminated Pascal matrix, Stirling matrix of the first and of the second kind matrices. We give factorizations and inverse factorizations of these matrices by virtue of the second order recurrence matrix. Moreover, we derive several combinatorial identities which are more general results of some earlier works.

Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine and sine functions with applications

In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions for real powers of the inverse cosine (sine) function and the inverse hyperbolic cosine (sine) function. By applying different series expansions for the square of the inverse cosine function, the author not only finds infinite series representations of the circular constant Pi and its square, but also derives two combinatorial identities involving central binomial coefficients.

Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine functions and applications

In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions for real powers of the inverse cosine function and the inverse hyperbolic cosine function. By applying different series expansions for the square of the inverse cosine function, the author not only finds infinite series representations of the circular constant Pi and its square, but also derives two combinatorial identities involving central binomial coefficients.

Degenerate poly-Bell polynomials and numbers

Numerous mathematicians have studied ‘poly’ as one of the generalizations to special polynomials, such as Bernoulli, Euler, Cauchy, and Genocchi polynomials. In relation to this, in this paper, we introduce the degenerate poly-Bell polynomials emanating from the degenerate polyexponential functions which are called the poly-Bell polynomials when $\lambda \rightarrow 0$ λ → 0 . Specifically, we demonstrate that they are reduced to the degenerate Bell polynomials if $k = 1$ k = 1 . We also provide explicit representations and combinatorial identities for these polynomials, including Dobinski-like formulas, recurrence relationships, etc.

A general inverse matrix series relation and associated polynomials – II

We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].