Jordan Canonical Forms of Riordan Arrays

We consider the possible Jordan canonical forms of Riordan arrays. We prove that there are, in fact, only two such forms. Moreover, the transition matrix is in the Riordan group only in the case when the given Riordan array has one of some three specific forms.

On identities involving generalized harmonic, hyperharmonic and special numbers with Riordan arrays

In this paper, by means of the summation property to the Riordan array, we derive some identities involving generalized harmonic, hyperharmonic and special numbers. For example, for n ≥ 0, ∑ k = 0 n B k k ! H ( n . k , α ) = α H ( n + 1 , 1 , α ) - H ( n , 1 , α ) , \sum\limits_{k = 0}^n {{{{B_k}} \over {k!}}H\left( {n.k,\alpha } \right) = \alpha H\left( {n + 1,1,\alpha } \right) - H\left( {n,1,\alpha } \right)} , and for n > r ≥ 0, ∑ k = r n - 1 ( - 1 ) k s ( k , r ) r ! α k k ! H n - k ( α ) = ( - 1 ) r H ( n , r , α ) , \sum\limits_{k = r}^{n - 1} {{{\left( { - 1} \right)}^k}{{s\left( {k,r} \right)r!} \over {{\alpha ^k}k!}}{H_{n - k}}\left( \alpha \right) = {{\left( { - 1} \right)}^r}H\left( {n,r,\alpha } \right)} , where Bernoulli numbers Bn and Stirling numbers of the first kind s (n, r).

A note on Eulerian numbers and Toeplitz matrices

This note presents a new formula of Eulerian numbers derived from Toeplitz matrices via Riordan array approach.

A Riordan array approach to Apostol type-Sheffer sequences

In this article, the generalized Apostol type-Sheffer sequences are introduced and their properties including the quasi-monomiality, determinant form and series and conjugate representations are derived via Riordan array techniques. The generalized Apostol-Bernoulli, Apostol-Euler and Apostol- Genocchi-Sheffer sequences are considered as their special cases. Certain examples are framed in terms of the generalized Apostol Bernoulli-associated Laguerre sequences, generalized Apostol-Euler-Hermite sequences and generalized Apostol-Genocchi-Legendre sequences to give the applications of main results. The numerical results to calculate the zeros and approximate solutions of these sequences are given and their graphical representations are shown.