with u0 = 0, where the dot indicates differentiation with respect to t. In this note, we shall show how solutions to this equation can be formulated in the context of exponential Riordan arrays. The Riordan arrays we shall consider may be considered as parameterised (or “time”-dependent) Riordan arrays. We have already considered parameterized Riordan arrays [1], exploring the links between the...

We use the Riordan Group to solve the linear first order differential equation. As a consequence we also get the formula for the change of variable in the integration.

Riordan paths are Motzkin paths without horizontal steps on the x-axis. We establish a correspondence between Riordan paths and (321, 31̄42)-avoiding derangements. We also present a combinatorial proof of a recurrence relation for the Riordan numbers in the spirit of the Foata-Zeilberger proof of a recurrence relation on the Schröder numbers.

The problem we consider in the present paper is how to find the closed form of a class of combinatorial sums, if it exists. The problem is well known in the literature, and is as old as Combinatorial Analysis is, since we can go back at least to Euler’s time. More recently, Riordan has tried to give a general approach to the subject, proposing a variety of methods, many of which are related to ...

An infinite matrix is called totally positive if its minors of all orders are nonnegative. A nonnegative sequence (an)n≥0 is called log-convex (logconcave, resp.) if aiaj+1 ≥ ai+1aj ( aiaj+1 ≤ ai+1aj , resp.) for 0 ≤ i < j . The object of this talk is to study various positivity properties of Riordan arrays, including the total positivity of such a matrix, the log-convexity of the 0th column an...

In this paper, we present some formulas of symbolic operator summation, which involving Generalization well-know number sequences or polynomial sequences, and mean while we obtain some identities about the sequences by employing M-R‘s substitution rule. Keywords—Generating functions; Operators sequence group; Riordan arrays; R.G Operator group; Combinatorial identities.

In this paper, we prove the q -analogue of the fundamental theorem of Riordan arrays. In particular, by defining two new binary operations ∗q and ∗1/q , we obtain a q -analogue of the Riordan representation of the q -Pascal matrix. In addition, by aid of the q -Lagrange expansion formula we get q -Riordan representation for its inverse matrix.

We study the relation between binary words excluding a pattern and proper Riordan arrays. In particular, we prove necessary and sufficient conditions under which the number of words counted with respect to the number of zeroes and ones bits are related to proper Riordan arrays. We also give formulas for computing the generating functions (d(x), h(x)) defining the Riordan array.

For a large class of ordered trees it’s possible to take results at the root and move them to an arbitrary vertex (the mutator) or to a leaf of the tree. This is the uplift principle. If you also sort by height and record the results in matrix form you end up with one or more elements of a group of infinite lower triangular matrices called the Riordan group. The main tools used are generating f...