In the realm of the Riordan group, we consider the characterization of Riordan arrays by means of the A-and Z-sequences. It corresponds to a horizontal construction of a Riordan array, whereas the traditional approach is through column generating functions. We show how the A-and Z-sequences of the product of two Riordan arrays are derived from those of the two factors; similar results are obtained for the inverse. Finally, we give the characterizations relative to the some subgroups of the Riordan group, in particular of the hitting-time subgroup.

We define the Sheffer group of all Sheffer-type polynomials and prove the isomorphism between the Sheffer group and the Riordan group. An equivalence of the Riordan array pair and generalized Stirling number pair is also presented. Finally, we discuss a higher dimensional extension of Riordan array pairs.

Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a general method to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.

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Under the slow varying ambient electric field, positive leader propagation exhibits steps characterized by intense reilluminations and abrupt elongations. These steps are currently not well understood. In this work, we investigate these steps in laboratory atmospheric discharges, using a high‐speed video camera and a synchronized electrical parameter measurement system. The discharge, emitting weak light and preceding the intense reillumination, is discovered. This finding suggests that the leader channel actually restarts and extends forward before the intense reillumination, which deepens our understanding of the dynamic process of the positive leader step. The discharge before the intense reillumination contributes to the corona inception from the electrode, leading to the intense reillumination of the leader channel and the emergence of an intense corona streamer burst from the leader tip.