(2025) Together with Ana Luzón and Manuel A. Morón, we studied differential equations in Ward’s calculus, an extension of the usual derivative defined via a formal power series h and acting on formal power series over a field of characteristic zero. Using an ultrametric framework on K[[x]], we applied Banach’s fixed point theorem, the fundamental theorem of h-calculus, and Barrow’s rule to establish existence and uniqueness results for various initial value problems. We developed operational methods, including a Heaviside-type calculus for h-derivatives, and characterized finite and infinite h-differential calculi through Sheffer’s expansion. Special cases connected to Pascal’s triangle, polylogarithms, Fibonacci numbers, and q-calculus yielded explicit solutions, often in hypergeometric form, and highlighted combinatorial interpretations. This work is a continuation of our previous article On Ward's differential calculus, Riordan matrices and Sheffer polynomials ( Linear Algebra Appl., 610 (2021), 440–473). These results are presented in our article Differential equations in Ward’s calculus, published in Journal of Mathematical Analysis and Applications. 

(2025) Together with Jean-Luc Baril and Pamela E. Harris, we studied flattened Catalan words, a new class of Catalan words whose maximal weakly increasing subwords have leading terms in weakly increasing order. We developed generating functions and derived closed formulas and asymptotic expressions for the number of such words classified by various statistics, including runs of (weak) ascents and descents, $\ell$-valleys, symmetric valleys, $\ell$-peaks, and symmetric peaks. Through bijective constructions, we related flattened Catalan words to order-consecutive partitions, binary words with dotted entries, Dyck paths with restricted patterns, and ordered trees with specified leaf configurations. Our findings offer new interpretations for several known sequences in the OEIS and provide fresh enumerative results for peaks and valleys in Catalan structures. These results are presented in our article Flattened Catalan Words, published in Bulletin of the ICA.

(2025) Together with Daniel F. Checa, Mohammed L. Nadji, and Moussa Ahmia, we studied Arndt compositions, a class of integer compositions with alternating descents between odd-indexed and even-indexed parts. We extended the theory in several directions: compositions with parts restricted to a finite set, palindromic Arndt compositions, and colored variants. Using generating functions, we established bijections with Padovan and Narayana’s cows sequences and derived explicit formulas for compositions with fixed numbers of parts. For palindromic compositions, we developed new generating function expressions based on alternating compositions and analyzed their asymptotic behavior. Finally, we introduced and enumerated colored Arndt compositions, providing rational and asymptotic generating functions. These results are detailed in our paper Arndt Compositions with Restricted Parts, Palindromes, and Colored Variants, published in the Journal of Integer Sequences.

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