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math.NT2025
Balanced Fibonacci word rectangles, and beyond
Jeffrey Shallit, Ingrid Vukusic
Following a recent paper of Anselmo et al., we consider rectangular matrices formed from the Fibonacci word, and we show that their balance properties can be solved wi…
math.NT2025
New properties of the -representation of integers
Jeffrey Shallit, Ingrid Vukusic
We prove a few new properties of the -representation of integers, where . In particular, we prove a 2012 conjecture of Kimberling. As software assistants, we…
math.NT2025
On the -adic valuations of values of Legendre polynomials
Max A. Alekseyev, Tewodros Amdeberhan, Jeffrey Shallit +1
We prove an explicit formula for the -adic valuation of the Legendre polynomials evaluated at a prime , and generalize an old conjecture of the third author. We also…