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On the \(p\)-Adic Properties of \(2\)-Sected Sums Involving Binomial Coefficients

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Abstract

For a prime \(p\), we determine the \(p\)-adic order of certain lacunary sums involving binomial coefficients. After forming sequences of the sums, we use various techniques, recurrence relations, multisecting ordinary generating functions, periodicity of linear sequences, divisibility sequences, Lucas and their companion sequences, and a \(2\)-adic analytic technique among other methods. We focus on \(2\)-sected sums and suggest extensions to other sections. We determine the rate of convergence of some subsequences to \(0\) or \(1\) in \( {\mathbb Z}_p \). The discovery phase of finding exact \(p\)-adic orders is followed by a series of suggestions in order to lead us to the predicted results, and several examples are offered to illustrate and highlight the differences. The current paper provides a significant extension to a method introduced by the author to fully characterize the \(p\)-adic orders of Fibonacci and Lucas numbers.

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  1. Mathematics Department, Occidental College, Los Angeles, USA

    T. Lengyel

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  1. T. Lengyel
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Lengyel, T. On the \(p\)-Adic Properties of \(2\)-Sected Sums Involving Binomial Coefficients. P-Adic Num Ultrametr Anal Appl 15, 23–47 (2023). https://doi.org/10.1134/S2070046623010028

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