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arXiv · 2407.19530

Periodic Column Partial Sums in the Riordan Array of a Polynomial

Nikolai A. Krylov
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Abstract

When $p(t)$ is a polynomial of degree $d$, $k$-th column of the Riordan array $\bigl(1/(1 - t^{d+1}), tp(t)\bigr)$ is an eventually periodic sequence with the repeating part beginning at the $1 + (k-1)(d+1)$-st term. The pre-periodic terms add up to the $(k-1)(d+1)$-st partial sum of the corresponding formal power series, and thus the Riordan array of $p(t)$ generates a sequence of column partial sums. We classify linear and quadratic polynomials, and present a particular family of polynomials of higher degrees, for which such sequences of column partial sums are eventually periodic.

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Nikolai A. Krylov. 2024-07-28. Periodic Column Partial Sums in the Riordan Array of a Polynomial. https://arxiv.org/abs/2407.19530

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