Orthorecursive expansion of unity

Alexander Kalmynin; Petr Kosenko

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arXiv · 1901.04044

Orthorecursive expansion of unity

Alexander Kalmynin·Petr Kosenko

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Abstract

We study the properties of a sequence cn defined by the recursive relation \[\frac{c_0}{n + 1}+\frac{c_1}{n + 2}+\ldots+\frac{c_n}{2n + 1}=0\] for $n>1$ and $c_0=1$. This sequence also has an alternative definition in terms of certain norm minimization in the space $L^2([0, 1])$. We prove estimates on growth order of $c_n$ and the sequence of its partial sums, infinite series identities, connecting $c_n$ with harmonic numbers $H_n$ and also formulate some conjectures based on numerical computations.

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Alexander Kalmynin, Petr Kosenko. 2019-01-13. Orthorecursive expansion of unity. https://arxiv.org/abs/1901.04044

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