On the Orthorecursive Expansion of Unity

Abstract

The orthorecursive expansion of unity with respect to the monomial system [Formula: see text] in [Formula: see text] produces a sequence of rational coefficients satisfying an explicit linear recurrence. Kalmynin and Kosenko proved that the coefficients decay at least as [Formula: see text] and that their partial sums are [Formula: see text], but the optimal rates remained open. We improve the partial sum bound to [Formula: see text] for every [Formula: see text], where [Formula: see text] is determined by the zeros of a transcendental function built from the digamma function, and the coefficient bound to [Formula: see text]. The method recasts the recurrence as a multiplicative Volterra integral equation and inverts it via a resolvent whose Mellin transform is controlled by the zero-free region of the associated kernel. We also propose a spectral form of the oscillatory asymptotic conjectured by Kalmynin and Kosenko, with a corrected decay exponent.