Fabius function: Difference between revisions

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There is a unique extension of {{mvar|f}} to the real numbers that satisfies the same differential equation for all ''x''. This extension can be defined by {{math|1=''f''{{hsp}}(''x'') = 0}} for {{math|''x'' ≤ 0}}, {{math|1=''f''{{hsp}}(''x'' + 1) = 1 − ''f''{{hsp}}(''x'')}} for {{math|0 ≤ ''x'' ≤ 1}}, and {{math|1=''f''{{hsp}}(''x'' + 2^''r'') = −''f''{{hsp}}(''x'')}} for {{math|0 ≤ ''x'' ≤ 2^''r''}} with {{mvar|r}} a positive [[integer]]. The sequence of intervals within which this function is positive or negative follows the same pattern as the [[Thue–Morse sequence]].

The ''Rvachëv up function''<ref>{{cite web | url=https://oeis.org/A288163 | title=A288163 - Oeis }}</ref> is closely related: {{<math|up display="block"> u(''x''t) =\begin{{=}cases} F(t+1),\quad -|t|<1 {{!}}''x''{{!}})\\ for0, {{!}}''x''{{!}}\quad ≤|t|\geq 1 \end{cases}},</math> which fulfills the [[Delay differential equation]]<ref>{{cite arXiv | eprint=1702.06487 | author1=Juan Arias de Reyna | title=Arithmetic of the Fabius function | year=2017 | class=math.NT }}</ref>

<math display="block">\frac{d}{dt}u(t)=2u(2t+1)-2u(2t-1).</math> (see [[Delay differential equation|Another example]]).

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with the numerators listed in {{OEIS2C|A272755}} and denominators in {{OEIS2C|A272757}}.

==Asymptotic==

<math>\begin{align}\log f(x)&=-\frac{\log^2x}{2\log2}+\frac{\log x\cdot\log(-\log x)}{\log2}-\left(\frac12+\frac{1+\log\log2}{\log2}\right)\log x -\frac{\log^2(-\log x)}{2\log2}+\frac{\log\log 2\cdot\log(-\log x)}{\log2}\\&+\left(\frac{6\gamma ^2+12\gamma_1-\pi^2-6\log^2\log2}{12\log 2}-\frac{7\log 2}{12}-\frac{\log\pi}2\right)+\frac{\log^2(-\log x)}{2\log2\cdot\log x}-\frac{\log\log2\cdot\log(-\log x)}{\log2\cdot\log x}+O\!\left(\frac1{\log x}\right)\end{align}</math>

for <math>x\to0^+,</math> where <math>\gamma</math> is [[Euler's constant]], and <math>\gamma_1</math> is the [[Stieltjes constants|Stieltjes constant]]. Equivalently,

<math>\log f\!\left(2^{-n}\right)=-\frac{n^2\log2}2-n\log n+\left(1+\frac{\log2}2\right)n -\frac{\log^2n}{2\log2}+\left(\frac{6\gamma ^2+12\gamma_1-\pi^2}{12\log 2}-\frac{7\log 2}{12}-\frac{\log\pi}2\right)-\frac{\log^2n}{2n\log^22}+O\!\left(\frac1n\right)</math>

for <math>n\to\infty.</math>

==References==