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Fabius function

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In mathematics, the Fabius function is an example of an <a href="Smoothness" title="Smoothness">infinitely differentiable function</a> that is nowhere <a href="Analytic_function" title="Analytic function">analytic</a>, found by Jaap Fabius (<a href="#CITEREFFabius1966">1966</a>).

This function satisfies the initial condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(0)=0}">

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<annotation encoding="application/x-tex">{\displaystyle f(0)=0}</annotation>

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</math><img src="./8d308c32c9894b88115262081194321ae7d9bbf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(0)=0}" loading="lazy">, the symmetry condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(1-x)=1-f(x)}">

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<annotation encoding="application/x-tex">{\displaystyle f(1-x)=1-f(x)}</annotation>

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</math><img src="./c17b8f10783d141cb06d8f25b4947ee9270cdd09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.939ex; height:2.843ex;" alt="{\displaystyle f(1-x)=1-f(x)}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq x\leq 1,}">

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<annotation encoding="application/x-tex">{\displaystyle 0\leq x\leq 1,}</annotation>

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</math><img src="./2f450acc3c8ef3db4d0ef19a9bc86b538da2e974.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.498ex; height:2.509ex;" alt="{\displaystyle 0\leq x\leq 1,}" loading="lazy"> and the <a href="Functional_differential_equation" title="Functional differential equation">functional differential equation</a>

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<annotation encoding="application/x-tex">{\displaystyle f'(x)=2f(2x)}</annotation>

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</math><img src="./eb72a6a2214bb8dfa32f2dad9dc53c873eb15dab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.985ex; height:3.009ex;" alt="{\displaystyle f'(x)=2f(2x)}" loading="lazy"></dd></dl>

for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq x\leq 1/2.}">

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<annotation encoding="application/x-tex">{\displaystyle 0\leq x\leq 1/2.}</annotation>

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</math><img src="./b25911162b6f9540ee20907abc2b235c0f9b7a1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.823ex; height:2.843ex;" alt="{\displaystyle 0\leq x\leq 1/2.}" loading="lazy"> It follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

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</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> is monotone increasing for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq x\leq 1,}">

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<annotation encoding="application/x-tex">{\displaystyle 0\leq x\leq 1,}</annotation>

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</math><img src="./2f450acc3c8ef3db4d0ef19a9bc86b538da2e974.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.498ex; height:2.509ex;" alt="{\displaystyle 0\leq x\leq 1,}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(1/2)=1/2}">

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<annotation encoding="application/x-tex">{\displaystyle f(1/2)=1/2}</annotation>

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</math><img src="./9c0cee3c2da7a07d0bb135a3e6d624f0ac48bccb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.161ex; height:2.843ex;" alt="{\displaystyle f(1/2)=1/2}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(1)=1}">

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<annotation encoding="application/x-tex">{\displaystyle f(1)=1}</annotation>

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</math><img src="./c23ec03a1dad7631fc47878cb66b800a538dff1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(1)=1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(1-x)=f'(x)}">

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<annotation encoding="application/x-tex">{\displaystyle f'(1-x)=f'(x)}</annotation>

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</math><img src="./520286c28105f505097af96b99043e672b9a6eef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.39ex; height:3.009ex;" alt="{\displaystyle f'(1-x)=f'(x)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(x)+f'({\tfrac {1}{2}}-x)=2.}">

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<annotation encoding="application/x-tex">{\displaystyle f'(x)+f'({\tfrac {1}{2}}-x)=2.}</annotation>

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It was also written down as the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of

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<annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(z)=\prod _{m=1}^{\infty }\left(\cos {\frac {\pi z}{2^{m}}}\right)^{m}}</annotation>

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</math><img src="./4cbb14d3a766e998ea42e30744ab0c9127e9a022.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.248ex; height:6.843ex;" alt="{\displaystyle {\hat {f}}(z)=\prod _{m=1}^{\infty }\left(\cos {\frac {\pi z}{2^{m}}}\right)^{m}}" loading="lazy"></dd></dl>

by Børge Jessen and Aurel Wintner (<a href="#CITEREFJessenWintner1935">1935</a>).

The Fabius function is defined on the unit interval, and is given by the <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }2^{-n}\xi _{n},}</annotation>

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</math><img src="./df1e3ffb68099d0d96c8de3d039e9d2e4370bac8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.285ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }2^{-n}\xi _{n},}" loading="lazy"></dd></dl>

where the ξn are <a href="Independence_(probability)" class="mw-redirect" title="Independence (probability)">independent</a> <a href="Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniformly distributed</a> <a href="Random_variable" title="Random variable">random variables</a> on the <a href="Unit_interval" title="Unit interval">unit interval</a>. That distribution has an expectation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation>

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</math><img src="./edef8290613648790a8ac1a95c2fb7c3972aea2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}" loading="lazy"> and a variance of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{36}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{36}}}</annotation>

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</math><img src="./654a30909bb4f8729fc533de1cae4a38657b0fd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.48ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{36}}}" loading="lazy">.

There is a unique extension of f to the real numbers that satisfies the same differential equation for all x. This extension can be defined by f (x) = 0 for x ≤ 0, f (x + 1) = 1 − f (x) for 0 ≤ x ≤ 1, and f (x + 2r) = −f (x) for 0 ≤ x ≤ 2r with r a positive <a href="Integer" title="Integer">integer</a>. The sequence of intervals within which this function is positive or negative follows the same pattern as the <a href="Thue%E2%80%93Morse_sequence" title="Thue–Morse sequence">Thue–Morse sequence</a>.

The Rvachëv up function<a href="#cite_note-1">[1]</a> is closely related: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(t)={\begin{cases}F(t+1),\quad |t|<1\\0,\quad |t|\geq 1\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle u(t)={\begin{cases}F(t+1),\quad |t|<1\\0,\quad |t|\geq 1\end{cases}}}</annotation>

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</math> which fulfills the <a href="Delay_differential_equation" title="Delay differential equation">Delay differential equation</a><a href="#cite_note-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}u(t)=2u(2t+1)-2u(2t-1).}</annotation>

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</math> (see <a href="Delay_differential_equation" title="Delay differential equation">Another example</a>).

<div class="mw-heading mw-heading2"><h2 id="Values">Values</h2></div>

The Fabius function is constant zero for all non-positive arguments, and assumes rational values at positive <a href="Dyadic_rational" title="Dyadic rational">dyadic rational</a> arguments. For example:<a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle f(1)=1}</annotation>

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</math><img src="./c23ec03a1dad7631fc47878cb66b800a538dff1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(1)=1}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle f({\tfrac {1}{2}})={\tfrac {1}{2}}}</annotation>

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</math><img src="./5b15415dd65ebb5017b00e467fea4b3808ff4845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.503ex; height:3.509ex;" alt="{\displaystyle f({\tfrac {1}{2}})={\tfrac {1}{2}}}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle f({\tfrac {1}{4}})={\tfrac {5}{72}}}</annotation>

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</math><img src="./72d813acc2fb8efdc630ceef1d49e9c2f037a0aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.325ex; height:3.676ex;" alt="{\displaystyle f({\tfrac {1}{4}})={\tfrac {5}{72}}}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle f({\tfrac {1}{8}})={\tfrac {1}{288}}}</annotation>

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</math><img src="./d9d8a61ba416c473928190a53e5d83dc5818f674.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.147ex; height:3.676ex;" alt="{\displaystyle f({\tfrac {1}{8}})={\tfrac {1}{288}}}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle f({\tfrac {1}{16}})={\tfrac {143}{2073600}}}</annotation>

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</math><img src="./e457287c6b956e5caf160312728c73f15a0f53d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.256ex; height:3.843ex;" alt="{\displaystyle f({\tfrac {1}{16}})={\tfrac {143}{2073600}}}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle f({\tfrac {1}{32}})={\tfrac {19}{33177600}}}</annotation>

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</math><img src="./8f3ec01ff05db61f1ebc3621cff668c083f91c62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.078ex; height:3.676ex;" alt="{\displaystyle f({\tfrac {1}{32}})={\tfrac {19}{33177600}}}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle f({\tfrac {1}{64}})={\tfrac {1153}{561842749440}}}</annotation>

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</math><img src="./4a2b55759917a6bfc6bc72f5544dfb41c1617a23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.366ex; height:3.676ex;" alt="{\displaystyle f({\tfrac {1}{64}})={\tfrac {1153}{561842749440}}}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle f({\tfrac {1}{128}})={\tfrac {583}{179789679820800}}}</annotation>

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</math><img src="./5e1e1b439bd5cc8a1e1e5d3a748dfdc524cd6b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:22.654ex; height:3.676ex;" alt="{\displaystyle f({\tfrac {1}{128}})={\tfrac {583}{179789679820800}}}" loading="lazy"></li></ul>

with the numerators listed in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A272755" class="extiw external" title="oeis:A272755">A272755</a> and denominators in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A272757" class="extiw external" title="oeis:A272757">A272757</a>.

<div class="mw-heading mw-heading2"><h2 id="Asymptotic">Asymptotic</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\log f(x)&=-{\frac {\log ^{2}x}{2\log 2}}+{\frac {\log x\cdot \log(-\log x)}{\log 2}}-\left({\frac {1}{2}}+{\frac {1+\log \log 2}{\log 2}}\right)\log x-{\frac {\log ^{2}(-\log x)}{2\log 2}}+{\frac {\log \log 2\cdot \log(-\log x)}{\log 2}}\\&+\left({\frac {6\gamma ^{2}+12\gamma _{1}-\pi ^{2}-6\log ^{2}\log 2}{12\log 2}}-{\frac {7\log 2}{12}}-{\frac {\log \pi }{2}}\right)+{\frac {\log ^{2}(-\log x)}{2\log 2\cdot \log x}}-{\frac {\log \log 2\cdot \log(-\log x)}{\log 2\cdot \log x}}+O\!\left({\frac {1}{\log x}}\right)\end{aligned}}}</annotation>

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</math><img src="./5ed01faa13097cfbee5cb57752a8065e531e10da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:118.33ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}\log f(x)&=-{\frac {\log ^{2}x}{2\log 2}}+{\frac {\log x\cdot \log(-\log x)}{\log 2}}-\left({\frac {1}{2}}+{\frac {1+\log \log 2}{\log 2}}\right)\log x-{\frac {\log ^{2}(-\log x)}{2\log 2}}+{\frac {\log \log 2\cdot \log(-\log x)}{\log 2}}\\&+\left({\frac {6\gamma ^{2}+12\gamma _{1}-\pi ^{2}-6\log ^{2}\log 2}{12\log 2}}-{\frac {7\log 2}{12}}-{\frac {\log \pi }{2}}\right)+{\frac {\log ^{2}(-\log x)}{2\log 2\cdot \log x}}-{\frac {\log \log 2\cdot \log(-\log x)}{\log 2\cdot \log x}}+O\!\left({\frac {1}{\log x}}\right)\end{aligned}}}" loading="lazy">

for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to 0^{+},}">

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<annotation encoding="application/x-tex">{\displaystyle x\to 0^{+},}</annotation>

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</math><img src="./49da54449023d4631c4ce27648fdafbd8a280fe4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.264ex; height:2.843ex;" alt="{\displaystyle x\to 0^{+},}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

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</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> is <a href="Euler's_constant" title="Euler's constant">Euler's constant</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma _{1}}</annotation>

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</math><img src="./6c2d6da217f4d6e937887bad1c90b371314830f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.259ex; height:2.176ex;" alt="{\displaystyle \gamma _{1}}" loading="lazy"> is the <a href="Stieltjes_constants" title="Stieltjes constants">Stieltjes constant</a>. Equivalently,

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<annotation encoding="application/x-tex">{\displaystyle \log f\!\left(2^{-n}\right)=-{\frac {n^{2}\log 2}{2}}-n\log n+\left(1+{\frac {\log 2}{2}}\right)n-{\frac {\log ^{2}n}{2\log 2}}+\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 ^{2}n}{2n\log ^{2}2}}+O\!\left({\frac {1}{n}}\right)}</annotation>

</semantics>

</math><img src="./7b6704394b90be04faa3dc733949f18324decf87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:123.877ex; height:6.843ex;" alt="{\displaystyle \log f\!\left(2^{-n}\right)=-{\frac {n^{2}\log 2}{2}}-n\log n+\left(1+{\frac {\log 2}{2}}\right)n-{\frac {\log ^{2}n}{2\log 2}}+\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 ^{2}n}{2n\log ^{2}2}}+O\!\left({\frac {1}{n}}\right)}" loading="lazy">

for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty .}">

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<annotation encoding="application/x-tex">{\displaystyle n\to \infty .}</annotation>

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</math><img src="./d8a34a9f62668de90200a6cbde865c27af2cdbb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.979ex; height:1.843ex;" alt="{\displaystyle n\to \infty .}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://oeis.org/A288163">"A288163 - Oeis"</a>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFJuan_Arias_de_Reyna2017" class="citation arxiv cs1">Juan Arias de Reyna (2017). "Arithmetic of the Fabius function". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1702.06487">1702.06487</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.NT">math.NT</a>].</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFSloane_"A272755"" class="citation web cs1"><a href="Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.). <a rel="nofollow" class="external text" href="https://oeis.org/A272755">"Sequence A272755 (Numerators of the Fabius function F(1/2^n).)"</a>. The <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a>. OEIS Foundation.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFSloane_"A272757"" class="citation web cs1"><a href="Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.). <a rel="nofollow" class="external text" href="https://oeis.org/A272757">"Sequence A272757 (Denominators of the Fabius function F(1/2^n).)"</a>. The <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a>. OEIS Foundation.</cite>

</li>

</ol></div></div>

<ul><li><cite id="CITEREFFabius1966" class="citation cs2">Fabius, J. (1966), "A probabilistic example of a nowhere analytic C ∞-function", Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 5 (2): 173–174, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf00536652">10.1007/bf00536652</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0197656">0197656</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122126180">122126180</a></cite></li>

<li><cite id="CITEREFJessenWintner1935" class="citation cs2">Jessen, Børge; Wintner, Aurel (1935), "Distribution functions and the Riemann zeta function", Trans. Amer. Math. Soc., 38: 48–88, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9947-1935-1501802-5">10.1090/S0002-9947-1935-1501802-5</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1501802">1501802</a></cite></li>

<li><cite id="CITEREFDimitrov2006" class="citation thesis cs1">Dimitrov, Youri (2006). <a rel="nofollow" class="external text" href="http://rave.ohiolink.edu/etdc/view?acc_num=osu1155149204">Polynomially-divided solutions of bipartite self-differential functional equations</a> (Thesis).</cite></li>

<li><cite id="CITEREFArias_de_Reyna2017" class="citation arxiv cs1">Arias de Reyna, Juan (2017). "Arithmetic of the Fabius function". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1702.06487">1702.06487</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.NT">math.NT</a>].</cite></li>

<li><cite id="CITEREFArias_de_Reyna2017" class="citation arxiv cs1">Arias de Reyna, Juan (2017). "An infinitely differentiable function with compact support: Definition and properties". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1702.05442">1702.05442</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.CA">math.CA</a>].</cite> (an English translation of the author's paper published in Spanish in 1982)</li>

<li>Alkauskas, Giedrius (2001), "Dirichlet series associated with Thue-Morse sequence", <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180412220529/https://www.pdf-archive.com/2018/04/13/thue-morse/thue-morse.pdf">preprint</a>.</li>

<li>Rvachev, V. L., Rvachev, V. A., "Non-classical methods of the approximation theory in boundary value problems", Naukova Dumka, Kiev (1979) (in Russian).</li></ul>

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