Fabius function: Difference between revisions

Revision as of 00:22, 11 June 2019

Revision as of 00:53, 11 June 2019

Line 20:

The Fabius function is constant zero for all non-positive arguments, and assumes rational values at positive [[dyadic rational]] arguments. These values can be computed using the formula:

:<math>

f\left(\frac nm{2^mn}\right)=\frac1{2^{mn^2}\,\left(\frac12;\,\frac12\right)_m_n}\,\sum _{k=0}^mn\,\frac{{\binom mn k}_{1/2}}{2^{k\,(k-1)}\,(mn+k)!} \sum_{\ellr=0}^{\lfloor 2^k\,nm\,-\,1\rfloor}\!(-1)^{s_2(\ellr)}\,\left(\ellr-2^k\,nm+\tfrac12\right)^{mn+k},</math>

where <math>n,\,m</math> are non-negative integers, <math>\left(a;\,q\right)_m</math> is the [[q-Pochhammer symbol]], <math>{\binom m k}_q</math> is the [[Gaussian binomial coefficient]], <math>\lfloor x\rfloor</math> is the [[floor function]], and <math>s_2(\ellr)</math> is the [[Digit sum|sum of digits]] of <math>\ell</math>''r'' in [[base-2]]. Note that <math>(-1)^{s_2(\ellr)}=f(2\ell2r+1)=t_\ellt_r</math> is just the signed [[Thue–Morse sequence]], satisfying the recurrence <math>t_0=1,\,\,t_\ellt_r=(-1)^\ellr\,t_{\lfloor\ell r/2\rfloor}.</math>

==References==