A generalization of Hata-Yamaguti’s results on the Takagi function

Digital Sum Problems for the Gray Code Representation of Natural Numbers

“…In this section, we prove Theorem 2 in a similar manner to the proof of Theorem 2.2 [18] which generalized Hata and Yamaguti's results connecting the Takagi function with Lebesgue's singular function [9]. We prepare some lemmas without proof.…”

A generalization of Hata-Yamaguti’s results on the Takagi function II: Multinomial case

“…Hata and Yamaguti showed that T is a unique continuous solution of the system (2). More generally, they obtained the following: In [15] we showed that the condition oo is a one Y]n=o [cn[ < oc necessary and gave a concrete representation of the solution. oo a unique continuous solution ifand on y if~n=o Ic~l < c~.…”

An explicit formula of the exponential sums of digital sums

“…In our previous paper [9], noticing the close connection between s(n) and the binomial measure, and using the result obtained in Sekiguchi and Shiota [13], we have obtained ah explicit formula of Sk(N). In this paper, by use of a method similar to that of [9], we study the exponential sum F(~, N).…”

“…This method is direct and does not require solving the inductive formula (see [9]). We notice that the higher order derivatives of L(r, x) with respect to r play ah important part in the explicit formula of Sk(N) and that 10L(r,x) 5 of ~=1/2 is the Takagi function (see Hata and Yamaguti [7], [13]). …”

“…Sekiguchi and Shiota[13]). L(r,x) is a continuous function valued analytic function of r 9 I and the equalityOkL(r, x) 0r k r=1/2 = k!T1/2,k(x) E(l)(1,t)+lE(2)(1,t)) (Coquet [3,Theorem 6], Osbaldestin [11, Theorem 4]), S3(N) = N + H3,2(t) + + H3,0(t) , n=0 and R(x) = lz,.o (x) -111.E(r,O) = 2r, E(r, 1-) = 1, E( 89 = 1, and E is continuous except for t E Z and periodic of period 1 ~ a function of t. Purthermore E is analytic in Sk ( N) = -o-~ F(~, N) ~=0 holds for k --1, 2, 3,...…”