Algebraic Independence of the Power Series Defined by Blocks of Digits

Linear relations between pattern sequences in a 〈q,r〉-numeration system

“…(6) follows from the definition of e q,r (w; n) and (7) is proved in [6,Lemma 4]. We show (8). Let w ∈ Σ * q,r with |w| = l and w = 0 l .…”

“…The second author [7] of the present paper proved that the generating functions of pattern sequences defined in distinct q-ary number systems are algebraically independent over C(z). Algebraic independence results of the values of these generating functions at an algebraic argument are also obtained in [6], [7], and [8].…”

“…Define the generating function of the pattern sequence e q,r (w; n) n 0 for w ∈ Σ * q,r by f (w; z) = n 0 e q,r (w; n)z n , which converges in |z| < 1, since e q,r (w; n) < n for sufficiently large n. In the case r = 0, Uchida [8] gave necessary and sufficient conditions for the generating functions f (w 1 ; z), . .…”

“…Let s = 1, so that 1 n < q − r. Then e q,r (wb; n) = 0 for any b ∈ Σ q,r and e q,r (w; n) = 1 if w = 0 l−1 n, = 0 otherwise. Since w = 0 l−1 n if and only if n = [w] q,r , we obtain (8). Suppose that (8) is valid for every integer m 1 with (m) q,r = s 1.…”

“…Then we apply (8) to get where α(n), β(n), γ(n) ∈ P l−1 (Z). Substituting these into (23), we obtain…”