Periodic sequences, binomials modulo a prime power, and a math/music application (Q6632120)

As anticipated by the authors [\textit{L. Fiorot} et al., Lect. Notes Comput. Sci. 13267, 376--382 (2022; Zbl 1492.00047)], this paper solves the questions posed by Andreatta, Vuza and Agon [\textit{M. Andreatta} et al., Soft Comput. 8, No. 9, 588--596 (2004; Zbl 1189.00025)] about the sequence \(V=[2,1,2,4,8,1,8,4]\) (whose coefficients are in \(\mathbb{Z} / 12 \mathbb{Z}\)) found by the musician Anatol Vieru, within the framework of twelve-tone serialism, together with the anti-difference operator plus the constant sequence [8] here called ``Vieru operator'' [after the Romanian composer Anatol Vieru (1926--1998)] and symbolized \(\mathscr{V}\).\N\NNamely, the authors establish a formula for the period of \(\mathscr{V}^{s}\), and they prove both the absence of the numbers \(3,6,9\) in \(\mathscr{V}^{s}\) and the proliferation of the values \(4\) and \(8\) among the coefficients of \(\mathscr{V}^{s}\). In addition to exhaustively concluding the algebraic investigation on Vieru's peculiar compositional technique, started by \textit{M. Andreatta} and \textit{D. T. Vuza} [Tatra Mt. Math. Publ. 23, 1--15 (2001; Zbl 1047.00006)], this paper enriches the \(p\)-adic valuation of sequences of binomial coefficients.\N\NIn fact, after recalling the uniqueness of the decomposition of a periodic sequence in its idempotent and nilpotent parts, the authors reduce the study of anti-differences of generic sequences to anti-differences of constant ones by using the Fundamental Theorem of finite calculus, the method of induction, the Stifel recursive formula, the Chinese Remainder Theorem, and a result concerning the periodicity modulo \(m\) of sequences of \(\binom{n}{k}\) from \textit{Ś. Zabek} [Ann. Univ. Mariae Curie-Skłodowska, Sect. A 10, 37--47 (1958; Zbl 0080.25902)]. Then, via Kummer's Theorem, they obtain new recurrence relations for binomial coefficients modulo a prime power with direct application to the mathematical-musical problems raised by \textit {A. Vieru} [The Book of Modes. Bucharest: Editura Muzicala (1993)]; their main recursive formula is defined as follows.\N\NFor \(k \geq 5\) and \(2^{k} \leq s<2^{k+1}\), denote: \N\begin{align*} \N\left(c_{1}, c_{2}, c_{3}, c_{4}\right) & :=(48,32,40,44) \\\N\left(c_{1}^{\prime}, c_{2}^{\prime}, c_{3}^{\prime}, c_{4}^{\prime}\right) & :=(48,40,44,48) \\\N\left(c_{1}^{\prime \prime}, c_{2}^{\prime \prime}, c_{3}^{\prime \prime}, c_{4}^{\prime \prime}\right) & :=(32,32,48,64) \\\N\mathcal{Z}_{k} & :=(Z(s))_{2^{k} \leq s<2^{k+1}}. \N\end{align*} \NThe initial condition of the \(2^{k}\)-tuple \(\mathcal{Z}_{k}\) is \N\begin{multline*}\N\mathcal{Z}_{5}=(88,64,80,88,92,64,80,88,104,92,104,108,94,78,88,96,108,96,\\ 104,108,110,102,108,112,118,114,118,120,64,64,96,128), \N\end{multline*}\Nwhereas, for \(k \geq 6\), \N\[ Z(s)= \N\begin{cases}2 Z\left(s-2^{k-1}\right) & \text { if } 2^{k} \leq s \leq 2^{k}+2^{k-2}-5 \\\NZ\left(s-2^{k-1}-2^{k-3}\right)+2^{k-5} c_{i} & \text { if } s=2^{k}+2^{k-2}-5+i, i=1,2,3,4 \\\N2 Z\left(s-2^{k-1}\right)-\mathfrak{d}_{k}(s) & \text { if } 2^{k}+2^{k-2} \leq s \leq 2^{k}+2^{k-1}-5 \\\NZ\left(s-2^{k-1}-2^{k-2}\right)+2^{k-5} c_{i}^{\prime} & \text { if } \left.s=2^{k}+2^{k-1}-5+i, i=1,2,3,4 \right) \\\NZ\left(s-2^{k}\right)+2^{k+1} & \text { if } 2^{k}+2^{k-1} \leq s \leq 2^{k+1}-5 \\\NZ\left(s-2^{k}\right)+2^{k-5} c_{i}^{\prime \prime} & \text { if } s=2^{k+1}-5+i, i=1,2,3,4 \N\end{cases} \] \Nwhere \(\mathfrak{d}_{k}(s)=2^{wt\left(2^{k}+2^{k-1}-4-s\right)+1}\), with \(wt(n)\) being the number of 1's in the binary expansion of \(n\), i.e., the Hamming weight of \(n\).