English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Consider a strongly $b$-multiplicative sequence and a prime $p$. Studying its $p$-rarefaction consists in characterizing the asymptotic behaviour of the sums of the first terms indexed by the multiples of $p$. The integer values of the "norm" $3$-variate polynomial $\mathcal N_{p,i_1,i_2}(Y_0,Y_1,Y_2)\!:=\!\prod_{j=1}^{p-1}\left(Y_0{+}\zeta_p^{i_1j}Y_1{+}\zeta_p^{i_2j}Y_2\right),$ where $\zeta_p$ is a primitive $p$-th root of unity, and $i_1,i_2{\in}\{1,2,\dots,p{-}1\},$ determine this asymptotic behaviour. It will be shown that a combinatorial method can be applied to $\mathcal N_{p,i_1,i_2}(Y_0,Y_1,Y_2).$ The method enables deducing functional relations between the coefficients as well as various properties of the coefficients of $\mathcal N_{p,i_1,i_2}(Y_0,Y_1,Y_2)$, in particular for $i_1{=}1,i_2{=}2,3.$ This method provides relations between binomial coefficients. It gives new proofs of the two identities $\prod_{j=1}^{p-1}\left(1{-}\zeta_p^j\right){=}p$ and $\prod_{j=1}^{p-1}\left(1{+}\zeta_p^j{-}\zeta_p^{2j}\right){=}L_p$ (the $p$-th Lucas number). The sign and the residue modulo $p$ of the symmetric polynomials of $1{+}\zeta_p{-}\zeta_p^2$ can also be obtained. An algorithm for computation of coefficients of $\mathcal N_{p,i_1,i_2}(Y_0,Y_1,Y_2)$ is developed.

Counting solutions without zeros or repetitions of a linear congruence and rarefaction in b-multiplicative sequences.