An odd-power identity involving discrete convolution - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2018

An odd-power identity involving discrete convolution

Kolosov Petro

Résumé

Let be a power function $f_{r,M}(s)$ defined for every $s$ within the finite set $M$ as follows $$f_{r,M}(s)= \begin{cases} s^r, \ &s\in M,\\ 0, \ &\mathrm{otherwise}. \end{cases} $$ Let a discrete convolution of $f_{r,M}(s)$ be denoted as follows $\mathrm{Conv}_{r,M}[n]=(f_{r,M}*f_{r,M})[n]$. Let a real coefficients $A_{m,j}$ be given by the following recurrence $$ A_{m,j} = \begin{cases} 0, & \mathrm{if } \ j<0 \ \mathrm{or } \ j>m, \\ (2j+1)\binom{2j}{j} \sum_{d=2j+1}^{m} A_{m,d} \binom{d}{2j+1} \frac{(-1)^{d-1}}{d-j} B_{2d-2j}, & \mathrm{if } \ 0 \leq j < m, \\ (2j+1)\binom{2j}{j}, & \mathrm{if } \ j=m. \end{cases} $$ In this paper we show that for every $n>0$ the following odd-power identities involving coefficients $A_{m,j}$ and convolution transform $\mathrm{Conv}_{r,M}[n]$ hold $$ \begin{split} n^{2m+1}+1&=\sum_{r=0}^{m}A_{m,r}\mathrm{Conv}_{r,\mathbb{N}}[n],\\ n^{2m+1}-1&=\sum_{r=0}^{m}A_{m,r}\mathrm{Conv}_{r,\mathbb{Z}_{>0}}[n],\\ n^{2m+1}&=\sum_{r=0}^{m}A_{m,r}\sum_{k=1}^{n} k^r(n-k)^r\\ &=\sum_{r=0}^{m}A_{m,r}\sum_{k=0}^{n-1} k^r(n-k)^r. \end{split} $$
Fichier principal
Vignette du fichier
an_odd_power_identity_involving_discrete_convolution.pdf (112.62 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-01283042 , version 1 (04-03-2016)
hal-01283042 , version 2 (08-04-2016)
hal-01283042 , version 3 (01-07-2016)
hal-01283042 , version 4 (08-03-2017)
hal-01283042 , version 5 (06-05-2017)
hal-01283042 , version 6 (01-08-2017)
hal-01283042 , version 7 (03-12-2017)
hal-01283042 , version 8 (12-01-2018)
hal-01283042 , version 9 (16-02-2018)
hal-01283042 , version 10 (29-05-2018)
hal-01283042 , version 11 (16-08-2018)
hal-01283042 , version 12 (18-10-2018)
hal-01283042 , version 13 (09-04-2019)
hal-01283042 , version 14 (14-04-2020)

Licence

Paternité - Pas d'utilisation commerciale

Identifiants

Citer

Kolosov Petro. An odd-power identity involving discrete convolution. 2018. ⟨hal-01283042v13⟩

Collections

TDS-MACS
1384 Consultations
936 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More