Preprint Article Version 1 This version is not peer-reviewed

# An Odd-Power Identity Involving Discrete Convolution

Version 1 : Received: 9 April 2019 / Approved: 10 April 2019 / Online: 10 April 2019 (10:27:59 CEST)

How to cite: Petro, K. An Odd-Power Identity Involving Discrete Convolution. Preprints 2019, 2019040126 (doi: 10.20944/preprints201904.0126.v1). Petro, K. An Odd-Power Identity Involving Discrete Convolution. Preprints 2019, 2019040126 (doi: 10.20944/preprints201904.0126.v1).

## Abstract

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 $$\begin{split} n^{2m+1}+1&=\sum_{r=0}^{m}A_{m,r}\mathrm{Conv}_{r,\mathbb{N}}[n],\\ n^{2m+1}-1&=\sum_{r=0}^{m}A_{m,r}\mathrm{Conv}_{r,\mathbb{Z}_{>0}}[n],\\ n^{2m+1}&=\sum_{r=0}^{m}A_{m,r}\sum_{k=1}^{n} k^r(n-k)^r\\ &=\sum_{r=0}^{m}A_{m,r}\sum_{k=0}^{n-1} k^r(n-k)^r. \end{split}$$

## Subject Areas

power identities; polynomials; convolution; convolution power; integral transforms; power function; binomial theorem; multinomial theorem; Faulhaber's formula; Bernoulli numbers; Worpitzky identity; Stirling numbers; falling factorial