Very interested in Computer Science, in particular, in .NET enviroment. During my free time I do Math, my interests in math are about Faulhaber's formula, Binomial theorem and other realted to power identities topics.
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
\begin{equation*}
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}
\end{equation*}
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{equation*}
\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}
\end{equation*}
Related OEIS sequences:
Numerators and Denominators of \(A_{m,r}\): A302971,
A304042
MSC classes: 11C08, 44A35
Comments: 5 pages
Keywords: Power Identities, Polynomials, Convolution, Convolution power, Integral transforms
The main aim of this paper to establish the relations between forward, backward and central finite difference and divided difference
(that is discrete analog of the derivative) and partial and ordinary high-order derivatives of the polynomials.
Calculating the value of \(C^{k\in\{1,\infty\}}\) class of smoothness real-valued function's derivative in point of \(\mathbb{R}^+\) in radius of convergence of its aylor polynomial (or series), applying an analog of Newton's binomial theorem and \(q\)-difference operator. \((P,q)\)-power difference introduced in section 5. Additionally, by means of Newton's interpolation formula, the discrete analog of Taylor series interpolation using \(q\)-difference and \(p,q\)-power difference is shown.
On the Eulerian numbers and Power Sums - In this short report we discuss a relation between Triangle of Eulerian Numbers and Power sums of the form \(\Sigma_k^n k^m\), where \(n,m\) are positive integers. This manuscript shows that arXiv:1805.11445 another time proves the Worpitzky Identity.
Contributions to the OEIS
OIES stands for the On-Line Encyclopedia of Integer Sequences, established by Neil J. Sloane in 1965 and since 1996 is published online. Here are some sequences contributed by me.
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Consider the identity (arXiv:1603.02468v15, p. 1.), that is
$$
(\star)\quad\quad P_m(\ell,T) := \sum_{k=1}^{\ell}\sum_{j=0}^m A_{m,j}k^j(T-k)^j=T^{2m+1},
$$
as \(\ell=T\). To show an example of use of above identity, consider the triangle of \(A_{m,j}\) coefficients, the sequences A302971, A304042 for the numerators and denominators, respectively. Recall the values of \(k(n-k), \ 1\leq k \leq n\), the sequence A094053. Consider the case \(T=n=3\) and \(m=2\) in \((\star)\). So, we keep our attention to the second row of triangle A302971/A304042 and to the third row of the the sequence \(k(n-k), \ 1\leq k \leq n\), A094053. The following GIF shows exactly the process.
For the case \(T=n=3, \ m=3 \) in \((\star)\), we have, respectively
And for the case \(T=n=4, \ m=3 \) in \((\star)\), we have
All of the above examples are aveilable on youtube as follows
Here is some important Mathematica codes, in general, concerning to the preprint arXiv:1603.02468.
Code 1 - displays the triangular array of coefficients \(A_{m,j}\), see arXiv:1603.02468v15, p. 12, table 4.
Code 2 - verifies the identity \(P_{m}(\ell,T)=\sum\limits_{1\leq k \leq\ell}\sum\limits_{0\leq j\leq m} A_{m,j}k^j(T-k)^j=T^{2m+1}\) as \(\ell=T\), see arXiv:1603.02468v15, p. 1, equation (1.1).
Code 3 - verifies the identity \(P_{m}(\ell,T)=\sum\limits_{0\leq k\leq m}(-1)^{m-k}U_{m}(\ell, k)\cdot T^k=T^{2m+1}\) as \(\ell=T\), see arXiv:1603.02468v15, p. 1, equation (1.1).
Code 4 - prints the tabular consisting the coefficients \(U_m(\ell,k)\), for particular \(m\in\mathbb{N}\). By default \(m=2\).
Code 5 - produces the list of polyomials \(P_{m}(\ell,T)=\sum\limits_{0\leq k\leq m}(-1)^{m-k}U_{m}(\ell, k)\cdot T^k\) for particular \(m\in\mathbb{N}\) and \(1\leq \ell \leq 40\), by default \(m=2\).
Code 6 - prints the tabular of the coefficients \(\mathcal{V}_{m}(\ell,k)\), defined by the identity \(\sum\limits_{0\leq k \leq\ell-1}\sum\limits_{0\leq j\leq m} A_{m,j}k^j(T-k)^j=\sum\limits_{0\leq k\leq m}(-1)^{m-k}\mathcal{V}_{m}(\ell,k)\cdot T^k=T^{2m+1}\) as \(\ell=T\), see arXiv:1603.02468v12, pp. 19-20, expression (4.17), (4.18), figure (4.19).
Code 7 - verifies the identity \(\sum\limits_{0\leq k\leq m}(-1)^{m-k}\mathcal{V}_{m}(\ell,k)\cdot T^k=T^{2m+1}\) as \(\ell=T\), see arXiv:1603.02468v12, p. 19, expression (4.18).
Code 8 - produces the list of polynomials \(\sum\limits_{0\leq k\leq m}(-1)^{m-k}\mathcal{V}_{m}(\ell,k)\cdot T^k=T^{2m+1}\) as \(\ell=T\) for \(\ell=1, ..., 40\), see arXiv:1603.02468v12, p. 20, figure (4.19).
Code 9 - produces the list of the "combined" coefficients \(\mathcal{U}_m(\ell,k)\), defined by \(\mathcal{U}_m(\ell,k)=\frac{1}{2}[U_m(\ell,k)+\mathcal{V}_{m}(\ell,k)]\), such that \(\sum\limits_{0\leq k\leq m}(-1)^{m-k}\mathcal{U}_{m}(\ell,k)\cdot T^k=T^{2m+1}\) as \(\ell=T\), see arXiv:1603.02468v12, p. 21, expression (4.22).
Code 10 - produces the list of polynomials \(\sum\limits_{0\leq k\leq m}(-1)^{m-k}\mathcal{U}_{m}(\ell,k)\cdot T^k=T^{2m+1}\) as \(\ell=T\) for \(\ell=1, ..., 40\), see arXiv:1603.02468v12, p. 21, figure (4.23).
Code 11 - verifies the identity \(\sum\limits_{0\leq k\leq m}(-1)^{m-k}\mathcal{U}_{m}(\ell,k)\cdot T^k=T^{2m+1}\) as \(\ell=T\), see arXiv:1603.02468v12, p. 21, expression (4.22).