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Petro Kolosov

2026-02-05

Abstract

Your abstract here.

Introduction

Include some references (Kolosov 2024, 2018c; Alekseyev, Max 2018). Lorem Ipsum is simply dummy text of the printing and typesetting industry. Lorem Ipsum has been the industry’s standard dummy text ever since the 1500s, when an unknown printer took a galley of type and scrambled it to make a type specimen book. It has survived not only five centuries, but also the leap into electronic typesetting, remaining essentially unchanged. It was popularised in the 1960s with the release of Letraset sheets containing Lorem Ipsum passages, and more recently with desktop publishing software like Aldus PageMaker including versions of Lorem Ipsum.

Image example

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Image example (from caption).
Coefficients \({\mathbf{A}}_{m,r}\). See OEIS sequences  (Kolosov 2018a, 2018b).
\(m/r\) 0 1 2 3 4 5 6 7
0 1
1 1 6
2 1 0 30
3 1 -14 0 140
4 1 -120 0 0 630
5 1 -1386 660 0 0 2772
6 1 -21840 18018 0 0 0 12012
7 1 -450054 491400 -60060 0 0 0 51480

\[\begin{equation*} \left\lceil\kern-3.5pt\left\lceil\genfrac{}{}{0pt}{}{a}{b} \right\rfloor\kern-3.5pt\right\rfloor_{m} \end{equation*}\] \[\begin{equation*} \left\lceil\kern-3.5pt\left\lceil\genfrac{}{}{0pt}{}{a}{b} \right\rfloor\kern-3.5pt\right\rfloor_{m} \end{equation*}\]

And for any natural \(m\) we have polynomial identity \[\begin{equation} x^m = \sum_{k=1}^{m} T(m, k) x^{[k]} \label{eq:knuth-power-identity} \end{equation}\] where \(x^{[k]}\) denotes central factorial defined by \[\begin{equation*} x^{[n]} = x \left(x+\frac{n}{2}-1\right)_{n-1} \end{equation*}\] where \(\left(n\right)_{k} = n (n-1) (n-2) \cdots (n-k+1)\) denotes falling factorial in Knuth’s notation. In particular, \[\begin{equation} \label{eq:falling-factorial} x^{[n]} = x \left( x+\frac{n}{2}-1 \right) \left( x+\frac{n}{2}-1 \right) \cdots \left (x+\frac{n}{2}-n-1 \right) = x \prod_{k=1}^{n-1} \left( x+\frac{n}{2}-k \right) \end{equation}\] This is an equation reference [eq:knuth-power-identity].

Continuing similarly, we are able to derive the formula for multifold sums of powers, which is

Theorem 1 (Multifold sums of powers via Newton’s series). For non-negative integers \(r,n,m\) and an arbitrary integer \(t\) \[\begin{align*} \Sigma^{r}\,{n}^{m} = \sum_{j=0}^{m} \Delta^{j} t^{m} \left[ \left( \sum_{s=1}^{r} (-1)^{j+s-1} \binom{j+t-1}{j+s} \Sigma^{r-s}\,{n}^{0} \right) + \binom{n-t+r}{j+r} \right] \end{align*}\]

Proof. By Newton’s series for power and repeated applications of the segmented hockey stick identity. ◻

Proposition 2 (Falling factorial). \[\begin{align*} \left(x\right)_{n} = x(x-1)(x-2)(x-3)\cdots(x-n+1) = \prod_{k=0}^{n-1}(x-k) \end{align*}\]

Proposition 3. \[\begin{align*} \frac{\left(x\right)_{n}}{n!} = \binom{x}{n} \end{align*}\]

Rising factorials

Proposition 4 (Rising factorial). \[\begin{align*} x^{\left(n\right)} = x(x+1)(x+2)(x+3)\cdots(x+n-1) = \prod_{k=0}^{n-1}(x+k) \end{align*}\]

Proposition 5. \[\begin{align*} \frac{x^{\left(n\right)}}{n!} = \binom{x+n-1}{n} \end{align*}\]

Central factorials

Lemma 6 (Central factorial). \[\begin{align*} n^{[k]} = n \left( n + \frac{k}{2} -1 \right)\left( n + \frac{k}{2} -2 \right) \cdots \left( n - \frac{k}{2} +1 \right) = n \prod_{j=1}^{k-1} \left( n + \frac{k}{2} -j \right) \end{align*}\]

Proposition 7. \[\begin{align*} n^{[k]} = n \left(n+\frac{k}{2}-1\right)_{k-1} \end{align*}\]

Derivatives

\[\begin{align*} \frac{\mathrm{d} x}{\mathrm{d} y} = \frac{f(x+h)-f(x)}{h} \end{align*}\]

\[\begin{align*} \frac{\mathrm{d}^{3} x}{\mathrm{d} y^{3}} = \frac{f(x+h)-f(x)}{h} \end{align*}\]

Conclusions

Conclusions of your manuscript.

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Acknowledgements

The author is grateful to [Full Name] for his valuable contribution [contribution] about the fact that [interesting claim].

Metadata

Alekseyev, Max. 2018. “MathOverflow answer 297916/113033.” In MathOverflow https://mathoverflow.net/a/297916/113033, 297916/113033. Https://mathoverflow.net/a/297916/113033.
Kolosov, Petro. 2018a. “Entry A302971 in The On-Line Encyclopedia of Integer Sequences.” In The On-Line Encyclopedia of Integer Sequences, No. A302971. Https://oeis.org/A302971.
Kolosov, Petro. 2018b. “Entry A304042 in The On-Line Encyclopedia of Integer Sequences.” In The On-Line Encyclopedia of Integer Sequences, No. A304042. Https://oeis.org/A304042.
Kolosov, Petro. 2018c. “The coefficients U(m, l, k), m = 3 defined by the polynomial identity.” In The On-Line Encyclopedia of Integer Sequences, No. A316387. Https://oeis.org/A316387.
Kolosov, Petro. 2024. “Finding the Derivative of Polynomials via Double Limit.” GitHub.