# Kolosov Petro

Just the person,
which likes Mathematics
kolosovp94@gmail.com

## Résumé

My primary research interests are lay in areas of Number Theory and Mathematical Analysis, particularly, in Binomial Theorem, Finite differences and derivatives, Pascal's Triangle and Binomial Coefficients. Although, my main education belongs to Marine Engineering, I try my best to write mathematical preprints, you could familiarize with in the next section.

My current research is devoted to generalization of OEIS sequence A287326, that is binomoial distributed triangle that shows perfect cube $$n$$, as sum of terms of corresponding row. For now, two analogs for power $$m=5$$ and $$m=7$$ are known.

UPDATE (1): Generalization of A287326 is found for each odd power $$2m+1, \ m=0,1,2...$$ and shown in arXiv:1603.02468, particularly, our solution is closely related to Faulhaber's formula. OEIS sequences related to arXiv:1603.02468 are A287326, A300656, A300785, A302971, A304042.

## Publications

• ### Series Representation of Power Function

In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube $$n$$ as sum of $$n$$-th row terms over $$k, \ 0\leq k\leq n-1$$ or $$1\leq k\leq n$$, by means of its symmetry. In this paper we derived a similar triangles in order to receive powers $$m=5,7$$ as row items sum and generalized obtained results in order to receive every odd-powered monomial $$n^{2m+1}, \ m\geq0$$ as sum of row terms of corresponding triangle.
• ### On the link between finite differences and derivatives of polynomials

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.
• UPDATED: Mon, 21 Aug 2017 17:22:31 GMT
• Comments: 12 pages, 2 figures
• arXiv version: 1608.00801
• ### On the quantum differentiation of smooth real-valued functions

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 Taylor 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.
• UPDATED: Mon, 21 Aug 2017 17:22:31 GMT
• Comments: 12 pages, 6 figures
• arXiv version: 1705.02516
• ### 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 $$\Sigma_k k^m$$, where $$k,m$$ are positive integers. This paper is concerning arXiv:1805.11445, and a short review of it. In this manuscript we have reviewed the results of arXiv:1805.11445 and found a relation with Worpitzky Identity. More detailed proof on connection with Worpitzky Identity could be found in Mathoverflow discussion. Essentially, this preprint is another proof of Worpitzky Identity.

## Contributions to On-line Encyclopedia of Integer Sequences

• Complete list of sequences contributed, below we briefly discuss most important of them.
• A287326 - Main sequence, triangle that show $$n=3$$ power as row sums.
Detailed derivation available in this short report.
• A300656 - Generalization for power $$n=5$$, provided by Albert Tkaczyk
at his article. Derivation of A300656 is shown here.
• A300785 - Generalization for power $$n=7$$, provided by Albert Tkaczyk
at his article.
• PDF Table, that shows coefficients $$A_{m, 0}, \ A_{m,1}, \ ..., \ A_{m,m}, \ m \leq 12$$ of polynomial $$A_{m,0}(n-k)^0k^0+A_{m,1}(n-k)^1k^1+...+A_{m,m}(n-k)^mk^m,$$ such that for every integer $$m\geq0$$ $$\sum_{k=0}^{n-1}\sum_{j=0}^{m}A_{m,j}(n-k)^jk^j=n^{2m+1}\qquad\quad(a)$$ Note that A287326, A300656, A300785 are partial cases of $$(a)$$ for $$m=1,2,3$$. One could verify the values of $$A_{m,j}$$ by Mathematica code def_2_12.txt. To verify results of $$(a)$$ one may use the code expression_2_1.txt.
• A302971 - Numerators of $$A_{m,j}, \ m\geq 0$$.
• A304042 - Denominators of $$A_{m,j}, \ m\geq 0$$.
• A303675 - Coefficients $$V_{m,k}$$ in Faulhaber's sum (see arXiv:9207222 p. 10) $$\forall m\geq 0: \ 1^{2m+1}+2^{2m+1}+\cdots+n^{2m+1}=\sum_{1\leq k \leq m}V_{m,k}\binom{n+k}{2k}$$
• A316349 - Fix $$m = 2$$. Sequence gives 3-column table read by rows, listing coefficients $$U_m(n,k), \ 0\leq k \leq m$$ defined by the identity $$n^{2m+1} = \sum_{0\leq k \leq m} (-1)^{m-k}U_m(n,k) \cdot n^k$$
Additional sequences, provided by Colin Barker, these sequences detail on the generating functions of the columns of A316349
• A316457 - Expansion of $$x(31+326x+336x^2+ 26x^3 + x^4)/(1 - x)^6$$, seems to be the first column of A316349
• A316458 - Expansion of $$60x(1 + 4x + x^2) / (1 - x)^5$$, seems to be the negative of the second column of A316349
• A316459 - Expansion of $$30x(1 + x) / (1 - x)^4$$, seems to be the third column of A316349
• A316387 - Fix $$m = 3$$. Sequence gives 4-column table read by rows, listing coefficients $$U_m(n,k), \ 0\leq k \leq m$$ defined by the identity $$n^{2m+1} = \sum_{0\leq k \leq m} (-1)^{m-k}U_m(n,k) \cdot n^k$$
Additional sequences, provided by Colin Barker, these sequences detail on the generating functions of the columns of A316387
• A317981 - Expansion of $$x(125+8028x+42237x^2+42272x^3+8007x^4+132x^5-x^6)/(1-x)^8$$, seems to be the negative of the first column of A316387
• A317982 - Expansion of $$14x(29+784x+1974x^2+784x^3+29x^4)/(1-x)^7$$, seems to be the second column of A316387
• A317983 - Expansion of $$420x(1+x)(1+10x+x^2)/(1-x)^6$$, seems to be the negative of the third column of A316387
• A317984 - Expansion of $$140x(1+4x+x^2)/(1-x)^5$$, seems to be the fourth column of A316387

## Support my research

Dear readers! If you found my research interesting and progressive, you are free to help me by donations. Since mathematics is not my main affiliation, I don't receive any money for my work - I work for idea. I never care about money, but I have to live :) Each collected donation will be spend only to proceed my research without delays.

## GIFs and Videos

• ### Concerning the preprint 1603.02468.

Here we show a few examples of theorem $$n^{2m+1}=\sum\limits_{1\leq k \leq n}D_{m}(n,k),$$ where $$D_m(n,k):=\sum\limits_{0\leq j \leq m}A_{m,j}k^j(n-k)^j$$ and $$A_{m,j}$$ is defined be division of sequences A302971/A304042, and $$T(n,k)=k(n-k)$$ is A094053. Detailed derivation in 1603.02468.
• Example of main result of the manuscript 1603.02468 for $$m=2, \ n=3$$

This exmaple is also available on Youtube.
• Example of main result of the manuscript 1603.02468 for $$m=3, \ n=3$$

This exmaple is also available on Youtube.
• Example of main result of the manuscript 1603.02468 for $$m=3, \ n=4$$

This exmaple is also available on Youtube.

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