# Petro Kolosov

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

I'm interested in Faulhaber's formula, Binomial theorem and realted topics.

## Publications

• ### Another Power Identity involving Binomial Theorem and Faulhaber's formula

In this paper, we derive and prove, by means of Binomial theorem and Faulhaber's formula, the following identity between $$m$$-order polynomials in $$T$$ $$\sum_{k=1}^{\ell}\sum_{j=0}^m A_{m,j}k^j(T-k)^j=\sum_{k=0}^{m}(-1)^{m-k}U_m(\ell,k)\cdot T^k=T^{2m+1}, \ \ell=T\in\mathbb{N}.$$

• Related OEIS sequences: A287326, A300656, A300785, A302971, A304042, A320047, A316349, A316387
• MSC classes: 11C08, 41A10
• Comments: 20 pages, 10 figures, 11 tables
• Keywords: Faulhaber's formula, Faulhaber's theorem, Binomial Theorem, Binomial coefficient, Binomial distribution, Binomial identities, Power Sums, Finite differences

• ### 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.

• MSC classes: 46G05, 30G25, 39-XX
• Comments: 12 pages, 2 figures
• Keywords: Finite difference, Divided difference, Derivative, Polynomials

• ### 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 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.

• MSC classes: 26A24, 05A30, 41A58
• Comments: 12 pages, 6 figures
• Keywords: Quantum algebra, Differential calculus, Taylor's formula, Approximations

## 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.

• A287326 - Numerical triangle, the $$n$$-th row sum starting from $$k=1$$ gives $$n^3$$. The triangle is symmetric. Triangle is generated by the partial case of polynomial $$P_{1}(n,n)$$, see arXiv:1603.02468v15, pp. 1, 3.

• A300656 - Numerical triangle, the $$n$$-th row sum starting from $$k=1$$ gives $$n^5$$. The triangle is symmetric. Triangle is generated by the partial case of polynomial $$P_{2}(n,n)$$, see arXiv:1603.02468v15, p. 7. Invented by Albert Tkaczyk after LinkedIn conversation, see his article.

• A300785 - Numerical triangle, the $$n$$-th row sum starting from $$k=1$$ gives $$n^7$$. The triangle is symmetric. Triangle is generated by the partial case of polynomial $$P_{3}(n,n)$$, see arXiv:1603.02468v15, p. 8. Invented by Albert Tkaczyk after LinkedIn conversation, see his article.

• Coefficients $$A_{m,j}$$ in the definition of the polynomial $$P_{m}(\ell,T)=\sum\limits_{1\leq k \leq\ell}\sum\limits_{0\leq j\leq m} A_{m,j}k^j(T-k)^j=T^{2m+1}$$ as $$\ell=T$$, above sequences represent the partial case of the $$P_{m}(\ell,T), \ \ell=T=n, \ m=1,2,3$$. Coefficients $$A_{m,j}$$ are real numbers, therefore are splited into two sequences, that show numerators and denominators:
• A302971 - Numerators of $$A_{m,j}$$.
• A304042 - Denominators of $$A_{m,j}$$.

• A303675 - Coefficients $$(2k-1)!T(2m,2k)$$ in the Faulhaber's sum $$\Sigma n^{2m-1}=\sum_{k=1}^{m}(2k-1)!T(2m,2k)\binom{n+k}{2k},$$ see Donald Knuth, "Johann Faulhaber and sums of powers" arXiv:9207222 p. 10.

• A320047 - Coefficients $$U_{m=1}(\ell, k)$$ defined by identity $$P_{m}(\ell,T)=\sum\limits_{1\leq k \leq\ell}\sum\limits_{0\leq j\leq m} A_{m,j}k^j(T-k)^j=\sum\limits_{0\leq k\leq m}(-1)^{m-k}U_{m}(\ell, k)\cdot T^k=T^{2m+1},$$ as $$\ell=T$$. This sequence gives 2-column table of coefficients $$U_{1}(\ell, k)$$ for $$k=0,1$$, see arXiv:1603.02468v15, p. 4.

• A316349 - Coefficients $$U_{m=2}(\ell, k)$$ defined by identity $$P_{m}(\ell,T)=\sum\limits_{1\leq k \leq\ell}\sum\limits_{0\leq j\leq m} A_{m,j}k^j(T-k)^j=\sum\limits_{0\leq k\leq m}(-1)^{m-k}U_{m}(\ell, k)\cdot T^k=T^{2m+1},$$ as $$\ell=T$$. This sequence gives 3-column table of coefficients $$U_{2}(\ell, k)$$ for $$k=0,1,2$$, see arXiv:1603.02468v15, p. 7.

• A316387 - Coefficients $$U_{m=3}(\ell, k)$$ defined by identity $$P_{m}(\ell,T)=\sum\limits_{1\leq k \leq\ell}\sum\limits_{0\leq j\leq m} A_{m,j}k^j(T-k)^j=\sum\limits_{0\leq k\leq m}(-1)^{m-k}U_{m}(\ell, k)\cdot T^k=T^{2m+1},$$ as $$\ell=T$$. This sequence gives 4-column table of coefficients $$U_{3}(\ell, k)$$ for $$k=0,1,2,3$$, see arXiv:1603.02468v15, p. 9.

• Sequences inspired by A316457, provided by Colin Barker:
• 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

• Sequences inspired by A317981, provided by Colin Barker:
• 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

• Complete list of sequences contributed, below we briefly discuss most important of them.

## 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.
• For more info and details concerning donation, proceed by the link

## GIFs and Videos

• ### Example of use of the identity reached at Another Power Identity involving Binomial Theorem and Faulhaber's formula

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

## Mathematica codes

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).

Updated: Nov, 2018.