The Experts below are selected from a list of 321 Experts worldwide ranked by ideXlab platform
Petro Kolosov - One of the best experts on this subject based on the ideXlab platform.
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another power identity involving Binomial Theorem and faulhaber s formula
viXra, 2017Co-Authors: Petro KolosovAbstract:In this paper, we derive and prove another odd power identity, by means of Binomial Theorem and Faulhaber's formula.
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On the link between Binomial Theorem and Discrete Convolution of Power Function
arXiv: Number Theory, 2016Co-Authors: Petro KolosovAbstract:In this manuscript we introduce and discuss the $2m+1$-degree integer valued polynomials $\mathbf{P}^{m}_{b}(n)$. These polynomials are in strong relation with discrete convolution of power function. It is also shown that odd Binomial expansion is partial case of $\mathbf{P}^{m}_{b}(n)$. Basis on above, we show the relation between Binomial Theorem and discrete convolution of power function.
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another power identity involving Binomial Theorem and faulhaber s formula
2016Co-Authors: Petro KolosovAbstract:Let $\mathbf{P}^{m}_{b}(x)$ be a $2m+1$-degree integer-valued polynomial in $b,x\in\mathbb{R}$ \[ \mathbf{P}^{m}_{b}(x) = \sum_{k=0}^{b-1} \sum_{r=0}^{m} \mathbf{A}_{m,r} k^r(x-k)^r, \] where $\mathbf{A}_{m,r}$ is a real coefficient. In this manuscript we establish a relation between Binomial Theorem and polynomial $\mathbf{P}^{m}_{b}(x)$. By means of relation between Binomial Theorem and polynomial $\mathbf{P}^{m}_{b}(x)$, we show a relation between Binomial Theorem and discrete convolution in terms of polynomials.
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on the link between Binomial Theorem and discrete convolution of polynomials
arXiv: Number Theory, 2016Co-Authors: Petro KolosovAbstract:Let $\mathbf{P}^{m}_{b}(x)$ be a $2m+1$-degree integer-valued polynomial in $b,x\in\mathbb{R}$ \[ \mathbf{P}^{m}_{b}(x) = \sum_{k=0}^{b-1} \sum_{r=0}^{m} \mathbf{A}_{m,r} k^r(x-k)^r, \] where $\mathbf{A}_{m,r}$ is a real coefficient. In this manuscript we establish a relation between Binomial Theorem and polynomial $\mathbf{P}^{m}_{b}(x)$. By means of relation between Binomial Theorem and polynomial $\mathbf{P}^{m}_{b}(x)$, we show a relation between Binomial Theorem and discrete convolution in terms of polynomials.
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on the relation between Binomial Theorem and discrete convolution of piecewise defined power function
2016Co-Authors: Petro KolosovAbstract:Let $\mathbf{P}^{m}_{b}(x)$ be a $2m+1$-degree integer-valued polynomial in $b,x\in\mathbb{R}$ \[ \mathbf{P}^{m}_{b}(x) = \sum_{k=0}^{b-1} \sum_{r=0}^{m} \mathbf{A}_{m,r} k^r(x-k)^r, \] where $\mathbf{A}_{m,r}$ is a real coefficient. In this manuscript we establish a relation between Binomial Theorem and polynomial $\mathbf{P}^{m}_{b}(x)$. By means of relation between Binomial Theorem and polynomial $\mathbf{P}^{m}_{b}(x)$, we show a relation between Binomial Theorem and discrete convolution in terms of polynomials.
Moa Apagodu - One of the best experts on this subject based on the ideXlab platform.
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new series representations for jacobi s triple product identity and more via the q markov method
Advances in Applied Mathematics, 2012Co-Authors: Moa ApagoduAbstract:We derive new series representations for Jacobi@?s triple product identity, the q-Binomial Theorem, q-analogs of the exponential function, and more with several special cases using the q-Markov-WZ method.
Kolosov Petro - One of the best experts on this subject based on the ideXlab platform.
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Series Representation of Power Function
arXiv: Number Theory, 2016Co-Authors: Kolosov PetroAbstract:In this paper described numerical expansion of natural-valued power function $x^n$, in point $x=x_0$ where $n, \ x_0$ - natural numbers. Applying numerical methods, that is calculus of finite differences, namely, discrete case of Binomial expansion is reached. Received results were compared with solutions according to Newton's Binomial Theorem and MacMillan Double Binomial sum. Additionally, in section 4 exponential function's $e^x$ representation is shown.
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series representation of power function
2016Co-Authors: Kolosov PetroAbstract:This paper presents the way to make expansion for the next form function: y = x, ∀(x, n) ∈ N to the numerical series. The most widely used methods to solve this problem are Newton’s Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are inverse operators). The paper provides the other kind of solution, except above described Theorems.
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series expansion for power function
2015Co-Authors: Kolosov PetroAbstract:This paper presents the way to make expansion for the next form function: = , ∈ N, ∈ N to the numerical series. The most widely used ways to solve this problem are Newton’s Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are inverse operators). The paper provides the other kind of solution, except above described Theorems. INTRODUCTION Let basically describe Newton’s Binomial Theorem and Fundamental Theorem of Calculus and some their properties. In elementary algebra, the Binomial Theorem (or Binomial expansion) describes the algebraic expansion of powers of a Binomial. According to the Theorem, it is possible to expand the power + into a sum involving terms of the form , where the exponents and are nonnegative integers with + = , and the coefficient of each term is a specific positive integer depending on and . The coefficient in the term of is known as the Binomial coefficient. The main properties of the binominal Theorem are next: I. the powers of go down until it reaches = 1starting value is (the in ( + ) ) II. the powers of go up from 0 ( = 1) until it reaches (also the in ( + ) ) III. the -th row of the Pascal's Triangle will be the coefficients of the expanded Binomial. IV. for each line, the number of products (i.e. the sum of the coefficients) is equal to 2 V. for each line, the number of product groups is equal to + 1 By using Binomial Theorem for our case we obtain next form function [1]: = + 2 + ⋯ + − 1 + 1 We can reach the same result by using Fundamental Theorem of Calculus, according it we have [2]:
Shahn Majid - One of the best experts on this subject based on the ideXlab platform.
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free braided differential calculus braided Binomial Theorem and the braided exponential map
Journal of Mathematical Physics, 1993Co-Authors: Shahn MajidAbstract:Braided differential operators ∂i are obtained by differentiating the addition law on the braided covector spaces introduced previously (such as the braided addition law on the quantum plane). These are affiliated to a Yang–Baxter matrix R. The quantum eigenfunctions expR(x‖v) of the ∂i (braided‐plane waves) are introduced in the free case where the position components xi are totally noncommuting. A braided R‐Binomial Theorem and a braided Taylor Theorem expR(a‖∂)f(x)=f(a+x) are proven. These various results precisely generalize to a generic R‐matrix (and hence to n dimensions) the well‐known properties of the usual one‐dimensional q‐differential and q‐exponential. As a related application, it is shown that the q‐Heisenberg algebra px−qxp=1 is a braided semidirect product C[x]×C[ p] of the braided line acting on itself (a braided Weyl algebra) and similarly for its generalization to an arbitrary R‐ matrix.
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free braided differential calculus braided Binomial Theorem and the braided exponential map
arXiv: High Energy Physics - Theory, 1993Co-Authors: Shahn MajidAbstract:Braided differential operators $\del^i$ are obtained by differentiating the addition law on the braided covector spaces introduced previously (such as the braided addition law on the quantum plane). These are affiliated to a Yang-Baxter matrix $R$. The quantum eigenfunctions $\exp_R(\vecx|\vecv)$ of the $\del^i$ (braided-plane waves) are introduced in the free case where the position components $x_i$ are totally non-commuting. We prove a braided $R$-Binomial Theorem and a braided-Taylors Theorem $\exp_R(\veca|\del)f(\vecx)=f(\veca+\vecx)$. These various results precisely generalise to a generic $R$-matrix (and hence to $n$-dimensions) the well-known properties of the usual 1-dimensional $q$-differential and $q$-exponential. As a related application, we show that the q-Heisenberg algebra $px-qxp=1$ is a braided semidirect product $\C[x]\cocross \C[p]$ of the braided line acting on itself (a braided Weyl algebra). Similarly for its generalization to an arbitrary $R$-matrix.
Andrew V Sills - One of the best experts on this subject based on the ideXlab platform.
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a refinement of the Binomial distribution using the quantum Binomial Theorem
Communications in Statistics-theory and Methods, 2021Co-Authors: Andrew V SillsAbstract:q-analogs of special functions, including hypergeometric functions, play a central role in mathematics and have numerous applications in physics. In the theory of probability, q-analogs of various ...
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a refinement of the Binomial distribution using the quantum Binomial Theorem
arXiv: Probability, 2020Co-Authors: Andrew V SillsAbstract:$q$-analogs of special functions, including hypergeometric functions, play a central role in mathematics and have numerous applications in physics. In the theory of probability, $q$-analogs of various probability distributions have been introduced over the years, including the Binomial distribution. Here, I propose a new refinement of the Binomial distribution by way of the quantum Binomial Theorem (also known as the the noncommutative $q$-Binomial Theorem), where the $q$ is a formal variable in which information related to the sequence of successes and failures in the underlying Binomial experiment is encoded in its exponent.
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hybrid proofs of the q Binomial Theorem and other identities
Electronic Journal of Combinatorics, 2011Co-Authors: Dennis Eichhorn, James Mclaughlin, Andrew V SillsAbstract:We give “hybrid” proofs of the q-Binomial Theorem and other identities. The proofs are “hybrid” in the sense that we use partition arguments to prove a restricted version of the Theorem, and then use analytic methods (in the form of the Identity Theorem) to prove the full version. We prove three somewhat unusual summation formulae, and use these to give hybrid proofs of a number of identities due to Ramanujan. Finally, we use these new summation formulae to give new partition interpretations of the Rogers-Ramanujan identities and the Rogers-Selberg identities.