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14

May 21, 2022
05/22

by
Petro Kolosov

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Polynomial identities involving Pascal's triangle rows, based on the famous Pascal's triangle property that each row sum is power of 2.

Topics: Binomial coefficients, Binomial theorem, Pascal's triangle, Binomial sums, Binomial distribution,...

149
149

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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 ∑ l k=1 ∑ j m=0 A m,j k j (T-k) j = ∑ m k=0 (-1) m-k U m (l,k) ∙ T k = T 2m+1

Topics: Faulhaber's formula, Faulhaber's theorem, Binomial Theorem, Binomial coefficient, Binomial...

16
16

May 22, 2022
05/22

by
Petro Kolosov

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In this manuscript we provide a new polynomial pattern. This pattern allows to find a polynomial expansion.

Topics: Binomial theorem, Power function, Polynomials, Polynomial identities, Multinomial theorem, Binomial...

99
99

Apr 7, 2016
04/16

by
Kolosov Petro

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Abstract : 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

Topics: Series Representation, Power function, monomial, Binomial Theorem, Multinomial theorem, Worpitzky...

115
115

Mar 16, 2016
03/16

by
Kolosov Petro

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Abstract : 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

Topics: Series Representation, Power function, monomial, Binomial Theorem, Multinomial theorem, Worpitzky...

84
84

Apr 30, 2017
04/17

by
Kolosov Petro

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Abstract : 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

Topics: Series Representation, Power function, monomial, Binomial Theorem, Multinomial theorem, Worpitzky...

115
115

Jul 8, 2016
07/16

by
Kolosov Petro

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Abstract : 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

Topics: Series Representation, Power function, monomial, Binomial Theorem, Multinomial theorem, Worpitzky...

20
20

May 21, 2022
05/22

by
Petro Kolosov

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A study on partial dynamic equation on time scales involving derivatives of polynomials

Topics: Dynamic equations on time scales, Partial differential equations on time scales, Partial dynamic...

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17

May 21, 2022
05/22

by
Petro Kolosov

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The relation between Binomial theorem And discrete convolution has been established

Topics: Dynamic equations on time scales, Partial differential equations on time scales, Partial dynamic...

100
100

May 1, 2017
05/17

by
Kolosov Petro

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Calculating the value of C k ∈ { 1 , ∞ } class of smoothness real-valued function's derivative in point of 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. ARXIV : 1705.02516 DOI :...

Topics: Mathematics, Quantum calculus, Quantum algebra, Power quantum calculus, Quantum difference,...