The Discrete Analog of the Newton-Leibniz Formula in the Problem of Summation over Simplex Lattice Points

Leinartas, Evgeniy K.; Shishkina, Olga A.; Лейнартас, Евгений К.

doi:10.17516/1997-1397-2019-12-4-503-508

Cited by 8 publications

(4 citation statements)

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“…For a treatment of diverse aspects of some summation formulas and their applications, the interested reader is referred to the relatively recent works [18][19][20].…”

Section: Background and Previous Resultsmentioning

confidence: 99%

A Look at Generalized Degenerate Bernoulli and Euler Matrices

2023

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In this paper, we consider the generalized degenerate Bernoulli/Euler polynomial matrices and study some algebraic properties for them. In particular, we focus our attention on some matrix-inversion formulae involving these matrices. Furthermore, we provide analytic properties for the so-called generalized degenerate Pascal matrix of the first kind, and some factorizations for the generalized degenerate Euler polynomial matrix.

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“…For a treatment of diverse aspects of some summation formulas and their applications, the interested reader is referred to the relatively recent works [18][19][20].…”

Section: Background and Previous Resultsmentioning

confidence: 99%

A Look at Generalized Degenerate Bernoulli and Euler Matrices

2023

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“…Recently, they have remarkable studies in operator theory [6][7][8][9], analytic function theory [10], and other fields [11,12]. Now, we define the Durrmeyer-type generalization of Szász operators involving confluent Appell polynomials…”

Section: Introductionmentioning

confidence: 99%

Szász–Durrmeyer Operators Involving Confluent Appell Polynomials

Kanat,

Erdal

2024

Axioms

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This article is concerned with the Durrmeyer-type generalization of Szász operators, including confluent Appell polynomials and their approximation properties. Also, the rate of convergence of the confluent Durrmeyer operators is found by using the modulus of continuity and Peetre’s K-functional. Then, we show that, under special choices of A(t), the newly constructed operators reduce confluent Hermite polynomials and confluent Bernoulli polynomials, respectively. Finally, we present a comparison of newly constructed operators with the Durrmeyer-type Szász operators graphically.

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“…and we use this function, expanded in fractional powers, in the generating function of the fractional form of the Bernoulli numbers and polynomials [12]. Note that the fractional exponential function above is a true exponential with respect to the operator D α x since it satisfies the following eigenvalue properties:…”

Section: Introductionmentioning

confidence: 99%

Fractional Bernoulli and Euler Numbers and Related Fractional Polynomials—A Symmetry in Number Theory

Caratelli,

Natalini,

Ricci

2023

Symmetry

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Bernoulli and Euler numbers and polynomials are well known and find applications in various areas of mathematics, such as number theory, combinatorial mathematics, series expansions, and the theory of special functions. Using fractional exponential functions, we extend the classical Bernoulli and Euler numbers and polynomials to introduce their fractional-index-based types. This reveals a symmetry in relation to the classical numbers and polynomials. We demonstrate some examples of these generalized mathematical entities, which we derive using the computer algebra system Mathematica©.

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The Discrete Analog of the Newton-Leibniz Formula in the Problem of Summation over Simplex Lattice Points

Abstract: Definition of the discrete primitive function is introduced in the problem of summation over simplex lattice points. The discrete analog of the Newton-Leibniz formula is found

Cited by 8 publications

References 1 publication

A Look at Generalized Degenerate Bernoulli and Euler Matrices

A Look at Generalized Degenerate Bernoulli and Euler Matrices

Szász–Durrmeyer Operators Involving Confluent Appell Polynomials

Fractional Bernoulli and Euler Numbers and Related Fractional Polynomials—A Symmetry in Number Theory

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