Generalized Bernoulli Polynomials and Numbers, Revisited

Elezović, Neven

doi:10.1007/s00009-014-0498-7

Cited by 8 publications

(7 citation statements)

References 9 publications

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“…Those defines a new class of polynomials in m called the reduced Bernoulli polynomials [8]. The representation of the ratio of two gamma functions (3.2) and different methods to derive the coefficients c (m) i and d…”

Section: Bernoulli Polynomial Identitiesmentioning

confidence: 99%

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Functional relations for hyperbolic cosecant series

Buzzegoli

2021

Preprint

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We study the function series ∞ n=1 φ 2m+2 cosch 2m+2 (nφ/2), and similar series, for integers m and complex φ. This hyperbolic series is linearly related to the Lambert series. The Lambert series is known to satisfy a functional equation which defines the Ramanujan polynomials. By using residue theorem (summation theorem) we find the functional equation satisfied by this hyperbolic series. The functional equation identifies a class of polynomials which can be seen as a generalization of the Ramanujan polynomials. These polynomials coincide with the asymptotic expansion of the hyperbolic series at the origin and they all vanish for φ = ±2πi. We furthermore derive several identities between Harmonic numbers and ordinary and generalized Bernoulli polynomials.

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Section: Bernoulli Polynomial Identitiesmentioning

confidence: 99%

“…The previous identity generates the reduced Bernoulli polynomials similarly to the formula in [8,Eq. (23)]…”

Section: In This Way We Could Obtain All the Values B (2m+1)mentioning

confidence: 99%

Functional relations for hyperbolic cosecant series

Buzzegoli

2021

Preprint

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“…For later use, let us denote G(α; p, λ; u; t) and H(α; p, a, b; u; t), (20) the generating functions of the left members in ( 15) and ( 17), respectively.…”

Section: Introductionmentioning

confidence: 99%

Two New Generalizations of Extended Bernoulli Polynomials and Numbers, and Umbral Calculus

Khan

Ghayasuddin

Kim

et al. 2022

Journal of Mathematics

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Among a remarkably large number of various extensions of polynomials and numbers, and diverse introductions of new polynomials and numbers, in this paper, we choose to introduce two new generalizations of some extended Bernoulli polynomials and numbers by using the Mittag–Leffler function and the confluent hypergeometric function. Then, we investigate certain properties and formulas of these newly introduced polynomials and numbers such as explicit representations, addition formulas, integral formulas, differential formulas, inequalities, and inequalities involving their integrals. Also, by using the theory of umbral calculus, five additional formulas regarding these new polynomials are provided. Furthermore, we propose to introduce four generalizations of the extended Euler and Genocchi polynomials. Finally, three natural problems are poised.

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“…The Bernoulli polynomials were first generalized by L. Carlitz [18], H.M. Srivastava et al [11,19,20]. More recently, several extensions have been made, as can be seen in, e.g., [21][22][23][24][25][26]. See also [11,22].…”

Section: Introductionmentioning

confidence: 99%

“…More recently, several extensions have been made, as can be seen in, e.g., [21][22][23][24][25][26]. See also [11,22].…”

Section: Introductionmentioning

confidence: 99%

A Family of the r-Associated Stirling Numbers of the Second Kind and Generalized Bernoulli Polynomials

Ricci

Srivastava

Natalini

2021

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Generalized Bernoulli Polynomials and Numbers, Revisited

Abstract: We describe with some new details the connection between generalized Bernoulli polynomials, Bernoulli polynomials and generalized Bernoulli numbers (Norlund polynomials). A new recursive and explicit formulae for these polynomials are derived.Mathematical Subject Classification. 33B15.

Cited by 8 publications

References 9 publications

Functional relations for hyperbolic cosecant series

Functional relations for hyperbolic cosecant series

Two New Generalizations of Extended Bernoulli Polynomials and Numbers, and Umbral Calculus

A Family of the r-Associated Stirling Numbers of the Second Kind and Generalized Bernoulli Polynomials

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