A new class of generalized polynomials involving Laguerre and Euler polynomials

Khan, Nabiullah; Usman, Talha; Choi, Junesang

doi:10.15672/hujms.555416

Cited by 7 publications

(5 citation statements)

References 22 publications

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“…Numerous polynomials, numbers, their extensions, degenerations, and new polynomials and new numbers have been developed and studied, owing primarily to their potential applications and use in a diverse variety of research fields (see, e.g., [66][67][68][69][70][71] and the references therein). For example, Bernoulli polynomials and numbers are among most important and useful ones (see, e.g., [5], pp.…”

Section: Sequences Of New Numbersmentioning

confidence: 99%

Reduction Formulas for Generalized Hypergeometric Series Associated with New Sequences and Applications

et al. 2021

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In this paper, by introducing two sequences of new numbers and their derivatives, which are closely related to the Stirling numbers of the first kind, and choosing to employ six known generalized Kummer’s summation formulas for 2F1(−1) and 2F1(1/2), we establish six classes of generalized summation formulas for p+2Fp+1 with arguments −1 and 1/2 for any positive integer p. Next, by differentiating both sides of six chosen formulas presented here with respect to a specific parameter, among numerous ones, we demonstrate six identities in connection with finite sums of 4F3(−1) and 4F3(1/2). Further, we choose to give simple particular identities of some formulas presented here. We conclude this paper by highlighting a potential use of the newly presented numbers and posing some problems.

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Section: Sequences Of New Numbersmentioning

confidence: 99%

Reduction Formulas for Generalized Hypergeometric Series Associated with New Sequences and Applications

et al. 2021

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“…It should be noted that the identities (10) and (12) can easily lead to the identities ( 9) and (11) when using the familiar addition theorems for the Bernoulli polynomials and the Euler polynomials described in [5,6]. For some new developments of identities of symmetry on these topics, one is referred to [18][19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Applications of Symmetric Identities for Apostol–Bernoulli and Apostol–Euler Functions

2023

Symmetry

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In this paper, we perform a further investigation on the Apostol–Bernoulli and Apostol–Euler functions introduced by Luo. By using the Fourier expansions of the Apostol–Bernoulli and Apostol–Euler polynomials, we establish some symmetric identities for the Apostol–Bernoulli and Apostol–Euler functions. As applications, some known results, for example, Raabe’s multiplication formula and Hermite’s identity, are deduced as special cases.

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“…(see, e.g., [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38] and the reference cited therein). 2.…”

Section: Introductionmentioning

confidence: 99%

A Family of Generalized Legendre-Based Apostol-Type Polynomials

Usman¹,

Khan

Aman

et al. 2022

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Numerous polynomials, their extensions, and variations have been thoroughly explored, owing to their potential applications in a wide variety of research fields. The purpose of this work is to provide a unified family of Legendre-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, with appropriate constraints for the Maclaurin series. Then we look at the formulae and identities that are involved, including an integral formula, differential formulas, addition formulas, implicit summation formulas, and general symmetry identities. We also provide an explicit representation for these new polynomials. Due to the generality of the findings given here, various formulae and identities for relatively simple polynomials and numbers, such as generalized Bernoulli, Euler, and Genocchi numbers and polynomials, are indicated to be deducible. Furthermore, we employ the umbral calculus theory to offer some additional formulae for these new polynomials.

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A new class of generalized polynomials involving Laguerre and Euler polynomials

Cited by 7 publications

References 22 publications

Reduction Formulas for Generalized Hypergeometric Series Associated with New Sequences and Applications

Reduction Formulas for Generalized Hypergeometric Series Associated with New Sequences and Applications

Applications of Symmetric Identities for Apostol–Bernoulli and Apostol–Euler Functions

A Family of Generalized Legendre-Based Apostol-Type Polynomials

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