• Journal of applied mathematics & informatics
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• 제38권3_4호
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• pp.211-220
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• 2020

In this paper, we introduce and investigate a new class of generalized polynomials associated with Hermite-Bernoulli polynomials of higher order. This generalization is a unification formula of Bernoulli numbers, Bernoulli polynomials, Hermite-Bernoulli polynomials of Dattoli, generalized Hermite-Bernoulli polynomials for two variables of order α and new other families of polynomials depending on any generating function f.

• Kyungpook Mathematical Journal
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• 제55권3호
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• pp.597-614
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• 2015

In this paper, we study some properties of the generalized Apostol type Hermite-based polynomials. which extend some known results. We also deduce some properties of the generalized Apostol-Bernoulli polynomials, the generalized Apostol-Euler polynomials and the generalized Apostol-Genocchi polynomials of high order. Numerous properties of these polynomials and some relationships between $F_n{^{({\alpha})}}(x;{\lambda};{\mu};{\nu};c)$ and $_HF_n{^{({\alpha})}}(x;{\lambda};{\mu};{\nu};c)$ are established. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions.

• 호남수학학술지
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• 제42권3호
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• pp.477-499
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• 2020

In this paper, we introduce a new class of generalized Apostol-type Frobenius-Euler-Hermite polynomials and derive some explicit and implicit summation formulae and symmetric identities by using different analytical means and applying generating functions. These results extend some known summations and identities of generalized Frobenius-Euler type polynomials and Hermite-based Apostol-Euler and Apostol-Genocchi polynomials studied by Pathan and Khan, Kurt and Simsek.

• 호남수학학술지
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• 제41권1호
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• pp.117-133
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• 2019

In the current investigation, we obtain the generating function for Hermite-based degenerate Bernoulli polynomials attached to ${\chi}$ of higher order using p-adic methods over the ring of integers. Useful identities, formulae and relations with well known families of polynomials and numbers including the Bernoulli numbers, Daehee numbers and the Stirling numbers are established. We also give identities of symmetry and additive property for Hermite-based generalized degenerate Bernoulli polynomials attached to ${\chi}$ of higher order. Results are supported by remarks and corollaries.

• 대한수학회논문집
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• 제33권2호
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• pp.651-669
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• 2018

The article is themed to classify new (fully) degenerate Hermite-Bernoulli polynomials with formulation in terms of p-adic fermionic integrals on $\mathbb{Z}_p$. The entire paper is designed to illustrate new properties in association with Daehee polynomials in a consolidated and generalized form.

• 대한수학회논문집
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• 제34권1호
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• pp.169-183
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• 2019

The main aim of this paper is to establish a new class of integrals involving the generalized Galu$Galu{\grave{e}}$-type Struve function with the different type of polynomials such as Jacobi, Legendre, and Hermite. Also, we derive the integral formula involving Legendre, Wright generalized Bessel and generalized Hypergeometric functions. The results obtained here are general in nature and can deduce many known and new integral formulas involving the various type of polynomials.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제24권1호
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• pp.35-44
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• 2017

The present paper aims at harnessing the technique of Lie Algebra and operational methods to derive and interpret generating relations for the three-variable Hermite Polynomials $H_n$(x, y, z) introduced recently in [2]. Certain generating relations for the polynomials related to $H_n$(x, y, z) are also obtained as special cases.

• 대한수학회논문집
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• 제33권2호
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• pp.473-484
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• 2018

In this paper, we discuss how the operational calculus can be exploited to the theory of generalized special functions of many variables and many indices. We obtained the generating relations for 3-index, 3-variable and 1-parameter Hermite polynomials. Some mixed type generating relations and bilateral generating relations of many indices and many variable like Lagurre-Hermite and Hermite-Sister Celine's polynomials are also obtained. Further we generalize some results on old symbolic notations using operational identities.

• 대한수학회논문집
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• 제32권1호
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• pp.29-38
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• 2017

The main objective of this paper is to establish certain unified integral formula involving the product of the generalized Mittag-Leffler type function $E^{({\gamma}_j),(l_j)}_{({\rho}_j),{\lambda}}[z_1,{\ldots},z_r]$ and the Srivastava's polynomials $S^m_n[x]$. We also show how the main result here is general by demonstrating some interesting special cases.

• 대한수학회논문집
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• 제31권2호
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• pp.343-353
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• 2016

A remarkably large number of integrals whose integrands are associated, in particular, with a variety of special functions, for example, the hypergeometric and generalized hypergeometric functions have been recorded. Here we aim at presenting certain (presumably) new and (potentially) useful integral formulas whose integrands are involved in a product of $_2F_1$, Srivastava polynomials, and $Kamp{\acute{e}}$ de $F{\acute{e}}riet$ functions. The main results are derived with the help of some known definite integrals obtained earlier by Qureshi et al. [4]. Some interesting special cases of our main results are also considered.