• Journal of Applied and Pure Mathematics
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• v.4 no.5_6
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• pp.221-231
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• 2022

In this paper, we study differential equations arising from the generating functions of (p, q)-Hermite polynomials. We use this differential equation to give explicit identities for (p, q)-Hermite polynomials.

• Honam Mathematical Journal
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• v.41 no.1
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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.

• Kyungpook Mathematical Journal
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• v.12 no.1
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• pp.115-117
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• 1972

The purpose of the present paper is to evaluate certain integrals involving Laguerre, Jacobi and Hermite polynomials. These integrals are very useful in case of expansion of any polynomial in a series of Orthogonal polynomials [1, Theo. 56].

• The Pure and Applied Mathematics
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• v.24 no.1
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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.

• Honam Mathematical Journal
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• v.42 no.2
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• pp.269-291
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• 2020

In this paper, we get acquainted to a new generalization of the modified Hermite matrix polynomials. An explicit representation and expansion of the Matrix exponential in a series of these matrix polynomials is obtained. Some important properties of Modified Hermite Matrix polynomials such as generating functions, recurrence relations which allow us a mathematical operations. Also we drive expansion formulae and some operational representations.

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.645-653
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• 1997

We study centrally symmetric orthogonal polynomials satisfying an admissible partial differential equation of the form $$ Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y = \lambda_n y, $$ where $A, B, \cdots, E$ are polynomials independent of n and $\lambda_n$ is the eignevalue parameter depending on n. We show that they are either the product of Hermite polymials or the circle polynomials up to a complex linear change of variables. Also we give some properties of them.

• East Asian mathematical journal
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• v.38 no.3
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• pp.365-377
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• 2022

In this paper, we present a new class of interpolatory Hermite subdivision schemes of order 2 reproducing polynomials. Each member in this class, denoted by H_n for n ≥ 1, preserves polynomials of degree up to 4n + 1 admitting the approximation order of 4n + 2. Furthermore, it has free parameters which provide flexibility in designing curves/surfaces. H₁, the simplest and the most attractive scheme in this class, achieves C⁴ smoothness with the parameters in certain ranges, and its performance is demonstrated with numerical examples.

• Honam Mathematical Journal
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• v.41 no.1
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• pp.35-49
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• 2019

In this work, it is shown that the combination of operational techniques and the use of the principle of quasi-monomiality can be a very useful tool for a more general insight into the theory of matrix polynomials and for their extension. We explore the formal properties of the operational rules to derive a number of properties of certain class of matrix polynomials and discuss the operational links with various known matrix polynomials.

• Communications of the Korean Mathematical Society
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• v.23 no.2
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• pp.295-305
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• 2008

This paper demonstrates how composite-exponential-fitting interpolation rules can be constructed to fit an oscillatory function using not only pointwise values of that function but also of that functions's derivative on a closed and bounded interval of interest. This is done in the framework of exponential-fitting techniques. These rules extend the classical composite cubic Hermite interpolating polynomials in the sense that they become the classical composite polynomials as a parameter tends to zero. Some examples are provided to compare the newly constructed rules with the classical composite cubic Hermite interpolating polynomials (or recently developed interpolation rules).

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.257-273
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• 2010

One can construct a theory of probability of fractional order in which the exponential function is replaced by the Mittag-Leffler function. In this framework, it seems of interest to generalize some useful classical mathematical tools, so that they are more suitable in fractional calculus. After a short background on fractional calculus based on modified Riemann Liouville derivative, one summarizes some definitions on probability density of fractional order (for the motive), and then one introduces successively fractional Euler's integrals (first and second kind) and fractional Hermite polynomials. Some properties of the Gaussian density of fractional order are exhibited. The fractional probability so introduced exhibits some relations with quantum probability.