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Journal articles on the topic 'Hermite's Polynomial'

Author: Grafiati

Published: 5 June 2025

Last updated: 25 June 2025

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1

Butt, Saad Ihsan, Khuram Ali Khan, and Josip Pečarić. "Popoviciu type inequalities via Hermite's polynomial." Mathematical Inequalities & Applications, no. 4 (2016): 1309–18. http://dx.doi.org/10.7153/mia-19-96.

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2

Cheng, Kaimin. "New Permutation Reversed Dickson Polynomials over Finite Fields." Algebra Colloquium 30, no. 01 (2022): 111–20. http://dx.doi.org/10.1142/s1005386723000093.

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Let [Formula: see text] be an odd prime, and [Formula: see text], [Formula: see text] be nonnegative integers. Let [Formula: see text] be the reversed Dickson polynomial of the [Formula: see text]-th kind. In this paper, by using Hermite's criterion, we study the permutational properties of the reversed Dickson polynomials [Formula: see text] over finite fields in the case of [Formula: see text] with [Formula: see text]. In particular, we provide some precise characterizations for [Formula: see text] being permutation polynomials over finite fields with characteristic [Formula: see text] when [Formula: see text], or [Formula: see text], or [Formula: see text].

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3

Khan, M. Adil, S. Ivelić Bra anović, and Josip Pečarić. "Generalizations of Sherman's inequality by Hermite's interpolating polynomial." Mathematical Inequalities & Applications, no. 4 (2016): 1181–92. http://dx.doi.org/10.7153/mia-19-87.

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4

Heinig, Georg, and Fadhel Al-Musallam. "Hermite's formula for vector polynomial interpolation with applications to structured matrices." Applicable Analysis 70, no. 3-4 (1998): 331–45. http://dx.doi.org/10.1080/00036819808840695.

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5

WAUBKE, HOLGER. "BOUNDARY ELEMENT METHOD FOR ISOTROPIC MEDIA WITH RANDOM SHEAR MODULI." Journal of Computational Acoustics 13, no. 01 (2005): 229–58. http://dx.doi.org/10.1142/s0218396x05002530.

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Green's functions for elastic solids with random properties are usually derived by means of the perturbation method. This paper deals with a new approach that has the potential to deal with a large variability of random shear modulus based on the transformation of a polynomial chaos. The deterministic Green's functions for stresses and displacements and the principal values in the boundary integrals caused by a pressure load on the surface are nonlinear transformations of the random variables. A series of transformations of the polynomial chaos is used to transform significant parts of the equation. The first operation is a projection of the log normal distributed shear modulus to a series of Hermite's polynomials based on a Gaussian variable. The second operation is the determination of an arbitrary potential of the wave velocity. The last operation, similar to the first one, consists in the determination of an exponential function depending on the inverse of the wave velocity. These operations, together with multiplications and summations, transform the complete relation from the random shear modulus to Green's functions and principal values. The inversion of the system matrix is already derived for the random finite element approach. The operations are independent of the specific problem and can be applied to almost all acoustic media and similar nonlinear problems.

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6

Kashpur, O. F. "UNDAMENTAL POLYNOMIALS OF HERMITE’SINTERPOLATION FORMULA IN LINEAR NORMAL AND INEUCLIDEAN SPACES." Journal of Numerical and Applied Mathematics, no. 2 (2022): 50–58. http://dx.doi.org/10.17721/2706-9699.2022.2.06.

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In a linear infinite-dimensional space with a scalar product and in a finite-dimensional Euclidean space the interpolation Hermite polynomial with a minimal norm, generated by a Gaussian measure, contains fundamental polynomials are shown. The accuracy of Hermit’s interpolation formulas on polynomials of the appropriate degree are researched.

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7

Chandra, Bahadur Khadka. "Transformation of Special Relativity into Differential Equation by Means of Power Series Method." International Journal of Basic Sciences and Applied Computing (IJBSAC) 10, no. 1 (2023): 10–15. https://doi.org/10.35940/ijbsac.B1045.0910123.

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Partial differential equations such as those involving Bessel differential function, Hermite’s polynomial, and Legendre polynomial are widely used during the separation of the wave equation in cylindrical and spherical coordinates. Such functions are quite applicable to solve the wide variety of physical problems in mathematical physics and quantum mechanics, but until now, there has been no differential equation capable for handling the problems involved in the realm of special relativity. In order to avert such trouble in physics, this article presents a new kind of differential equation of the form: , where c is the speed of light in a vacuum. In this work, the solution of this equation has been developed via the power series method, which generates a formula that is completely compatible with relativistic phenomena happening in nature. In this highly exciting topic, the particular purpose of this paper is to define entirely a new differential equation to handle physical problems happening in the realm of special relativity.

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8

Abro, Goh André-Pascal, Kidjégbo Augustin Touré, and Gossrin Jean-Marc Bomisso. "Influence of control parameters on the stabilization of an Euler-Bernoulli flexible beam." Journal of Numerical Analysis and Approximation Theory 53, no. 1 (2024): 109–23. http://dx.doi.org/10.33993/jnaat531-1384.

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In this work, we numerically study the influence of control parameters on the stabilization of a flexible Euler-Bernoulli beam fixed at one end and subjected at the other end to a force control and a punctual moment control proportional respectively to velocity and rotation velocity. First, we analyze the displacement stabilization and the asymptotic behavior of the beam energy using a stable numerical scheme, resulting from the Crank-Nicholson algorithm for time discretization and the finite element method based on the approximation by Hermite's cubic polynomial functions, for discretization in space. Then, by means of the finite element method, we represent the spectrum of the operator associated with this beam problem and we carry out a qualitative study of thelocus of the eigenvalues according to the positive control parameters. From these studies we conclude that rotation velocity control has more effect on the stabilization of the beam compared to velocity control. Finally, this result is confirmed by a sensitivity study on the control parameters involved in the stabilization of the beam.

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9

Brackx, F., H. De Schepper, N. De Schepper, and F. Sommen. "Hermitean Clifford-Hermite Polynomials." Advances in Applied Clifford Algebras 17, no. 3 (2007): 311–30. http://dx.doi.org/10.1007/s00006-007-0032-0.

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10

Blel, Mongi. "On some m-symmetric generalized hypergeometric d-orthogonal polynomials." Filomat 38, no. 4 (2024): 1279–89. http://dx.doi.org/10.2298/fil2404279b.

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In [9] I. Lamiri and M. Ouni state some characterization theorems for d-orthogonal polynomials of Hermite, Gould-Hopper and Charlier type polynomials. In [3] Y. Ben Cheikh I. Lamiri and M.Ouni give a characterization theorem for some classes of generalized hypergeometric polynomials containing for example, Gegenbauer polynomials, Gould-Hopper polynomials, Humbert polynomials, a generalization of Laguerre polynomials and a generalization of Charlier polynomials. In this work, we introduce a new class D of generalized hypergeometric m-symmetric polynomial sequence containing the Hermite polynomial sequence and Laguerre polynomial sequence. Then we consider a characterization problem consisting in finding the d-orthogonal polynomial sequences in the class D, m ? d. The solution provides new d-orthogonal polynomial sequences to be classified in d-Askey-scheme and also having a m-symmetry property with m ? d. This class contains the Gould-Hopper polynomial sequence, the class considered by Ben Cheikh-Douak, the class considered in [3]. This class contains new d-orthogonal polynomial sequences not belonging to the classA. We derive also in this work the d-dimensional functional vectors ensuring the d-orthogonality of these polynomials. We also give an explicit expression of the d-dimensional functional vector.

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11

Kabluchko, Zakhar. "Lee–Yang zeroes of the Curie–Weiss ferromagnet, unitary Hermite polynomials, and the backward heat flow." Annales Henri Lebesgue 8 (May 20, 2025): 1–34. https://doi.org/10.5802/ahl.227.

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The backward heat flow on the real line started from the initial condition z n results in the classical n th Hermite polynomial whose zeroes are distributed according to the Wigner semicircle law in the large n limit. Similarly, the backward heat flow with the periodic initial condition (sinθ 2) n leads to trigonometric or unitary analogues of the Hermite polynomials. These polynomials are closely related to the partition function of the Curie–Weiss model and appeared in the work of Mirabelli on finite free probability. We relate the n th unitary Hermite polynomial to the expected characteristic polynomial of a unitary random matrix obtained by running a Brownian motion on the unitary group U(n). We identify the global distribution of zeroes of the unitary Hermite polynomials as the free unitary normal distribution. We also compute the asymptotics of these polynomials or, equivalently, the free energy of the Curie–Weiss model in a complex external field. We identify the global distribution of the Lee–Yang zeroes of this model. Finally, we show that the backward heat flow applied to a high-degree real-rooted polynomial (respectively, trigonometric polynomial) induces, on the level of the asymptotic distribution of its roots, a free Brownian motion (respectively, free unitary Brownian motion).

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12

Ftorek, Branislav, and Mariana Marˇcokov´A. "Markov type polynomial inequality for some generalized Hermite weight." Tatra Mountains Mathematical Publications 49, no. 1 (2011): 111–18. http://dx.doi.org/10.2478/v10127-011-0030-4.

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ABSTRACT In this paper we study some weighted polynomial inequalities of Markov type in L2-norm. We use the properties of the system of generalized Hermite polynomials . The polynomials H(α)n (x) are orthogonal in ℝ = (−∞,∞) with respect to the weight function . The classical Hermite polynomials Hn(x) present the special case for α = 0.

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13

Plucińska, A., and E. Plucińskii. "Polynomial normal densities generated by Hermite polynomials." Journal of Mathematical Sciences 92, no. 3 (1998): 3921–25. http://dx.doi.org/10.1007/bf02432364.

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14

FAN, HONG-YI, LI-YUN HU, and XUE-XIANG XU. "LEGENDRE POLYNOMIALS AS THE NORMALIZATION OF PHOTON-SUBTRACTED SQUEEZED STATES." Modern Physics Letters A 24, no. 20 (2009): 1597–603. http://dx.doi.org/10.1142/s021773230902996x.

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By converting the photon-subtracted squeezed state (PSSS) to a squeezed Hermite-polynomial excitation state we find that the normalization factor of PSSS is an m-order Legendre polynomial of the squeezing parameter, where m is the number of subtracted photons. Some new relations about the Legendre polynomials are obtained by this analysis. We also show that the PSSS can also be treated as a Hermite-polynomial excitation on squeezed vacuum state.

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15

ATAKISHIYEV, NATIG M. "ON q-EXTENSIONS OF MEHTA'S EIGENVECTORS OF THE FINITE FOURIER TRANSFORM." International Journal of Modern Physics A 21, no. 23n24 (2006): 4993–5006. http://dx.doi.org/10.1142/s0217751x06031673.

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Mehta has shown that eigenvectors [Formula: see text] of the finite Fourier transform with the matrix [Formula: see text], 0 ≤ j, k ≤ N-1, can be defined in terms of the classical Hermite functions [Formula: see text] as [Formula: see text], where [Formula: see text]. We argue that the finite Fourier transform [Formula: see text] does actually govern also some q-extensions of Mehta's eigenvectors [Formula: see text], associated with certain well-known orthogonal q-polynomial families. For the pairs of the continuous q-Hermite and q-1-Hermite polynomials, the Rogers–Szegő and Stieltjes–Wigert polynomials, and the discrete q-Hermite polynomials of types I and II such links are explicitly derived. In the limit when the base q → 1 these q-extensions coincide with Mehta's eigenvectors [Formula: see text], whereas in the continuous limit (i.e. when the parameter N → ∞) they correspond to the classical Fourier integral transforms between the above-mentioned pairs of q-polynomial families.

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16

Alam, Noor, Shahid Ahmad Wani, Waseem Ahmad Khan, et al. "On a Class of Generalized Multivariate Hermite–Humbert Polynomials via Generalized Fibonacci Polynomials." Symmetry 16, no. 11 (2024): 1415. http://dx.doi.org/10.3390/sym16111415.

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This paper offers a thorough examination of a unified class of Humbert’s polynomials in two variables, extending beyond well-known polynomial families such as Gegenbauer, Humbert, Legendre, Chebyshev, Pincherle, Horadam, Kinnsy, Horadam–Pethe, Djordjević, Gould, Milovanović, Djordjević, Pathan, and Khan polynomials. This study’s motivation stems from exploring polynomials that lack traditional nomenclature. This work presents various expansions for Humbert–Hermite polynomials, including those involving Hermite–Gegenbauer (or ultraspherical) polynomials and Hermite–Chebyshev polynomials. The proofs enhanced our understanding of the properties and interrelationships within this extended class of polynomials, offering valuable insights into their mathematical structure. This research consolidates existing knowledge while expanding the scope of Humbert’s polynomials, laying the groundwork for further investigation and applications in diverse mathematical fields.

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17

Leventeli, Yasemin, Yilmaz Simsek, and Ilyas Yilmazer. "Curve fitting for seismic waves of earthquake with Hermite polynomials." Publications de l'Institut Math?matique (Belgrade) 115, no. 129 (2024): 101–16. http://dx.doi.org/10.2298/pim2429101l.

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We investigate and study on mathematical structures involving mathematical models and others associated with seismic waves in an earthquake. Our first aim is to give some novel formulas and certain finite sums including the Bernoulli numbers and the Hermite polynomials with the aid of generating functions, the Riemann integral, and the Volkenborn integral. The second aim is to examine the seismic wave propagation in different geological units with the help of special polynomials containing the Hermite polynomials and their graph fitting of functions. To evaluate the shape of the seismic waves propagating within the ground (rock and/or soil), we use comparing method with the graph of the Hermite polynomials and functions and the polynomial Rocking Bearings. Furthermore, we also define generating function for the polynomial type Rocking Bearings. We give open problems on this generating function and earthquake facts. By applying partial derivative operator to the generating function for the m-parametric Hermite type polynomials, we give a novel recurrence relation and derivative formulas for these polynomials. We also give a new general formula for monomials in terms of these polynomials. Moreover, for the purpose of visualizing curve fitting approach to the seismic waves, we draw many plots of the Hermite functions with Mathematica (Version 12.0.0) with their codes. Finally, with the aid of these graphs, we give useful evaluation on the shapes of the seismic waves propagated in the ground (rock and/or soil).

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18

Feng, Xiao, and Yaotang Li. "Brauer-Type Inclusion Sets of Zeros for Chebyshev Polynomial." Mathematics 7, no. 2 (2019): 155. http://dx.doi.org/10.3390/math7020155.

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The generalized polynomials such as Chebyshev polynomial and Hermite polynomial are widely used in interpolations and numerical fittings and so on. Therefore, it is significant to study inclusion regions of the zeros for generalized polynomials. In this paper, several new inclusion sets of zeros for Chebyshev polynomials are presented by applying Brauer theorem about the eigenvalues of the comrade matrix of Chebyshev polynomial and applying the properties of ovals of Cassini. Some examples are given to show that the new inclusion sets are tighter than those provided by Melman (2014) in some cases.

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19

Prasad, J. "On the degree of approximation of the Hermite and Hermite-Fejer interpolation." International Journal of Mathematics and Mathematical Sciences 15, no. 1 (1992): 47–56. http://dx.doi.org/10.1155/s0161171292000061.

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Here we find the order of convergence of the Hermite and Hermite-Fejér interpolation polynomials constructed on the zeros of(1−x2)Pn(x)wherePn(x)is the Legendre polynomial of degreenwith normalizationPn(1)=1.

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20

Komlov, A. V. "The polynomial Hermite-Padé -system for meromorphic functions on a compact Riemann surface." Sbornik: Mathematics 212, no. 12 (2021): 1694–729. http://dx.doi.org/10.1070/sm9577.

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Abstract Given a tuple of germs of arbitrary analytic functions at a fixed point, we introduce the polynomial Hermite-Padé -system, which includes the Hermite-Padé polynomials of types I and II. In the generic case we find the weak asymptotics of the polynomials of the Hermite-Padé -system constructed from the tuple of germs of functions that are meromorphic on an -sheeted compact Riemann surface . We show that if for some meromorphic function on , then with the help of the ratios of polynomials of the Hermite-Padé -system we recover the values of on all sheets of the Nuttall partition of , apart from the last sheet. Bibliography: 18 titles.

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21

Komlov, A. V. "The polynomial Hermite-Padé -system for meromorphic functions on a compact Riemann surface." Sbornik: Mathematics 212, no. 12 (2021): 1694–729. http://dx.doi.org/10.1070/sm9577.

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Abstract Given a tuple of germs of arbitrary analytic functions at a fixed point, we introduce the polynomial Hermite-Padé -system, which includes the Hermite-Padé polynomials of types I and II. In the generic case we find the weak asymptotics of the polynomials of the Hermite-Padé -system constructed from the tuple of germs of functions that are meromorphic on an -sheeted compact Riemann surface . We show that if for some meromorphic function on , then with the help of the ratios of polynomials of the Hermite-Padé -system we recover the values of on all sheets of the Nuttall partition of , apart from the last sheet. Bibliography: 18 titles.

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22

Metwally, M. S., M. T. Mohamed, and A. Shehata. "On Hermite-Hermite matrix polynomials." Mathematica Bohemica 133, no. 4 (2008): 421–34. http://dx.doi.org/10.21136/mb.2008.140630.

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23

XU, XUE-XIANG, LI-YUN HU, and HONG-YI FAN. "ON THE NORMALIZED TWO-MODE PHOTON-SUBTRACTED SQUEEZED VACUUM STATE." Modern Physics Letters A 24, no. 32 (2009): 2623–30. http://dx.doi.org/10.1142/s0217732309031168.

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We show that the two-mode photon-subtracted squeezed state (TPSSS) is a squeezed two-variable Hermite polynomial excitation state, and we can therefore determine its normalization as a Jacobi polynomial of the squeezing parameter. Some new relations about the Jacobi polynomials are obtained by this analysis. We also show that the TPSSS can be treated as a two-variable Hermite-polynomial excitation on squeezed vacuum state. The technique of integration within an ordered product of operators brings convenience in our derivation.

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24

He, Tian-Xiao, and José L. Ramírez. "The dual of number sequences, Riordan polynomials, and Sheffer polynomials." Special Matrices 10, no. 1 (2021): 153–65. http://dx.doi.org/10.1515/spma-2021-0153.

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Abstract In this paper we introduce different families of numerical and polynomial sequences by using Riordan pseudo involutions and Sheffer polynomial sequences. Many examples are given including dual of Hermite numbers and polynomials, dual of Bell numbers and polynomials, among other. The coefficients of some of these polynomials are related to the counting of different families of set partitions and permutations. We also studied the dual of Catalan numbers and dual of Fuss-Catalan numbers, giving several combinatorial identities.

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25

V, Pandichelvi, and Umamaheswari B. "Establishment of Order of 4-Tuples Concerning Familiar Polynomials with Captivating Condition." Indian Journal of Science and Technology 16, no. 36 (2023): 2982–87. https://doi.org/10.17485/IJST/v16i36.1578.

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Abstract <strong>Objectives:</strong> Finding the feature of a number pattern is a fascinating area of current research. There are numerous features, one of which is that the Area by Perimeter of a Pythagorean triangle is a nasty number, figurate number, and so on. This manuscript aims to identify the precise sorts of 4-tuples combining Legendre polynomials and Probabilist’s Hermite polynomials in which the product of any two polynomials is added by one square to another polynomial. <strong>Method:</strong> By using specific transformation, the assumption can be turned into a universal second-degree equation with two variables known as the Pell equation. The solution to this equation yields 3-tuples and their extension into 4-tuples from 2-tuples in a precise manner, as explained.<strong> Findings:</strong> This paper describes some types of 4-tuples involving Legendre polynomials and Probabilist’s Hermite polynomials in which the product of any two polynomials is added by the number one in any 4-tuple results in the square of an alternate polynomial. <strong>Novelty:</strong> A Python program is achieved for displaying all 4-tuples with the numerical values of the specified unknown combination with the special property. <strong>Keywords:</strong> Legendre polynomials; Probabilist’s Hermite polynomials; Pell equation; Integer solutions; Diophantine m-tuples

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26

Alqahtani, Awatif Muflih, Saleem Yousuf, Shahid Ahmad Wani, and Roberto S. Costas-Santos. "Investigating Multidimensional Degenerate Hybrid Special Polynomials and Their Connection to Appell Sequences: Properties and Applications." Axioms 13, no. 12 (2024): 859. https://doi.org/10.3390/axioms13120859.

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This paper explores the operational principles and monomiality principles that significantly shape the development of various special polynomial families. We argue that applying the monomiality principle yields novel results while remaining consistent with established findings. The primary focus of this study is the introduction of degenerate multidimensional Hermite-based Appell polynomials (DMHAP), denoted as An[r]H(l1,l2,l3,…,lr;ϑ). These DMHAP forms essential families of orthogonal polynomials, demonstrating strong connections with classical Hermite and Appell polynomials. Additionally, we derive symmetric identities and examine the fundamental properties of these polynomials. Finally, we establish an operational framework to investigate and develop these polynomials further.

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27

Al-Salam, Waleed A. "A characterization of the Rogersq-hermite polynomials." International Journal of Mathematics and Mathematical Sciences 18, no. 4 (1995): 641–47. http://dx.doi.org/10.1155/s0161171295000810.

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28

SRIVASTAVA, HARI MOHAN. "Some generating functions of the Bessel and related orthogonal polynomials." Applicable Nonlinear Analysis 1, no. 1 (2024): 1–19. http://dx.doi.org/10.69829/apna-024-0101-ta01.

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Our main object in this article is to present several families of generating functions for the simple Bessel polynomials and the generalized Bessel polynomials . We also investigate and present various potentially useful generating-function relationships involving such other orthogonal polynomial systems as (for example) the Jacobi, Laguerre and Hermite polynomials, together with their specialized and parametrically-varied forms.

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29

Aras-Gazic, Gorana, Josip Pečaric, and Ana Vukelic. "INTEGRAL ERROR REPRESENTATION OF HERMITE INTERPOLATING POLYNOMIAL AND RELATED INEQUALITIES FOR QUADRATURE FORMULAE." Mathematical Modelling and Analysis 21, no. 6 (2016): 836–51. http://dx.doi.org/10.3846/13926292.2016.1247755.

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We consider integral error representation related to the Hermite interpolating polynomial and derive some new estimations of the remainder in quadrature formulae of Hermite type, using Holder’s inequality and some inequalities for the Čebyšev functional. As a special case, generalizations for the zeros of orthogonal polynomials are considered.

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Wang, Jinsheng, Muhannad Aldosary, Song Cen, and Chenfeng Li. "Hermite polynomial normal transformation for structural reliability analysis." Engineering Computations 38, no. 8 (2021): 3193–218. http://dx.doi.org/10.1108/ec-05-2020-0244.

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Purpose Normal transformation is often required in structural reliability analysis to convert the non-normal random variables into independent standard normal variables. The existing normal transformation techniques, for example, Rosenblatt transformation and Nataf transformation, usually require the joint probability density function (PDF) and/or marginal PDFs of non-normal random variables. In practical problems, however, the joint PDF and marginal PDFs are often unknown due to the lack of data while the statistical information is much easier to be expressed in terms of statistical moments and correlation coefficients. This study aims to address this issue, by presenting an alternative normal transformation method that does not require PDFs of the input random variables. Design/methodology/approach The new approach, namely, the Hermite polynomial normal transformation, expresses the normal transformation function in terms of Hermite polynomials and it works with both uncorrelated and correlated random variables. Its application in structural reliability analysis using different methods is thoroughly investigated via a number of carefully designed comparison studies. Findings Comprehensive comparisons are conducted to examine the performance of the proposed Hermite polynomial normal transformation scheme. The results show that the presented approach has comparable accuracy to previous methods and can be obtained in closed-form. Moreover, the new scheme only requires the first four statistical moments and/or the correlation coefficients between random variables, which greatly widen the applicability of normal transformations in practical problems. Originality/value This study interprets the classical polynomial normal transformation method in terms of Hermite polynomials, namely, Hermite polynomial normal transformation, to convert uncorrelated/correlated random variables into standard normal random variables. The new scheme only requires the first four statistical moments to operate, making it particularly suitable for problems that are constraint by limited data. Besides, the extension to correlated cases can easily be achieved with the introducing of the Hermite polynomials. Compared to existing methods, the new scheme is cheap to compute and delivers comparable accuracy.

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31

Alakija, Temitope O., Ismaila S. Amusa, Bolanle O. Olusan, and Ademola A. Fadiji. "On New Probabilistic Hermite Polynomials." International Journal of Research and Innovation in Applied Science VIII, no. VII (2023): 14–20. http://dx.doi.org/10.51584/ijrias.2023.8702.

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In the theory of differential equation and probability, Probabilistic Hermite polynomials Hr(x) = {r=0,1,2,…,n} are the polynomials obtained from derivatives of the standard normal probability density function (pdf) of the form α(x)=1/√2π e^(-1/2 x^2 ). These polynomials played an important role in the Gram-Charlier series expansion of type A and the Edgeworth’s form of the type A series (see [18]). In this paper, we obtained new Probabilistic Hermite polynomials by considering a standard normal distribution with probability density function (pdf) given as β(x)=1/(2√π) e^(-1/4 x^2 ). The generating function, recurrence relations and orthogonality properties are studied. Finally, a differential equation governing these polynomials was presented which enables us to obtain the expression of the polynomial in a closed form.

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Brackx, Fred, Hennie De Schepper, Nele De Schepper, David Eelbode, and Frank Sommen. "Orthogonality of the generalized Hermitean Clifford–Hermite polynomials." Integral Transforms and Special Functions 19, no. 10 (2008): 687–707. http://dx.doi.org/10.1080/10652460802166799.

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33

Cesarano, Clemente, Yamilet Quintana, and William Ramírez. "A Survey on Orthogonal Polynomials from a Monomiality Principle Point of View." Encyclopedia 4, no. 3 (2024): 1355–66. http://dx.doi.org/10.3390/encyclopedia4030088.

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This survey highlights the significant role of exponential operators and the monomiality principle in the theory of special polynomials. Using operational calculus formalism, we revisited classical and current results corresponding to a broad class of special polynomials. For instance, we explore the 2D Hermite polynomials and their generalizations. We also present an integral representation of Gegenbauer polynomials in terms of Gould–Hopper polynomials, establishing connections with a simple case of Gegenbauer–Sobolev orthogonality. The monomiality principle is examined, emphasizing its utility in simplifying the algebraic and differential properties of several special polynomial families. This principle provides a powerful tool for deriving properties and applications of such polynomials. Additionally, we review advancements over the past 25 years, showcasing the evolution and extensive applicability of this operational formalism in understanding and manipulating special polynomial families.

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Foster, William H., and Ilia Krasikov. "Explicit Bounds for Hermite Polynomials in the Oscillatory Region." LMS Journal of Computation and Mathematics 3 (2000): 307–14. http://dx.doi.org/10.1112/s1461157000000310.

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AbstractWe apply a method of positive quadratic forms based on polynomial inequalities to establish sharp explicit bounds on the envelope of Hermite polynomials in the oscillatory region |x| < (2k – 3/2)1/2.

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Hwang, Gyung Won, Cheon Seoung Ryoo, and Jung Yoog Kang. "Some properties for 2-variable modified partially degenerate Hermite (MPDH) polynomials derived from differential equations and their zeros distributions." AIMS Mathematics 8, no. 12 (2023): 30591–609. http://dx.doi.org/10.3934/math.20231564.

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<abstract><p>The 2-variable modified partially degenerate Hermite (MPDH) polynomials are the subject of our study in this paper. We found basic properties of these polynomials and obtained several types of differential equations related to MPDH polynomials. Based on the MPDH polynomials, we looked at the structures of the approximation roots for a particular polynomial and checked the values of the approximate roots. Further, we presented some conjectures for MPDH polynomials.</p></abstract>

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36

Abduganiev, Mukhriddin, and Sultan Gafurov. "Application of two variable hermite splines in digital image processing." InterConf, no. 34(159) (June 20, 2023): 308–20. http://dx.doi.org/10.51582/interconf.19-20.06.2023.030.

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During the research, the use of piece-polynomial methods in digital processing of images was considered. The Hermite spline function is chosen from piece-polynomials as a mathematical model in digital processing of signals, and the construction of a two-variable third-order Hermite spline function is presented. An image restoration algorithm was developed based on the constructed mathematical model.

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37

Özat, Zeynep, Bayram Çekim, Mehmet Ali Özarslan, and Francesco Aldo Costabile. "Truncated-Exponential-Based General-Appell Polynomials." Mathematics 13, no. 8 (2025): 1266. https://doi.org/10.3390/math13081266.

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In this paper, a new and general form of truncated-exponential-based general-Appell polynomials is introduced using the two-variable general-Appell polynomials. For this new polynomial family, we present an explicit representation, recurrence relation, shift operators, differential equation, determinant representation, and some other properties. Finally, two special cases of this family, truncated-exponential-based Hermite-type and truncated-exponential-based Laguerre–Frobenius Euler polynomials, are introduced and their corresponding properties are obtained.

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Wani, Shahid Ahmad, Ibtehal Alazman та Badr Saad T. Alkahtani. "Certain Properties and Applications of Convoluted Δh Multi-Variate Hermite and Appell Sequences". Symmetry 15, № 4 (2023): 828. http://dx.doi.org/10.3390/sym15040828.

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This study follows the line of research that by employing the monomiality principle, new outcomes are produced. Thus, in line with prior facts, our aim is to introduce the Δh multi-variate Hermite Appell polynomials ΔhHAm[r](q1,q2,⋯,qr;h). Further, we obtain their recurrence sort of relations by using difference operators. Furthermore, symmetric identities satisfied by these polynomials are established. The operational rules are helpful in demonstrating the novel characteristics of the polynomial families and thus operational principle satisfied by these polynomials is derived and will prove beneficial for future observations. Further, a few members of the Δh Appell polynomial family are considered and their corresponding results are derived accordingly.

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Alazman, Ibtehal, Badr Saad T. Alkahtani та Shahid Ahmad Wani. "Certain Properties of Δh Multi-Variate Hermite Polynomials". Symmetry 15, № 4 (2023): 839. http://dx.doi.org/10.3390/sym15040839.

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The research described in this paper follows the hypothesis that the monomiality principle leads to novel results that are consistent with past knowledge. Thus, in line with prior facts, our aim is to introduce the Δh multi-variate Hermite polynomials ΔhHm(q1,q2,⋯,qr;h). We obtain their recurrence relations by using difference operators. Furthermore, symmetric identities satisfied by these polynomials are established. The operational rules are helpful in demonstrating the novel characteristics of the polynomial families, and thus the operational principles satisfied by these polynomials are derived and will prove beneficial for future observations.

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40

Habbachi, Yahia. "A NEW CHARACTERIZATION OF SYMMETRIC DUNKL AND \(q\)-DUNKL-CLASSICAL ORTHOGONAL POLYNOMIALS." Ural Mathematical Journal 9, no. 2 (2023): 109. http://dx.doi.org/10.15826/umj.2023.2.009.

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In this paper, we consider the following \(\mathcal{L}\)-difference equation$$\Phi(x) \mathcal{L}P_{n+1}(x)=(\xi_nx+\vartheta_n)P_{n+1}(x)+\lambda_nP_{n}(x),\quad n\geq0,$$where \(\Phi\) is a monic polynomial (even), \(\deg\Phi\leq2\), \(\xi_n,\,\vartheta_n,\,\lambda_n,\,n\geq0\), are complex numbers and \(\mathcal{L}\) is either the Dunkl operator \(T_\mu\) or the the \(q\)-Dunkl operator \(T_{(\theta,q)}\). We show that if \(\mathcal{L}=T_\mu\), then the only symmetric orthogonal polynomials satisfying the previous equation are, up a dilation, the generalized Hermite polynomials and the generalized Gegenbauer polynomials and if \(\mathcal{L}=T_{(\theta,q)}\), then the \(q^2\)-analogue of generalized Hermite and the \(q^2\)-analogue of generalized Gegenbauer polynomials are, up a dilation, the only orthogonal polynomials sequences satisfying the \(\mathcal{L}\)-difference equation.

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41

C. Kayelvizhi and A. Emimal Kanaga Pushpam. "Solving neutral delay differential equations using least square method based on successive integration technique." International Journal of Science and Research Archive 11, no. 2 (2024): 509–17. http://dx.doi.org/10.30574/ijsra.2024.11.2.0474.

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The main objective of this work is to propose the Least square method (LSM) using successive integration technique for solving Neutral delay differential equations (NDDEs). Continuous LSM and Discrete LSM have been presented by adopting different orthogonal polynomials as weighted basis functions. In this study, the most widely used classical orthogonal polynomials, namely, the Bernoulli polynomial, the Chebyshev polynomial, the Hermite polynomial, and the Fibonacci polynomial are considered. Numerical examples of linear and nonlinear NDDEs have been provided to demonstrate the efficiency and accuracy of the method. Approximate solutions obtained by the proposed method are well comparable with exact solutions. From the results it is observed that the accuracy of the numerical solutions by the proposed method increases as N (order of the polynomial) increases. The proposed method is very effective, simple, and suitable for solving the linear and nonlinear NDDEs in real-world problems.

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42

Bretti, Gabriella, Pierpaolo Natalini, and Paolo E. Ricci. "Generalizations of the Bernoulli and Appell polynomials." Abstract and Applied Analysis 2004, no. 7 (2004): 613–23. http://dx.doi.org/10.1155/s1085337504306263.

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We first introduce a generalization of the Bernoulli polynomials, and consequently of the Bernoulli numbers, starting from suitable generating functions related to a class of Mittag-Leffler functions. Furthermore, multidimensional extensions of the Bernoulli and Appell polynomials are derived generalizing the relevant generating functions, and using the Hermite-Kampé de Fériet (or Gould-Hopper) polynomials. The main properties of these polynomial sets are shown. In particular, the differential equations can be constructed by means of the factorization method.

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FAN, XIANG. "PERMUTATION POLYNOMIALS OF DEGREE 8 OVER FINITE FIELDS OF ODD CHARACTERISTIC." Bulletin of the Australian Mathematical Society 101, no. 1 (2019): 40–55. http://dx.doi.org/10.1017/s0004972719000674.

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We give an algorithmic generalisation of Dickson’s method of classifying permutation polynomials (PPs) of a given degree $d$ over finite fields. Dickson’s idea is to formulate from Hermite’s criterion several polynomial equations satisfied by the coefficients of an arbitrary PP of degree $d$. Previous classifications of PPs of degree at most 6 were essentially deduced from manual analysis of these polynomial equations, but this approach is no longer viable for $d>6$. Our idea is to calculate some radicals of ideals generated by the polynomials, implemented by a computer algebra system. Our algorithms running in SageMath 8.6 on a personal computer work very fast to determine all PPs of degree 8 over an arbitrary finite field of odd order $q>8$. Such PPs exist if and only if $q\in \{11,13,19,23,27,29,31\}$ and are explicitly listed in normalised form.

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C, Kayelvizhi, and Emimal Kanaga Pushpam A. "Application of Polynomial Collocation Method Based on Successive Integration Technique for Solving Delay Differential Equation in Parkinson's Disease." Indian Journal of Science and Technology 17, no. 2 (2024): 112–19. https://doi.org/10.17485/IJST/v17i2.2573.

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Abstract <strong>Background/Objective:</strong> Parkinson’s disease (PD) is a neurological disorder that is often prevalent in elderly people. This is induced by the reduction or loss of dopamine secretion. The main objective of this work is to apply the polynomial collocation method using successive integration technique for solving delay differential equations (DDEs) arising in PD models. <strong>Methods:</strong> The polynomial collocation method based on successive integration techniques is proposed to obtain approximate solutions of the PD models. In this study, the most widely used classical orthogonal polynomials, namely, the Bernoulli polynomials, the Chebyshev polynomials, the Hermite polynomials, and the Fibonacci polynomials are considered. <strong>Findings:</strong> Numerical examples of two PD models have been considered to demonstrate the efficiency of the proposed method. Numerical simulations of the proposed method are well comparable to the simulation by step method using Picard approximation. <strong>Novelty:</strong> The numerical simulation demonstrates the reliability and efficiency of the proposed polynomial collocation method. The proposed method is very effective, simple, and suitable for solving the nonlinear DDEs model of PD and similar real-world problems exist in different fields of science and engineering. <strong>Keywords:</strong> Polynomial collocation method, Successive Integration Technique, Delay differential equation, Parkinson's disease, Simulation

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Swain, A. K., J. K. Parida, D. K. Bisoyi, S. Mazumder, and A. K. Mohanty. "Modelling of small-angle X-ray scattering data using Hermite polynomials." Journal of Applied Crystallography 34, no. 4 (2001): 510–18. http://dx.doi.org/10.1107/s0021889801006951.

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A new algorithm, called the term-selection algorithm (TSA), is derived to treat small-angle X-ray scattering (SAXS) data by fitting models to the scattering intensity using weighted Hermite polynomials. This algorithm exploits the orthogonal property of the Hermite polynomials and introduces an error-reduction ratio test to select the correct model terms or to determine which polynomials are to be included in the model and to estimate the associated unknown coefficients. With noa prioriinformation about particle sizes, it is possible to evaluate the real-space distribution function as well as three- and one-dimensional correlation functions directly from the models fitted to raw experimental data. The success of this algorithm depends on the choice of a scale factor and the accuracy of orthogonality of the Hermite polynomials over a finite range of SAXS data. An algorithm to select a weighted orthogonal term is therefore derived to overcome the disadvantages of the TSA. This algorithm combines the properties and advantages of both weighted and orthogonal least-squares algorithms and is numerically more robust for the estimation of the parameters of the Hermite polynomial models. The weighting feature of the algorithm provides an additional degree of freedom to control the effects of noise and the orthogonal feature enables the reorthogonalization of the Hermite polynomials with respect to the weighting matrix. This considerably reduces the error in orthogonality of the Hermite polynomials. The performance of the algorithm has been demonstrated considering both simulated data and experimental data from SAXS measurements of dewaxed cotton fibre at different temperatures.

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46

Provatidis, Christopher G. "Equivalence between C1 -continuous Cubic B-splines and Cubic Hermite Polynomials in Finite Element and Collocation Methods." WSEAS TRANSACTIONS ON SYSTEMS 23 (December 31, 2024): 585–97. https://doi.org/10.37394/23202.2024.23.60.

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In this paper, we will show that the C 1 -continuous B-spline functional set of polynomial degree p = 3 , can be written as a linear transformation of the well-known piecewise cubic Hermite polynomials. This change of functional basis means that the global B-spline finite element solution is equivalent to that of usual piecewise finite elements in conjunction with cubic Hermite polynomials, with two degrees per nodal point, like those used in beam-bending analysis. In this context, we validate the equivalence between the global Bspline solution and the piecewise solution in boundary-value and eigenvalue problems, for collocation and RitzGalerkin methods.

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Berriochoa, Elías, Alicia Cachafeiro, and Jaime Díaz. "Hermite Interpolation on the Unit Circle Considering up to the Second Derivative." ISRN Mathematical Analysis 2014 (March 10, 2014): 1–10. http://dx.doi.org/10.1155/2014/808519.

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The paper is devoted to study the Hermite interpolation problem on the unit circle. The interpolation conditions prefix the values of the polynomial and its first two derivatives at the nodal points and the nodal system is constituted by complex numbers equally spaced on the unit circle. We solve the problem in the space of Laurent polynomials by giving two different expressions for the interpolation polynomial. The first one is given in terms of the natural basis of Laurent polynomials and the remarkable fact is that the coefficients can be computed in an easy and efficient way by means of the Fast Fourier Transform (FFT). The second expression is a barycentric formula, which is very suitable for computational purposes.

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Mino, Lukáš, Imrich Szabó, and Csaba Török. "Bicubic splines and biquartic polynomials." Open Computer Science 6, no. 1 (2016): 1–7. http://dx.doi.org/10.1515/comp-2016-0001.

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AbstractThe paper proposes a new efficient approach to computation of interpolating spline surfaces. The generalization of an unexpected property, noticed while approximating polynomials of degree four by cubic ones, confirmed that a similar interrelation property exists between biquartic and bicubic polynomial surfaces as well. We prove that a 2×2-component C1 -class bicubic Hermite spline will be of class C2 if an equispaced grid is used and the coefficients of the spline components are computed from a corresponding biquartic polynomial. It implies that a 2×2 uniform clamped spline surface can be constructed without solving any equation. The applicability of this biquartic polynomials based approach to reducing dimensionalitywhile computing spline surfaces is demonstrated on an example.

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Molla, Hasib Uddin, and Goutam Saha. "Numerical Approximation of Fredholm Integral Equation (FIE) of 2nd Kind using Galerkin and Collocation Methods." GANIT: Journal of Bangladesh Mathematical Society 38 (January 14, 2019): 11–25. http://dx.doi.org/10.3329/ganit.v38i0.39782.

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In this research work, Galerkin and collocation methods have been introduced for approximating the solution of FIE of 2nd kind using LH (product of Laguerre and Hermite) polynomials which are considered as basis functions. Also, a comparison has been done between the solutions of Galerkin and collocation method with the exact solution. Both of these methods show the outcome in terms of the approximate polynomial which is a linear combination of basis functions. Results reveal that performance of collocation method is better than Galerkin method. Moreover, five different polynomials such as Legendre, Laguerre, Hermite, Chebyshev 1st kind and Bernstein are also considered as a basis functions. And it is found that all these approximate solutions converge to same polynomial solution and then a comparison has been made with the exact solution. In addition, five different set of collocation points are also being considered and then the approximate results are compared with the exact analytical solution. It is observed that collocation method performed well compared to Galerkin method. GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 11-25

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C. Kayelvizhi and A. Emimal Kanaga Pushpam. "Subdomain collocation method based on successive integration technique for solving delay differential equations." International Journal of Science and Research Archive 11, no. 2 (2024): 382–90. http://dx.doi.org/10.30574/ijsra.2024.11.2.0429.

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The main objective of this work is to propose the polynomial based Subdomain collocation method using successive integration technique for solving delay differential equations (DDEs). In this study, the most widely used classical orthogonal polynomials, namely, the Bernoulli polynomial, the Chebyshev polynomial, the Hermite polynomial, and the Fibonacci polynomial are considered. Numerical examples of linear and nonlinear DDEs have been considered to demonstrate the efficiency and accuracy of the method. Approximate solutions obtained by the proposed method are well comparable with exact solutions. From the results it is observed that the accuracy of the numerical solutions by the proposed method increases as N (order of the polynomial) increases. The proposed method is very effective, simple, and suitable for solving the linear and nonlinear DDEs in real-world problems.

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