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Relevant bibliographies by topics / Chebyshev polynomials of the first kind

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Academic literature on the topic 'Chebyshev polynomials of the first kind'

Author: Grafiati

Published: 4 June 2021

Last updated: 27 April 2022

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Journal articles on the topic "Chebyshev polynomials of the first kind"

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Kim, Taekyun, Dae San Kim, Dmitry V. Dolgy, and Jongkyum Kwon. "Representing Sums of Finite Products of Chebyshev Polynomials of the First Kind and Lucas Polynomials by Chebyshev Polynomials." Mathematics 7, no. 1 (2018): 26. http://dx.doi.org/10.3390/math7010026.

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In this paper, we study sums of finite products of Chebyshev polynomials of the first kind and Lucas polynomials and represent each of them in terms of Chebyshev polynomials of all kinds. Here, the coefficients involve terminating hypergeometric functions 2F1 and these representations are obtained by explicit computations.

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Pavlyuk, A. M. "Generalized Equidistant Chebyshev Polynomials and Alexander Knot Invariants." Ukrainian Journal of Physics 63, no. 6 (2018): 488. http://dx.doi.org/10.15407/ujpe63.6.488.

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We introduce the generalized equidistant Chebyshev polynomials T(k,h) of kind k of hyperkind h, where k, h are positive integers. They are obtained by a generalization of standard and monic Chebyshev polynomials of the first and second kinds. This generalization is fulfilled in two directions. The horizontal generalization is made by introducing hyperkind ℎ and expanding it to infinity. The vertical generalization proposes expanding kind k to infinity with the help of the method of equidistant coefficients. Some connections of these polynomials with the Alexander knot and link polynomial invariants are investigated.

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Rababah, Abedallah. "Conformable chebyshev differential Equation of first kind." International Journal of Electrical and Computer Engineering (IJECE) 11, no. 1 (2021): 628. http://dx.doi.org/10.11591/ijece.v11i1.pp628-635.

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In this paper, the Chebyshev-I conformable differential equation is considered. A proper power series is examined; there are two solutions, the even solution and the odd solution. The Rodrigues’ type formula is also allocated for the conformable Chebyshev-I polynomials.

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Wang, Xiao, and Jiayuan Hu. "An Identity Involving the Integral of the First-Kind Chebyshev Polynomials." Mathematical Problems in Engineering 2018 (May 31, 2018): 1–5. http://dx.doi.org/10.1155/2018/7186940.

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We used the algebraic manipulations and the properties of Chebyshev polynomials to obtain an interesting identity involving the power sums of the integral of the first-kind Chebyshev polynomials and solved an open problem proposed by Wenpeng Zhang and Tingting Wang.

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Cesarano, Clemente, and Claudio Fornaro. "A note on two-variable Chebyshev polynomials." Georgian Mathematical Journal 24, no. 3 (2017): 339–49. http://dx.doi.org/10.1515/gmj-2016-0034.

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AbstractIn this paper we discuss generalized two-variable Chebyshev polynomials and their relevant relations; in particular, by using their integral representations, we prove some operational identities. The approach is based on the generalized two-variable Hermite polynomials and the integral representations of ordinary Chebyshev polynomials of first and second kind. In addition, we discuss how the families of generalized Chebyshev polynomials can be used to prove some interesting properties related to ordinary Chebyshev polynomials of first and second kind. A fundamental role, as we see, is played by the powerful operational techniques verified by the families of generalized Hermite polynomials.

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Cesarano, Clemente, Sandra Pinelas, and Paolo Ricci. "The Third and Fourth Kind Pseudo-Chebyshev Polynomials of Half-Integer Degree." Symmetry 11, no. 2 (2019): 274. http://dx.doi.org/10.3390/sym11020274.

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New sets of orthogonal functions, which correspond to the first, second, third, and fourth kind Chebyshev polynomials with half-integer indexes, have been recently introduced. In this article, links of these new sets of irrational functions to the third and fourth kind Chebyshev polynomials are highlighted and their connections with the classical Chebyshev polynomials are shown.

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AlQudah, Mohammad. "Characterization of the generalized Chebyshev-type polynomials of first kind." International Journal of Applied Mathematical Research 4, no. 4 (2015): 519. http://dx.doi.org/10.14419/ijamr.v4i4.4788.

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<p>Orthogonal polynomials have very useful properties in the mathematical problems, so recent years have seen a great deal in the field of approximation theory using orthogonal polynomials. In this paper, we characterize a sequence of the generalized Chebyshev-type polynomials of the first kind \(\left\{\mathscr{T}_{n}^{(M,N)}(x)\right\}_{n\in\mathbb{N}\cup\{0\}},\) which are orthogonal with respect to the measure \(\frac{\sqrt{1-x^{2}}}{\pi}dx+M\delta_{-1}+N\delta_{1},\) where \(\delta_{x}\) is a singular Dirac measure and \(M,N\geq 0.\) Then we provide a closed form of the constructed polynomials in term of the Bernstein polynomials \(B_{k}^{n}(x).\)</p><p>We conclude the paper with some results on the integration of the weighted generalized Chebyshev-type with the Bernstein polynomials.</p>

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Berriochoa, E., A. Cachafeiro, and J. M. Garcia-Amor. "A characterization of the four Chebyshev orthogonal families." International Journal of Mathematics and Mathematical Sciences 2005, no. 13 (2005): 2071–79. http://dx.doi.org/10.1155/ijmms.2005.2071.

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We obtain a property which characterizes the Chebyshev orthogonal polynomials of first, second, third, and fourth kind. Indeed, we prove that the four Chebyshev sequences are the unique classical orthogonal polynomial families such that their linear combinations, with fixed length and constant coefficients, can be orthogonal polynomial sequences.

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Cesarano, Clemente, Pierpaolo Natalini, and Paolo Ricci. "Pseudo-Lucas Functions of Fractional Degree and Applications." Axioms 10, no. 2 (2021): 51. http://dx.doi.org/10.3390/axioms10020051.

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In a recent article, the first and second kinds of multivariate Chebyshev polynomials of fractional degree, and the relevant integral repesentations, have been studied. In this article, we introduce the first and second kinds of pseudo-Lucas functions of fractional degree, and we show possible applications of these new functions. For the first kind, we compute the fractional Newton sum rules of any orthogonal polynomial set starting from the entries of the Jacobi matrix. For the second kind, the representation formulas for the fractional powers of a r×r matrix, already introduced by using the pseudo-Chebyshev functions, are extended to the Lucas case.

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Jacobs, David P., Mohamed O. Rayes, and Vilmar Trevisan. "The Resultant of Chebyshev Polynomials." Canadian Mathematical Bulletin 54, no. 2 (2011): 288–96. http://dx.doi.org/10.4153/cmb-2011-013-1.

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AbstractLet Tn denote the n-th Chebyshev polynomial of the first kind, and let Un denote the n-th Chebyshev polynomial of the second kind. We give an explicit formula for the resultant res(Tm, Tn). Similarly, we give a formula for res(Um, Un).

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Dissertations / Theses on the topic "Chebyshev polynomials of the first kind"

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Tinawi, Félix. "Solution de C. Hyltén-Cavallius pour un problème de P. Turán concernant des polynômes." Thèse, 2008. http://hdl.handle.net/1866/7876.

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Hachani, Mohamed Amine. "Sur les comportements locaux de polynômes et polynômes trigonométriques." Thèse, 2008. http://hdl.handle.net/1866/7877.

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Book chapters on the topic "Chebyshev polynomials of the first kind"

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Saw, Vijay, and Sushil Kumar. "The Approximate Solution for Multi-term the Fractional Order Initial Value Problem Using Collocation Method Based on Shifted Chebyshev Polynomials of the First Kind." In Advances in Intelligent Systems and Computing. Springer Singapore, 2018. http://dx.doi.org/10.1007/978-981-10-7590-2_4.

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Bozzini, M., and M. Rossini. "An Approximation Method Based on the Second Kind Chebyshev Polynomials." In Nonlinear Numerical Methods and Rational Approximation II. Springer Netherlands, 1994. http://dx.doi.org/10.1007/978-94-011-0970-3_16.

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Sultanakhmedov, M. S. "Some Properties of Sobolev Orthogonal Polynomials Associated with Chebyshev Polynomials of the Second Kind." In Trends in Mathematics. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-49763-7_18.

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Andrews, George E. "Dyson’s “Favorite” Identity and Chebyshev Polynomials of the Third and Fourth Kind." In Trends in Mathematics. Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-57050-7_6.

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"Chebyshev Polynomials of the First Kind." In The Classical Orthogonal Polynomials. WORLD SCIENTIFIC, 2015. http://dx.doi.org/10.1142/9789814704045_0005.

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Grosjean, Carl C. "MISCELLANEOUS PROBLEMS SOLVED IN TERMS OF CHEBYSHEV'S ORTHOGONAL POLYNOMIALS OF THE FIRST AND THE SECOND KIND." In Topics in Polynomials of One and Several Variables and Their Applications. WORLD SCIENTIFIC, 1993. http://dx.doi.org/10.1142/9789814360296_0015.

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"Chebyshev Polynomials of the Second Kind." In The Classical Orthogonal Polynomials. WORLD SCIENTIFIC, 2015. http://dx.doi.org/10.1142/9789814704045_0006.

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"Chebyshev Polynomials of the Third Kind." In The Classical Orthogonal Polynomials. WORLD SCIENTIFIC, 2015. http://dx.doi.org/10.1142/9789814704045_0007.

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"Chebyshev Polynomials of the Fourth Kind." In The Classical Orthogonal Polynomials. WORLD SCIENTIFIC, 2015. http://dx.doi.org/10.1142/9789814704045_0008.

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"The Functions F1 - Elliptic Polynomials of the First Kind." In Elliptic Polynomials. Chapman and Hall/CRC, 2000. http://dx.doi.org/10.1201/9781482285765-11.

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Conference papers on the topic "Chebyshev polynomials of the first kind"

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Duru, Hatice Kübra, and Elçin Yusufoğlu. "Solution of the system of Cauchy-type singular integral equations of the first kind by third- and fourth- kind Chebyshev polynomials." In PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2014 (ICNAAM-2014). AIP Publishing LLC, 2015. http://dx.doi.org/10.1063/1.4912683.

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Pandiyan, R., and S. C. Sinha. "Techniques for Periodic Control of Flapping of a Helicopter Blade." In ASME 1995 Design Engineering Technical Conferences collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium. American Society of Mechanical Engineers, 1995. http://dx.doi.org/10.1115/detc1995-0262.

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Abstract The primary objective of this paper is to present new periodic control strategies for the control of flapping motion of an individual helicopter rotor blade in forward flight which is represented by a differential equation with periodic coefficients. First, an algebraic procedure based on Chebyshev polynomial expansion is employed to control the periodic flapping motion. In this approach, the state vector and the elements of the periodic system matrix have been expanded in terms of shifted Chebyshev polynomials of the first kind over the principal period. Later, optimal control theory in conjunction with Floquet Theory has been used to design full state and observer state feedback controllers. In the second method, the feedback controllers have been designed in the time-invariant domain through an application of a linear, invertible, periodic transformation known as Liapunov-Floquet (L-F) transformation to the periodic system model. It is shown that by using both the methods the periodic control gains can be obtained as explicit functions of time and therefore, a control scheme more suitable for real-time implementation can be achieved. The design procedures have been found to be much simpler in character when compared to those techniques that have been appeared in the literature.

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Szabó, Zsolt, S. C. Sinha, and Gábor Stépán. "Dynamics of Pipes Containing Pulsative Flow." In ASME 1997 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1997. http://dx.doi.org/10.1115/detc97/vib-4022.

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Abstract Several mechanical models exist on elastic pipes containing fluid flow. In this paper those models are considered, where the fluid is incompressible, frictionless and its velocity relative to the pipe has the same but time-periodic magnitude along the pipe at a certain time instant. The pipe can be modelled either as a chain of articulated rigid pipes or as a continuum. The dynamic behaviour of the system strongly depends on the different kinds of boundary conditions and on the fact whether the pipe is considered to be inextensible, i.e. the cross-sectional area of the pipe is constant. The equations of motion are derived via Lagrangian equations and Hamilton’s principle. These systems are non-conservative, and the amount of energy carried in and out by the flow appears in the model. It is well-known, that intricate stability problems arise when the flow pulsates and the corresponding mathematical model, a system of ordinary or partial differential equations, becomes time-periodic. There are several standard techniques, like perturbation method, harmonic balance, finite difference, etc., to analyze these models. The method which constructs the state transition matrix used in Floquet theory in terms of the shifted Chebyshev polynomials of the first kind is especially effective for stability analysis of large systems. The implementation of this method using computer algebra enables us to obtain more accurate results and to investigate more complex models. The stability charts are created with respect to three important parameters: the forcing frequency ω, the perturbation amplitude υ and the mean flow velocity U.

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Stosic, Biljana P., and Vlastimir D. Pavlovic. "Chebyshev polynomials of the second kind in filter design." In 2017 13th International Conference on Advanced Technologies, Systems and Services in Telecommunications (TELSIKS). IEEE, 2017. http://dx.doi.org/10.1109/telsks.2017.8246260.

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Dewi, I. P., S. Utama, and S. Aminah. "Deriving the explicit formula of Chebyshev polynomials of the third kind and the fourth kind." In PROCEEDINGS OF THE 3RD INTERNATIONAL SYMPOSIUM ON CURRENT PROGRESS IN MATHEMATICS AND SCIENCES 2017 (ISCPMS2017). Author(s), 2018. http://dx.doi.org/10.1063/1.5064199.

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Ng, Guan Shen, Yuhao Leong, Sovuthy Cheab, Isnani B. Alias, and Socheatra Soeung. "Design of Dual-Band Bandpass Filter Based on Chained Chebyshev Polynomials of the Second Kind." In 2020 IEEE Asia-Pacific Microwave Conference (APMC 2020). IEEE, 2020. http://dx.doi.org/10.1109/apmc47863.2020.9331442.

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Zagorowska, Marta, Nina Thornhill, and Charlotte Skourup. "Dynamic Modelling and Control of a Compressor Using Chebyshev Polynomial Approximation." In ASME Turbo Expo 2018: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/gt2018-76399.

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The aim of this study is to apply a Chebyshev polynomial approximation of the compressor map for dynamic modelling and control of centrifugal compressors. The results are compared to those from an approximation based on the third order polynomials and a compressor map derived from first principles. In the analysis of centrifugal compressors, a combination of dynamic conservation laws and static compressor map provides an insight into the surge phenomenon, whose avoidance remains one of the objectives of compressor control. The compressor maps based on the physical laws provide accurate results, but require a detailed knowledge about the properties of the system, such as the geometry of the compressor and gas quality. Third order polynomials are usually used as an approximation for the compressor map, providing simplified models at the expense of accuracy. Chebyshev polynomial approximation provides a trade-off between the accuracy of physical modelling with the ease of use provided by third order polynomial approximation.

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Sarkar, Dripta, Emiliano Renzi, and Frederic Dias. "Oscillating Wave Surge Converters: Interactions in a Wave Farm." In ASME 2014 33rd International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/omae2014-23393.

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The hydrodynamic behaviour of a wave farm comprising of Oscillating Wave Surge Converters (OWSC) is investigated using a mathematical model based on linear potential flow theory. The developed method can analyse a large number of wave energy converters in arbitrary configurations with oblique wave incidence and considers the hydrodynamic interactions amongst all the devices. The highly efficient novel method is based on Greens Integral Equation formulation, yielding hypersingular integrals which are finally solved using Chebyshev polynomials of the second kind. Using the semi -analytical approach, some possible configurations of a wave farm are studied. In the case of an inline configuration of the OWSCs with normal wave incidence, the occurrence of a near resonant behaviour already observed for 3 flaps is confirmed. A strong wave focussing effect is observed in some special configurations comprising of a large number of such devices. In general, the flaps located on the front of the wave farm are found to exhibit an enhanced performance behaviour in average, due to the mutual interactions arising within the array. A special case of two back to back flaps, oscillating independently, is also analysed using the above approach.

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Deshmukh, Venkatesh. "Stability Analysis and Computation of Solutions of Nonlinear Delay Differential Algebraic Equations With Time Periodic Coefficients." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-35263.

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Stability theory of Nonlinear Delay Differential Algebraic Equations (DDAE) with periodic coefficients is proposed with a geometric interpretation of the evolution of the linearized system. First, a numerical algorithm based on direct integration by expansion in terms of Chebyshev polynomials is derived for linear analysis. The proposed algorithm is shown to have deeper connections with and computationally less cumbersome than the solution of the underlying semi-explicit system via a similarity transformation. The stability of time periodic DDAE systems is characterized by the spectral radius of a finite dimensional approximation or a “monodromy matrix” of a compact infinite dimensional operator. The monodromy operator is essentially a map of the Chebyshev coefficients of the state form the delay interval to the next adjacent interval of time. The monodromy matrix is obtained by a similarity transformation of the momodromy matrix of the associated semi-explicit system. The computations are entirely performed in the original system form to avoid cumbersome transformations associated with the semi-explicit system. Next, two computational algorithms are detailed for obtaining solutions of nonlinear DDAEs with periodic coefficients for consistent initial functions.

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Yuceoglu, U., V. O¨zerciyes, and K. C¸il. "Free Flexural Vibrations of Bonded and Centrally Doubly Stiffened Composite Orthotropic Base Plates or Panels." In ASME 2004 International Mechanical Engineering Congress and Exposition. ASMEDC, 2004. http://dx.doi.org/10.1115/imece2004-59769.

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The problem of the “Free Flexural Vibrations of Bonded and Centrally Doubly Stiffened Composite Orthotropic Base Plates or Panels” is formulated and investigated. The composite plate or panel system is made up of an orthtropic base plate reinforced or doubly stiffened by the upper and lower stiffening orthotropic plate strips. The stiffening plate strips are at the mid-center and are adhesively bonded to the base plate. The base plate and the stiffening plate strips are considered as dissimilar orthotropic Mindlin plates. Thus, the analysis is based on a “First Order Shear Deformation Plate Theory (FSDPT)” of Mindlin type. In the very thin, linearly elastic adhesive layers, the transverse normal and shear stresses are included. The sets of the dynamic equations and other equations of plates and adhesive layers are finally reduced to a “Governing System of the First Order Ordinary Differential Equations.” Then, this system is integrated by means of the “Modified Transfer Matrix Method (with Interpolation Polynomials and/or Chebyshev Polynomials).” The mode shapes and the associated natural frequencies are calculated and some parametric studies are presented. Also, the influences of the “hard” and the “soft” adhesive layers on the natural frequencies and the mode shapes are shown.

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