A non-classical, orthogonal polynomial family

1985 ◽  
pp. 506-513 ◽  
Author(s):  
A. G. Law ◽  
M. B. Sledd
Keyword(s):  
Orthogonal Polynomial ◽  
Classical Orthogonal Polynomial ◽  
Polynomial Family

scholarly journals A characterization of the four Chebyshev orthogonal families

2005 ◽  
Vol 2005 (13) ◽  
pp. 2071-2079 ◽  
Author(s):  
E. Berriochoa ◽  
A. Cachafeiro ◽  
J. M. Garcia-Amor
Keyword(s):  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Linear Combinations ◽  
Classical Orthogonal Polynomial ◽  
Polynomial Sequences ◽  
Fixed Length ◽  
Constant Coefficients ◽  
Fourth Kind

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.

scholarly journals HAHN'S PROBLEM WITH RESPECT TO SOME PERTURBATIONS OF THE RAISING OPERATOR \(X-c\)

2020 ◽  
Vol 6 (2) ◽  
pp. 15
Author(s):  
Baghdadi Aloui ◽  
Jihad Souissi
Keyword(s):  
Orthogonal Polynomial ◽  
Complex Number ◽  
Classical Orthogonal Polynomial ◽  
Arbitrary Complex Number ◽  
Raising Operators ◽  
Charlier Polynomial ◽  
Arbitrary Complex ◽  
Raising Operator

In this paper, we study the Hahn's problem with respect to some raising operators perturbed of the operator \(X-c\), where \(c\) is an arbitrary complex number. More precisely, the two following characterizations hold: up to a normalization, the \(q\)-Hermite (resp. Charlier) polynomial is the only \(H_{\alpha,q}\)-classical (resp. \(\mathcal{S}_{\lambda}\)-classical) orthogonal polynomial, where \(H_{\alpha, q}:=X+\alpha H_q\) and \(\mathcal{S}_{\lambda}:=(X+1)-\lambda\tau_{-1}.\)

scholarly journals L-classical d-orthogonal polynomial sets of Sheffer type

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 881-895
Author(s):  
Youssèf Cheikh ◽  
Inès Gam
Keyword(s):  
Differential Equation ◽  
Orthogonal Polynomial ◽  
Order Differential Equation ◽  
Derivative Operator ◽  
Classical Orthogonal Polynomial ◽  
First Case ◽  
Lowering Operators ◽  
Lowering Operator

In this paper, we characterize L-classical d-orthogonal polynomial sets of Sheffer type where L being a lowering operator commutating with the derivative operator D and belonging to {D,eD-1, sin(D)}. For the first case we state a (d+1)-order differential equation satisfied by the corresponding polynomials. We, also, show that, with these three lowering operators, all the orthogonal polynomial sets are classified as L-classical orthogonal polynomial sets.

scholarly journals Fast associated classical orthogonal polynomial transforms

2021 ◽  
pp. 113831
Author(s):  
Brock Klippenstein ◽  
Richard Mikaël Slevinsky
Keyword(s):  
Orthogonal Polynomial ◽  
Classical Orthogonal Polynomial

Identities connecting the Chebyshev polynomials

2016 ◽  
Vol 100 (549) ◽  
pp. 450-459 ◽  
Author(s):  
Jonny Griffiths
Keyword(s):  
Differential Equations ◽  
Weight Function ◽  
Orthogonal Polynomial ◽  
Recurrence Relation ◽  
Approximation Theory ◽  
Chebyshev Polynomials ◽  
Pell Equation ◽  
Fibonacci Polynomials ◽  
The Third ◽  
Polynomial Family

There are many families of polynomials in mathematics, and they often occur naturally in pairs. The Fibonacci polynomials and the Lucas polynomials, for example, are generated by the same recurrence relation but with different starting values, and there are many identities that link the two families [1]. The same is true for the Chebyshev polynomials of the first and second kinds, Tn (x) and Un (x) [2], respectively. There are two further polynomial families that are less well-known, the Chebyshev polynomials of the third and fourth kinds, Vn (x) and Wn (x) [3], respectively. Each of the four kinds is an example of an orthogonal polynomial family Pn (x), where for some appropriate weight function W (x), whenever n ≠ m. The families Tn (x) and Un (x) in particular are ubiquitous in their mathematical uses, in approximation theory, in differential equations, and in solving the Pell equation, to name but three. There are also many connections between Tn (x), Un (x), Vn (x) and Wn (x), some of which are explored here, and some of which we hope are new.

Recurrence equations and their classical orthogonal polynomial solutions

2002 ◽  
Vol 128 (2-3) ◽  
pp. 303-327 ◽  
Author(s):  
Wolfram Koepf ◽  
Dieter Schmersau
Keyword(s):  
Orthogonal Polynomial ◽  
Classical Orthogonal Polynomial ◽  
Polynomial Solutions ◽  
Recurrence Equations

Pengaruh Pemberian Kunyit (Curcuma domestica) terhadap Beberapa Kualitas Fisik dan Organoleptik Bakso Daging Itik

2013 ◽  
Vol 8 (1) ◽  
pp. 16-24
Author(s):  
Sari Murti ◽  
Suharyanto Suharyanto ◽  
Desia Kaharuddin
Keyword(s):  
Orthogonal Polynomial

ABSTRAKPenelitian ini bertujuan untuk mengevaluasi level kunyit yang tepat sehingga dapat meningkatkan kualitas bakso daging itik. Percobaan ini menggunakan Rancangan Acak Lengkap, yang terdiri dari empat perlakuan yaitu P0 (tanpa pemberian kunyit), P1 (pemberian kunyit taraf 2,5%), P2 (pemberian kunyit taraf 5%), dan P3 (pemberian kunyittaraf 7,5%), masing-masing perlakuan terdiri dari tiga ulangan. Hasil penelitian menunjukkan bahwa pemberian kunyit berpengaruh tidak nyata (P>0,05) terhadap variabel susut masak, daya mengikat air, pH dan tekstur bakso daging itik, tetapi nyata (P<0,05) meningkatkan skor warna, menurunkan derajat keamisan dan variabel sifat organoleptik rasa. Hasil yang berpengaruh nyata diuji lanjut menggunakan Orthogonal polynomial dengan menghasilkan persamaan linear dan kuadratik. Dari hasil penelitian disimpulkan bahwa penambahan kunyit 2,5% hingga 7,5% tidak menurunkan kualitas fisik (susut masak, pH, dan DMA). Penambahan kunyit 2,5% dapat mempertahankan cita rasa dan penerimaan umum panelis terhadap bakso daging itik.Kata kunci : Kunyit (Curcuma domestica), kualitas fisik, daging itik

A Useful Curve Fitting Method for Concrete Uniaxial Compressive Experiment Data

2011 ◽  
Vol 291-294 ◽  
pp. 1015-1020 ◽  
Author(s):  
Chong Jin ◽  
Hong Wang ◽  
Xiao Zhou Xia
Keyword(s):  
Experimental Data ◽  
Orthogonal Polynomial ◽  
Least Square ◽  
Base Function ◽  
Concrete Material ◽  
Fitting Method ◽  
Orthogonal Base ◽  
The Matrix ◽  
Uniform Function ◽  
Curve Fitting Method

Based on the superiority avoiding the matrix equation to be morbid for those fitting functions constructed by orthogonal base, the Legendre orthogonal polynomial is adopted to fit the experimental data of concrete uniaxial compression stress-strain curves under the frame of least-square. With the help of FORTRAN programming, 3 series of experimental data is fitted. And the fitting effect is very satisfactory when the item number of orthogonal base is not less than 5. What’s more, compared with those piecewise fitting functions, the Legendre orthogonal polynomial fitting function obtained can be introduced into the nonlinear harden-soften character of concrete constitute law more convenient because of its uniform function form and continuous derived feature. And the fitting idea by orthogonal base function will provide a widely road for studying the constitute law of concrete material.

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