Diophantine equations for Morgan-Voyce and other modified orthogonal polynomials

2008 ◽  
Vol 58 (1) ◽  
Author(s):  
Thomas Stoll ◽  
Robert Tichy
Keyword(s):  
Orthogonal Polynomials ◽  
Liouville Equation ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Second Order ◽  
Recurrence Equation ◽  
Classical Orthogonal Polynomials ◽  
The Third ◽  
Sturm Liouville

AbstractIt is well-known that Morgan-Voyce polynomials B n(x) and b n(x) satisfy both a Sturm-Liouville equation of second order and a three-term recurrence equation ([SWAMY, M.: Further properties of Morgan-Voyce polynomials, Fibonacci Quart. 6 (1968), 167–175]). We study Diophantine equations involving these polynomials as well as other modified classical orthogonal polynomials with this property. Let A, B, C ∈ ℚ and {pk(x)} be a sequence of polynomials defined by $$\begin{gathered} p_0 (x) = 1 \hfill \\ p_1 (x) = x - c_0 \hfill \\ p_{n + 1} (x) = (x - c_n )p_n (x) - d_n p_{n - 1} (x), n = 1,2,..., \hfill \\ \end{gathered} $$ with $$(c_0 ,c_n ,d_n ) \in \{ (A,A,B),(A + B,A,B^2 ),(A,Bn + A,\tfrac{1}{4}B^2 n^2 + Cn)\} $$ with A ≠ 0, B > 0 in the first, B ≠ 0 in the second and C > −¼B 2 in the third case. We show that the Diophantine equation with m > n ≥ 4, ≠ 0 has at most finitely many solutions in rational integers x, y.

scholarly journals Connection Problem for Sums of Finite Products of Chebyshev Polynomials of the Third and Fourth Kinds

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 617 ◽  
Author(s):  
Dmitry Dolgy ◽  
Dae Kim ◽  
Taekyun Kim ◽  
Jongkyum Kwon
Keyword(s):  
Orthogonal Polynomials ◽  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Linear Combinations ◽  
The Third ◽  
Connection Problem ◽  
Finite Products

This paper treats the connection problem of expressing sums of finite products of Chebyshev polynomials of the third and fourth kinds in terms of five classical orthogonal polynomials. In fact, by carrying out explicit computations each of them are expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials which involve some terminating hypergeometric functions F 0 2 , F 1 2 , and F 2 3 .

scholarly journals On Basis Property of Root Functions For a Class Second Order Differential Operator

2020 ◽  
Vol 5 (1) ◽  
pp. 361-368
Author(s):  
Volkan Ala ◽  
Khanlar R. Mamedov
Keyword(s):  
Differential Operator ◽  
Boundary Conditions ◽  
Spectral Parameter ◽  
Liouville Equation ◽  
Basis Property ◽  
Second Order ◽  
Order Differential Operator ◽  
Root Functions ◽  
Sturm Liouville

AbstractIn this work we investigate the completeness, minimality and basis properties of the eigenfunctions of one class discontinuous Sturm-Liouville equation with a spectral parameter in boundary conditions.

scholarly journals Connection Problem for Sums of Finite Products of Chebyshev Polynomials of the Third and Fourth Kinds

2018 ◽  
Author(s):  
Dmitry Victorovich Dolgy ◽  
Dae San Kim ◽  
Taekyun Kim ◽  
Jongkyum Kwon
Keyword(s):  
Orthogonal Polynomials ◽  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Linear Combinations ◽  
The Third ◽  
Connection Problem ◽  
Finite Products

This paper treats the connection problem of expressing sums of finite products of Chebyshev polynomials of the third and fourth kinds in terms of five classical orthogonal polynomials. In fact, by carrying out explicit computations each of them are expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials which involve some terminating hypergeometric functions ${}_2 F_0, {}_2 F_1$, and ${}_3 F_2$.

scholarly journals Some properties of orthogonal polynomials satisfying fourth order differential equations

1995 ◽  
Vol 37 (1) ◽  
pp. 105-113 ◽  
Author(s):  
R. G. Campos ◽  
L. A. Avila
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Order Differential Equation ◽  
Polynomial Solution ◽  
Second Order ◽  
System Of Nonlinear Equations ◽  
Singular Boundary ◽  
Classical Orthogonal Polynomials ◽  
Second Order Differential Equation

In the last few years, there has been considerable interest in the properties of orthogonal polynomials satisfying differential equations (DE) of order greater than two, their connection to singular boundary value problems, their generalizations, and their classification as solutions of second order DE (see for instance [2–8]). In this last interesting problem, some known facts about the classical orthogonal polynomials can be incorporated to connect these two sets of families and yield some nontrivial results in a very simple way. In this paper we only work with the nonclassical Jacobi type, Laguerre type and Legendre type polynomials, and we show how they can be connected with the classical Jacobi, Laguerre and Legendre polynomials, respectively; at the same time we obtain certain bounds for the zeros of the first ones by using a system of nonlinear equations satisfied by the zeros of any polynomial solution of a second order differential equation which, for the classical polynomials is known since Stieltjes and concerns the electrostatic interpretation of the zeros [10, Section 6.7; 9,1]. We also correct an expression for the second order differential equation of the Legendre type polynomials that circulates through the literature.

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