Incomplete symmetric orthogonal polynomials of finite type generated by a generalized Sturm–Liouville theorem

The classical Kramer sampling theorem is, in the subject of self-adjoint boundary value problems, one of the richest sources to obtain sampling expansions. It has become very fruitful in connection with discrete Sturm-Liouville problems. In this paper a discrete version of the analytic Kramer sampling theorem is proved. Orthogonal polynomials arising from indeterminate Hamburger moment problems as well as polynomials of the second kind associated with them provide examples of Kramer analytic kernels.

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Two Finite Sequences of Symmetric q-Orthogonal Polynomials Generated by Two q-Sturm–Liouville Problems

Reports on Mathematical Physics ◽

10.1016/s0034-4877(20)30009-4 ◽

2020 ◽

Vol 85 (1) ◽

pp. 41-55 ◽

Cited By ~ 1

Author(s):

Mohammad Masjed-Jamei ◽

Fatemeh Soleyman ◽

Wolfram Koepf

Keyword(s):

Orthogonal Polynomials ◽

Finite Sequences ◽

Sturm Liouville

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A basic class of symmetric orthogonal functions using the extended Sturm–Liouville theorem for symmetric functions

Journal of Computational and Applied Mathematics ◽

10.1016/j.cam.2007.04.025 ◽

2008 ◽

Vol 216 (1) ◽

pp. 128-143 ◽

Cited By ~ 7

Author(s):

Mohammad Masjed-Jamei

Keyword(s):

Symmetric Functions ◽

Liouville Theorem ◽

Orthogonal Functions ◽

Basic Class ◽

Sturm Liouville

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Diophantine equations for Morgan-Voyce and other modified orthogonal polynomials

Mathematica Slovaca ◽

10.2478/s12175-007-0051-2 ◽

2008 ◽

Vol 58 (1) ◽

Cited By ~ 3

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.

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