Second-order Sturm-Liouville difference equations and orthogonal polynomials

1995 ◽  
Vol 113 (542) ◽  
pp. 0-0 ◽  
Author(s):  
Alouf Jirari
Keyword(s):  
Orthogonal Polynomials ◽  
Difference Equations ◽  
Second Order ◽  
Sturm Liouville

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 Special cases of critical linear difference equations

2021 ◽  
pp. 1-17
Author(s):  
Jan Jekl
Keyword(s):  
Difference Equations ◽  
Adjoint Equation ◽  
Second Order ◽  
Linear Difference Equations ◽  
Order Linear ◽  
Special Cases ◽  
Even Order ◽  
Nonnegative Coefficients ◽  
Sturm Liouville ◽  
Second Order Equations

In this paper, we investigate even-order linear difference equations and their criticality. However, we restrict our attention only to several special cases of the general Sturm–Liouville equation. We wish to investigate on such cases a possible converse of a known theorem. This theorem holds for second-order equations as an equivalence; however, only one implication is known for even-order equations. First, we show the converse in a sense for one term equations. Later, we show an upper bound on criticality for equations with nonnegative coefficients as well. Finally, we extend the criticality of the second-order linear self-adjoint equation for the class of equations with interlacing indices. In this way, we can obtain concrete examples aiding us with our investigation.

scholarly journals On the Neumann eigenvalues for second-order Sturm–Liouville difference equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yan-Hsiou Cheng
Keyword(s):  
Difference Equations ◽  
Order Relation ◽  
Second Order ◽  
Dirichlet Eigenvalue ◽  
Eigenvalue Gap ◽  
First Dirichlet Eigenvalue ◽  
Constant Potential ◽  
Sturm Liouville ◽  
Neumann Eigenvalue ◽  
Neumann Eigenvalues

Abstract The paper is concerned with the Neumann eigenvalues for second-order Sturm–Liouville difference equations. By analyzing the new discriminant function, we show the interlacing properties between the periodic, antiperiodic, and Neumann eigenvalues. Moreover, when the potential sequence is symmetric and symmetric monotonic, we show the order relation between the first Dirichlet eigenvalue and the second Neumann eigenvalue, and prove that the minimum of the first Neumann eigenvalue gap is attained at the constant potential sequence.

scholarly journals On Second Order q-Difference Equations Satisfied by Al-Salam–Carlitz I-Sobolev Type Polynomials of Higher Order

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1300
Author(s):  
Carlos Hermoso ◽  
Edmundo J. Huertas ◽  
Alberto Lastra ◽  
Anier Soria-Lorente
Keyword(s):  
Orthogonal Polynomials ◽  
Real Number ◽  
Difference Equations ◽  
Arbitrary Number ◽  
Recurrence Formula ◽  
Second Order ◽  
Inner Product ◽  
Sobolev Type ◽  
Ladder Operators ◽  
Connection Formulas

This contribution deals with the sequence {Un(a)(x;q,j)}n≥0 of monic polynomials in x, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam–Carlitz I orthogonal polynomials, and involving an arbitrary number j of q-derivatives on the two boundaries of the corresponding orthogonality interval, for some fixed real number q∈(0,1). We provide several versions of the corresponding connection formulas, ladder operators, and several versions of the second order q-difference equations satisfied by polynomials in this sequence. As a novel contribution to the literature, we provide certain three term recurrence formula with rational coefficients satisfied by Un(a)(x;q,j), which paves the way to establish an appealing generalization of the so-called J-fractions to the framework of Sobolev-type orthogonality.

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