A solution is given for un + 1 in terms of u1 and u0, where the elements of the sequence {un} satisfy the linear difference equation Two linearly independent solutions of the equation are written as determinants and relations are given which can be used to check the evaluation of these determinants.

Determinants, First Minors, and Cofactors.- A Summary of Basic Determinant Theory.- Intermediate Determinant Theory.- Particular Determinants.- Further Determinant Theory.- Applications of Determinants in Mathematical Physics.

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Solution of a linear difference equation

Determinants and their applications in mathematical physics

Sturmian comparison theorems for three-term recurrence equations

We express a weigthed generalization of the Delannoy numbers in terms of shifted Jacobi polynomials. A specialization of our formulas extends a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago, to all Delannoy numbers and certain Jacobi polynomials. Another specialization provides a weighted lattice path enumeration model for shifted Jacobi polynomials and allows the presentation of a combinatorial, non-inductive proof of the orthogonality of Jacobi polynomials with natural number parameters. The proof relates the orthogonality of these polynomials to the orthogonality of (generalized) Laguerre polynomials, as they arise in the theory of rook polynomials. We also find finite orthogonal polynomial sequences of Jacobi polynomials with negative integer parameters and expressions for a

background

Shifted Jacobi polynomials and Delannoy numbers

In Section 3, since relation (6) is valid only for n ≥ 2 r , the condition n ≥ r in relation (9) should be replaced by n ≥ 2 r . When r ≤ n ≤ 2 r − 1, relation (9) still holds but now relations (6) and (7) should be replaced, respectively, by the relations and the asterisk again denoting omission of the last column. The proof of (9) for r ≤ n ≤ 2 r − 1 is exactly similar to its proof for n ≥ 2 r .

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Solution of a general homogeneous linear difference equation

A matrix solution and a determinantal solution are obtained for a general linear recurrence relation.

Additional excerpts

/pdf/solution-of-a-general-liner-difference-equation-2oyhetb47a.pdf

Solution of a general liner difference equation

/pdf/solution-of-a-linear-difference-equation-gjnl6ay50h.pdf

Solution of a linear difference equation

Citations

Determinants and their applications in mathematical physics

Sturmian comparison theorems for three-term recurrence equations

Shifted Jacobi polynomials and Delannoy numbers

Solution of a general homogeneous linear difference equation

Solution of a general liner difference equation

Related Papers (5)

The Linear Difference Equation of the First Order

The Solution of the Homogeneous Linear Difference Equation of the Second Order

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On the solution of a third order linear homogeneous difference equation with variable coefficents