Abstract
In this chapter we complete the work initiated in Section 3.2 of [8] (see also Problems 5, page 79, and 6, page 31), showing how to solve a linear recurrence relation with constant coefficients and arbitrary order. We first need to properly define the objects involved, and we do this now.
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Notes
- 1.
The reader must pay attention to the fact that we shall sometimes consider sequences (z n )n≥0 of complex numbers, so that the corresponding series will be denoted by ∑k≥0 z k .
- 2.
Actually, one can show that f is an example of a holomorphic (i.e., complex differentiable) function. However, since we shall not need this concept, and in order to maintain the elementary character of these notes, we will only establish the continuity of functions defined by power series. The interested reader can find the relevant results and definitions in [11].
- 3.
Karl Weierstrass , German mathematician of the nineteenth century.
- 4.
This also follows from basic Linear Algebra, for the transpose of the matrix of the coefficients of the linear system is a Vandermonde matrix of nonvanishing determinant.
References
A. Caminha, An Excursion Through Elementary Mathematics I - Real Numbers and Functions (Springer, New York, 2017)
A. Caminha, An Excursion Through Elementary Mathematics II - Euclidean Geometry (Springer, New York, 2018)
J.B. Conway, Functions of One Complex Variable I (Springer, New York, 1978)
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Caminha Muniz Neto, A. (2018). Linear Recurrence Relations. In: An Excursion through Elementary Mathematics, Volume III. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-77977-5_21
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DOI: https://doi.org/10.1007/978-3-319-77977-5_21
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