Abstract
Laplace transforms provide one of the means for solving homogeneous and inhomogeneous differential equations. Generating functions provide the corresponding transform for difference equations. A z-transform, also called a Laurent transform, is a generating function in which the variable \(\omega \) is replaced by \(z=1/\omega \).
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Notes
- 1.
See Weixlbaumer [42] for a mathematical analysis about the state of the art concerning the search for solutions of linear difference equations. Algorithms are presented for finding polynomial, rational, hypergeometric and d’Alembertian solutions.
- 2.
For \(d_{1} =0\) we have \(\beta _{1}= -\frac{c_{1}}{\Delta }+ \frac{1}{2}\left( \frac{c_{0}}{d_{0}}-\frac{c_{2}}{d_{2}}\right) , \quad \beta _{2}= \frac{c_{1}}{\Delta }+ \frac{1}{2}\left( \frac{c_{0}}{d_{0}}-\frac{c_{2}}{d_{2}}\right) \).
- 3.
As noted in Appendix A of [22], Eqs. (13)–(15) in [13] are incorrect.
- 4.
The alternate transformation, \(t=\frac{s}{s-1}\), leads to functions which follow from those obtained from \(t=\frac{s-1}{s}\) using Kummer’s transformation [36, Sect. 13.2(vii), Eq. 13.2.39].
- 5.
We note that, in terms of the difference operator \(\Delta \) defined in Chap. 1, the difference equation for w(x) can be written in the form \((\gamma _{2}+x+1)\Delta ^{2}w(x-1) + (2-\beta _{1})\Delta w(x-1)+zw(x)=0\), corresponding to the confluent hypergeometric function F5t given in Appendix I.
- 6.
If \(a=0,-1,-2\ldots \), then from [36, Sect. 13.2(i), Eq. 13.2.4], \(\frac{\Gamma (c-a+x)}{\Gamma (c+x)}{_{1}F_{1}}(a;c+x;z) = (-1)^{a}U(a;c+x;z)\); hence \(y_{1}(x) = (-1)^{a}y_{2}(x)\).
- 7.
If \(\frac{d_{0}}{c_{2}}<0\) we can choose \(\lambda = - \text {sgn}\left( \frac{c_{1}}{c_{2}}\right) \sqrt{-\frac{d_{0}}{c_{2}}}\) so that \(z>0\).
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Maximon, L.C. (2016). Generating Functions, Z-Transforms, Laplace Transforms and the Solution of Linear Differential and Difference Equations. In: Differential and Difference Equations. Springer, Cham. https://doi.org/10.1007/978-3-319-29736-1_7
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DOI: https://doi.org/10.1007/978-3-319-29736-1_7
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