Difference Equations: An Introduction with Applications

By W. Kelley, Allan Peterson. 2000

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The H-Laplace and Q-Laplace Transforms (2010, background): “The classical Z -transform of a sequence {x(k)}∞ k=0 is defined by (see [14]) Z{x}(z) = ∞ ∑ x(k) zk .”
Stability for time varying linear dynamic systems on time scales (2005, background): “When T = Z, definite integrals correspond to definite sums from the difference calculus; see [21].”
Hille and Nehari type criteria for third-order delay dynamic equations (2013, background): “The three most popular examples of calculus on time scales are differential calculus, difference calculus (see [17]), and quantum calculus (see Kac and Cheung [16]), i.”
Initial value problems in discrete fractional calculus (2008, background): “Recall [6] that for linear difference equations with constant coefficients, the family of functions (1 + α) plays the same role that the family of functions e plays for linear ordinary differential equations with constant coefficients.”
The Theory of Discrete Fractional Calculus: Development and Application (2011, unknown): “43 the following well-known formula (see [13]):”
Unmasking the Theta Method (2003, background): “We note that (1) is a second-order difference equation and has the solution (see Kelley, 2000) Yt,θ = aθ + bθ(t − 1) +θXt (2) where aθ and bθ are constants and t = 1, . . . , n.”