Skip to main content

Table 3 Basic Operations of DTM

From: Mathematical analysis of a generalized epidemic model with nonlinear incidence function

Original function f(x) Transformed function F(k)
f(x) = g(x) ± h(x) F(k) = G(k) ± H(k)
f(x) = cg(x) F(k) = cG(k), where c is a constant
\( f(x)=\frac{dg(x)}{dx} \) F(k) = (k + 1)G(k + 2)
\( f(x)=\frac{d^mg(x)}{dx^m} \) F(k) = (k + 1)(k + 2)…(k + m)G(k + m)
f(x) = 1 F(k) = δ(k)
f(x) = x F(k) = δ(k − 1)
f(x) = xm \( F(k)=\delta \left(k-m\right)=\Big\{{}_{0,\kern0.5em if\ k\ne m}^{1,\kern0.5em if\ k=m} \)
f(x) = g(x)h(x) \( F(k)={\sum}_{m=0}^kH(m)G\left(k-m\right) \)
f(x) = emx \( F(k)=\frac{m^k}{k!} \)
f(x) = (1 + x)m \( F(k)=\frac{m\left(m-1\right)\left(m-2\right)\dots \left(m-k+1\right)}{k!} \)