Difference between revisions of "Delta gk"
| Line 26: | Line 26: | ||
[[Delta hk]] | [[Delta hk]] | ||
| − | {{:Delta special functions footer}} | + | <center>{{:Delta special functions footer}}</center> |
Revision as of 22:31, 2 June 2016
Let $\mathbb{T}$ be a time scale and let $t,s \in \mathbb{T}$. The $g_k$ monomials are defined by the recurrence $$\left\{ \begin{array}{ll} g_0(t,s)=1 \\ g_{k+1}(t,s)=\displaystyle\int_s^t g_k(\sigma(\tau),s) \Delta \tau. \end{array} \right.$$
Graph of $g_2(t,0;\mathbb{Z})$.
Graph of $g_3(t,0;\mathbb{Z})$.
Graph of $g_4(t,0;\mathbb{Z})$.
Graph of $g_5(t,0;\mathbb{Z})$.
Contents
Properties
Theorem
Let $\mathbb{T}$ be a time scale, let $t,s \in \mathbb{T}$, and let $k$ be a nonnegative integer. Then for all $0 \leq n \leq k-1$, $$g_k(\rho^n(t),t)=0,$$ where $g_n$ denotes the $g_k$ monomial and $\rho^k$ denotes compositions of the backward jump.
Proof
References
Theorem
Let $\mathbb{T}$ be a time scale, let $t,s \in \mathbb{T}$, and let $k$ be a nonnegative integer. Then the following formula holds: $$h_k(t,s;\mathbb{T})=(-1)^kg_k(s,t;\mathbb{T}),$$ where $h_k$ denotes the delta hk and $g_k$ denotes the delta gk.
Proof
References
Examples
| $\mathbb{T}=$ | $g_k(t,t_0)=$ |
| $\mathbb{R}$ | $g_k(t,t_0)=\dfrac{(t-t_0)^k}{k!}$ |
| $\mathbb{Z}$ | $g_k(t,t_0)= $ |
| $h\mathbb{Z}$ | $g_k(t,t_0)=$ |
| $\mathbb{Z}^2$ | $g_k(t,t_0)=$ |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | $g_k(t,t_0)=$ |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | $g_k(t,t_0)=$ |
| $\mathbb{H}$ | $g_k(t,t_0)=$ |
See also
$\Delta$-special functions on time scales | ||||||
 $\cos_p$ |
 $\cosh_p$ |
 $e_p$ |
 $g_k$ |
 $h_k$ |
 $\sin_p$ |
 $\sinh_p$ |
