论文标题
一些递归定义的函数的封闭式公式
Closed-form formula for some recursively-defined integro-difference sequence of functions
论文作者
论文摘要
本文的主要目的是得出封闭式解决方案的序列$（g_n）_ {n \ in \ mathbb {n}} $的integro-difference方程的$递归定义如下：\ begin {align*} g_1（x）＆=χ_ {（ - 1/2，1/2）}（x）， g_ {n + 1}（x）＆= g_n（x + 1/2） - g_n（x- 1/2） + \ int_ {x- \ frac {1} {2}}}}^{x + \ \\ frac {1} {2} {2} {2}} {2}} g_n（s）其中$ g_1（x）=χ_ {（ - 1/2，1/2）}（x）$是单位间隔$（-1/2，1/2）$的特征函数，其值等于$ 1 $ in $（1/2，1/2）$上的$ 1 $，而$ 0 $ 0 $ in $ \ mathbb {r} $。
The main purpose of this paper is to derive the closed form solution the sequence $(g_n)_{n\in \mathbb{N}}$ of integro-difference equations that is defined recursively as follows: \begin{align*} g_1(x) & = χ_{(-1/2, 1/2)} (x), g_{n+1}(x) & = g_n(x + 1/2)- g_n(x- 1/2) + \int_{x-\frac{1}{2}}^{x + \frac{1}{2}} g_n(s)ds, \, n\in \mathbb{N}, \end{align*} where $ g_1(x)= χ_{(-1/2, 1/2)} (x) $ is the characteristic function of the unit interval $(-1/2, 1/2) $ has value equal to $ 1 $ on $(-1/2, 1/2) $ and $0$ elsewhere in $ \mathbb{R} $.
