FANDOM


Continuous

$ U(n,x) = \sum_{k=0}^{\lfloor x\rfloor}\frac{(-1)^k(x-k)^n}{k!(n-k)!} $

$ S(n,x) = \sum_{k=0}^{\lfloor x\rfloor}\frac{(-1)^k(n-x)(x-k)^{n-1}}{k!(n-k)!} = U(n-1,x)-U(n,x) $

$ \left(\sum_{n=\lceil x \rceil}^{\infty}S(n,x)\right) = 1 $

$ E(x) = \left(\sum_{n=\lceil x \rceil}^{\infty}S(n,x)\times n\right) = \left(\sum_{j=0}^{\left \lfloor x \right \rfloor} \frac{e^{x-j}(j-x)^j}{j!} \right) \approx 2x + \frac{2}{3} $

$ E(x) = \begin{cases} e^x & x \le 1 \\ 1 + \int_{0}^{1}E(x-u)\, du & x > 1 \end{cases} $

Discrete

$ D(m,h)=\sum_{k=0}^{\lceil \frac{h}{m+1}\rceil - 1}(-1)^k \binom{h-km-1}{k} \frac{(m+1)^{h-k(m+1)-1}}{m^{h-km-1}} $

$ D(m,h) = \begin{cases} 0 & h < 1 \\ 1 & h = 1 \\ D(m,h-1)+D(m,h-1)/m-D(m,h-m-1)/m = 1 + \frac{1}{m}\sum_{k=1}^{m}D(m,h-k)& h > 1 \end{cases} $

$ P(m,h,r) = \frac{\binom{h-1}{r-1}m-\binom{h-1}{r}}{m^r} $aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa