Harmonic Number for $H_{n+1}$ and $H_{n+\frac{1}{2}}$ According to the definition of harmonic number $H_n = \sum\limits_{k=1}^n\frac{1}{k}$.
How we can define $H_{n+1}$ and $H_{n+\frac{1}{2}}$? 

$$H_{n+1} = \sum_{k=1}^{n+1} \dfrac{1}{k} = H_n + \frac{1}{n+1}$$
$H_{n+1/2}$ is not defined by the formula for $H_n$, unless you want to just say it's the same as $H_n$.  However, one function  whose restriction to the positive integers is $H_n$ is $\psi(n+1)+\gamma$, where $\psi$ is the digamma function and $\gamma$ is the Euler-Mascheroni constant.  This would make
$$ H_{n+1/2} = \psi(n+3/2)+ \gamma$$
EDIT: Since $$\psi(n+3/2) = 2 \psi(2n+2) - \psi(n+1) - 2 \ln 2$$
you could also write
$$ H_{n+1/2} = 2 H_{2n+1} - H_n - 2 \ln 2 $$
