Does anyone know a function that can describe a harmonic series? I want to find a function that satisfies the following functional equation:
$F(z+1)=1/z+F(z)$
This is a generalization of harmonic series 1 + 1/2 + 1/3 + 1/4 + ..., and is similar to the gamma function which satisfies the relation $F(z+1)=zF(z)$.
I searched the internet. However, I didn't find such a function. Could anyone tell me something about this function?

In Mathematica, such a recursion may be solved with
RSolve[{f[z + 1] == f[z] + 1/z, f[1] == 1}, f[z], z]

where I assumed an initial condition of $f[1]=1$. The solution returned is
f[z] -> 1 + EulerGamma + PolyGamma[0, z]

