Look at Ramanujan Summation, also called indefinite summation. Ramanujan extended The Euler Mclaurin summation method using Bernoulli numbers and a whole stack of ideas. This is a well studied subject. In general, if your $g$ is defined in the right half plane and satisfies for $z = x + iy$
$$|g(z)| < C e^{\rho|x| + \kappa|y|}$$

where $\kappa < \pi$ and $\rho$ is arbitrary, then a unique solution exists, where $f(0) = 0$ and $f(z+1) - f(z) = g(z)$. Ramanujan's formula is rather simple

$$f(z) = C + \int_0^zg(t)\,dt + \frac{1}{2}g(z) + \sum_{n=1}^\infty \dfrac{B_{2k}}{2k!}g^{(2k-1)}(z)$$

where $C$ is a constant making $f(0) =0$ and $B_{2k}$ are the bernoulli numbers. Since this is a linear operator on $g$ it is common to write

$$\sum_z g = f$$
$$\sum_{j=1}^z g(j) = f(z)$$
$$\sum_1^z g(p)\Delta p = f(z)$$

where each draws the similarity between it and the integral.

I can give you a formula for the more restrictive case $\kappa < \pi/2$, which is a modification of Ramanujan's method.

$$f(z) = \frac{1}{\Gamma(1-z)}\Big{(}\sum_{n=0}^\infty \sum_{j=1}^{n+1}g(j)\frac{(-1)^n}{n!(n+1-z)} + \int_1^\infty\big{(}\sum_{n=0}^\infty \sum_{j=1}^{n+1}g(j)\frac{(-w)^n}{n!}\big{)}w^{-z}\,dw\Big{)}$$

which follows by Ramanujan's master theorem. All in all the subject has four names: Indefinite summation, continuum sums, Ramanujan summation, Euler-Mclaurin summation. I can't remember the names of the books, but there are some books on the calculus of finite differences that deal with this subject.

The idea goes hand in hand with Newton series as well, where in such a case it is rather trivial to produce an indefinite sum if the function is expanded in a Newton series. I.e: if $$g(z) = \sum_{n=0}^\infty a_n\dbinom{z}{n}$$
In this case it's much easier to work with $g$ bounded in the right half plane, so it's more restrictive.

Also, if $g(z)$ has nice decay properties we can always take the wondrous classical equation

$$f(z) = \sum_{k=0}^\infty g(z-k) - g(-k)$$
or

$$f(z) = \sum_{k=0}^\infty g(z+k) - g(k)$$

which was Ramanujan's motivation, but these are more volatile.