A diagonal operation on power series Given a formal power series $f(x,y)=\sum_{n,m\ge 0}f(n,m)\: x^n y^m$ in two variables $x$ and $y$ over some field of characteristic zero, e.g. the field of complex numbers $\mathbb C$, we define a new formal power series in one variable $t$:
$\Delta(f)(t):=\sum_{i\ge 0} f(i,i) \:t^i$.
It is known that if $f(x,y)$ is rational $\Delta(f)(t)$ is in general not rational (I think it is algebraic though).
Example: $f(x,y)=\frac{1}{1-x-y}$ leads to $\Delta(f)(t)=(1-4t)^{-1/2}$.
The question is: Is there a constructive way (i.e. algorithm or explicit formula) to calculate 
$\Delta(f)(t)$ for rational (or algebraic) $f(x,y)$ ?
 A: There is a discussion of taking diagonals of rational functions in Stanley's Enumerative Combinatorics Volume II (section 6.3). It contains a reasonably explicit description of how to take diagonals using Puiseux series as well as a description of how to do it using residues. 
A: The reference to Stanley's book given by Qiaochu Yuan provides indeed an enlightening exposition on diagonals of bivariate rational functions. Note that the method using a residue mentioned at the end of example 6.3.4 is better suited for computations than the one using Puiseux series (Theorem 6.3.3).
This being said, it is hard (impossible?) to find a reference with a complete algorithm for the computation of algebraic equations for bivariate diagonals in characteristic 0. This is one of the motivations for a recent paper of Alin Bostan, Bruno Salvy and myself on that subject: Algebraic Diagonals and Walks: Algorithms, Bounds, Complexity. 
H.-C. Herbig's request is met there by Algorithm 3 and formula (4).
A: Yes, there is a constructive way for both rational and algebraic $f.$ You should check out the very nice paper by Adamszewski and Bell, and references there in -- the formula for rational functions is given by Deligne (ref [13] in the cited paper), the result for algebraic functions appears very deep (see papers by Andre and Christol cited in the reference), but anyway, just read the introduction to the paper.
