  This paper is a plea for diagonals and telescopers of rational, or algebraic,
functions using creative telescoping, in a computer algebra experimental
mathematics learn-by-examples approach. We show that diagonals of rational
functions (and this is also the case with diagonals of algebraic functions) are
left invariant when one performs an infinite set of birational transformations
on the rational functions. These invariance results generalize to telescopers.
We cast light on the almost systematic property of homomorphism to their
adjoint of the telescopers of rational, or algebraic, functions. We shed some
light on the reason why the telescopers, annihilating the diagonals of rational
functions of the form P/Q^k and 1/Q, are homomorphic. For telescopers with
solutions (periods) corresponding to integration over non-vanishing cycles, we
have a slight generalization of this result. We introduce some challenging
examples of generalization of diagonals of rational functions, like diagonals
of transcendental functions, yielding simple $_2F_1$ hypergeometric functions
associated with elliptic curves, or (differentially algebraic) lambda-extension
of correlation of the Ising model.
