If thedomain of the unknown function is supposed to be thenatural numbers, the function is generally viewed as asequence, and, in this case, a functional equation (in the narrower meaning) is called arecurrence relation. Thus the termfunctional equation is used mainly forreal functions andcomplex functions. Moreover asmoothness condition is often assumed for the solutions, since without such a condition, most functional equations have highly irregular solutions. For example, thegamma function is a function that satisfies the functional equation and the initial value There are many functions that satisfy these conditions, but the gamma function is the unique one that ismeromorphic in the whole complex plane, andlogarithmically convex forx real and positive (Bohr–Mollerup theorem).
Recurrence relations can be seen as functional equations in functions over the integers or natural numbers, in which the differences between terms' indexes can be seen as an application of theshift operator. For example, the recurrence relation defining theFibonacci numbers,, where and
Thecommutative andassociative laws are functional equations. In its familiar form, the associative law is expressed by writing thebinary operation ininfix notation, but if we writef(a,b) instead ofa ○b then the associative law looks more like a conventional functional equation,
The functional equation wherea,b,c,d areintegers satisfying, i.e. = 1, definesf to be amodular form of orderk.
One feature that all of the examples listed above have in common is that, in each case, two or more known functions (sometimes multiplication by a constant, sometimes addition of two variables, sometimes theidentity function) are inside the argument of the unknown functions to be solved for.
When it comes to asking forall solutions, it may be the case that conditions frommathematical analysis should be applied; for example, in the case of theCauchy equation mentioned above, the solutions that arecontinuous functions are the 'reasonable' ones, while other solutions that are not likely to have practical application can be constructed (by using aHamel basis for thereal numbers asvector space over therational numbers). TheBohr–Mollerup theorem is another well-known example.
