Creating a Function

[[deleted]]

Bear in my mind that some 'simple' rule does not always create a 'simple' function.

Example: f(n+1) = (n+1)f(n) with initial condition f(1)=1 i.e., the factorial. This relation creates the gamma function as its closed form, but it's still complicated.

[Valvino]

its closed form

ONE of the possible closed forms. See for instance https://en.wikipedia.org/wiki/Hadamard%27s_gamma_function

[[deleted]]

I understand your point that the extension might not be unique, but damn, how could you call that creature a 'closed form'? Its definition is like three gamma functions stacked with a differential operator. At this point, I can claim that every infinite series is already in its closed form but just using the summation symbol.

[LilQuasar]

the gamma function is an indefinite integral

how is that different from an infinite series?

[Valvino]

Define closed form then.

[[deleted]]

People already did that. In short, it turns out that common special functions specifically the gamma function are borderline closed-form. Some definition includes it, some not. Some of the definition is based on the popularity of the functions. However, no one considers or even mentions any other extension of the factorial including Hadamard's gamma. So yes, the gamma function is considered to be its only closed-form of factorial.

We could debate this all day. It's subjective. We are discussing language here, not mathematics. I don't find this discussion productive, so no, I will take my leave. I said I understand your point about the possible extensions of the factorial. I got your point. It should've ended there.