User blog:Vel!/Factorials

Consider the lowly factorial, the saddest excuse for a fast-growing function ever to exist:

\[n! = 1 \cdot 2 \cdot 3 \cdots n.\]

You can use the factorial to define the exponential function:

\[e^x = \frac{1}{0!} + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\]

and the trigonometric functions:

\[\sin x = \frac{x}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\]

\[\cos x = \frac{1}{0!} - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots\]

Factorials also form the basis for the binomial coefficients and many other basic mathematical concepts.

From this observation springs a curious fact: if you replace the factorial function with another function, you can create variants of many fundamental mathematical functions. For example, you can create hyperfactorial-based variants on \(e^x\), sine, cosine, and tangent. As you might expect, this fact is totally useless.

...okay, maybe not totally useless. q-theory uses this principle by using a q-factorial, and q-theory has many natural applications.

Of the many substitutes we can make, this one is perhaps the most interesting: define \(n!^*\) as \(-n!\) when \(n\) is 2 or 3 mod 4, and \(n!\) otherwise. (Starting at 0, this forms the sequence \(1, 1, -2, -6, 120, 720, -5040, -40320, \ldots\)) If we plug this function into the infinite series above, we find that \(\sin^* = \sinh\) and \(\cos^* = \cosh\). It has the effect of converting the circular functions into hyperbolic ones.