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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, if useless, fact: if you replace the factorial function with another function, you can create variants of many fundamental mathematical functions.

Example 1

Define

\[H(n) = 1^1 \cdot 2^2 \cdot 3^3 \cdots n^n.\]

This gives us a variant of the exponential function

\[e_H^x = \frac{1}{H(0)} + \frac{x}{H(1)} + \frac{x^2}{H(2)} + \frac{x^3}{H(3)} + \cdots\]

and the trigs as well:

\[\sin_H x = \frac{x}{H(1)} - \frac{x^3}{H(3)} + \frac{x^5}{H(5)} - \frac{x^7}{H(7)} + \cdots\]

\[\cos_H x = \frac{1}{H(2)} - \frac{x^2}{H(4)} + \frac{x^4}{H(6)} - \frac{x^6}{H(8)} + \cdots\]

The hyperfactorial trig functions have some pretty damn weird graphs, but that's the only interesting thing I see here.

Example 2

Suppose we define \(n!^*\) as \(-n!\) when \(n\) is 2 or 3 mod 4, and \(n!\) otherwise. (This is, like, \(n!^* = n! \sqrt{2} \sin ((2n + 1)\pi/4)\) or something.) Then:

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

and

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

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

That these are actually the hyperbolic sine and cosine! Our function has the effect of converting the circular functions into hyperbolic ones.

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