Factorial in power series; intuitive/combinatorial interpretation?

Question

It is well known that the terms of the power series of exponential and trigonometric functions often involve the factorial function, essentially as a consequence of iterating the power rule.

My question is whether there is another way to view the occurrence of factorials so often; factorials are often involved in counting/combinatorics, so is there a combinatorial interpretation of why this happens, or some other interesting interpretation?

More generally, is there any way I can look at the power series of $e^x$ or some other function and intuitively understand why factorials are involved, rather than just thinking of the power rule and derivatives?

Answer 1

We have $\sin^2t+\cos^2t=1$ and $\cosh^2t-\sinh^2t=1$, both of which are born of the implicit algebraic equations of the circle and hyperbola, $x^2\pm y^2=r^2$. In other words, these two geometric shapes, studied since the time of the ancient Greeks, are defined by constant or bounded sums of powers. And where are factorials known to occur ? In the expression of binomial coefficients, which famously characterize the binomial theorem, which expands the power of a sum into a sum of powers. So the intrinsic umbilical link between trigonometric or hyperbolic functions and factorials or binomial coefficients is as natural and intuitive as can be. Thus, it should come as no surprise that the beta and $\Gamma$ functions of argument $1/n$ are inherently connected to geometric figures described by equations of the form $x^n+y^n=r^n$, yielding, for instance, the celebrated identities $\Gamma\bigg(\dfrac12\bigg)=\sqrt\pi$ and $B\bigg(\dfrac12~,~\dfrac12\bigg)=\pi$.

Answer 2

To take a different starting point, you could think about the formula:

$$ e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n $$

which falls out from thoughts of compound interest compounded infinitely often.

Answer 3

Look at chapter 2 of "generatingfunctionology", available here: https://www.math.upenn.edu/~wilf/DownldGF.html

There is a discussion of ordinary generating functions and exponential generating functions, the ones with factorials.

Wilf says "We remarked earlier that the convolution of sequences that is shown in (2.3.4) is useful in counting problems where structures of size n are obtained by stitching together structures of sizes r and n − r in all possible ways. Correspondingly, the convolution (2.3.3) is useful in combinatorial situations where we not only stitch together two such structures, but we also relabel the structures. For then, roughly speaking, there are $\binom{n}{r}$ ways to choose the new labels of the elements of the structure of size r, as well as $a_r$ ways to choose that structure and $b_{n−r}$ ways to choose the other one."

Answer 4

Many classes of functions can be developed into a function series. There are different kinds of function series possible. Each analytic function e.g. can locally be given by a convergent power series. A simple form of such a power series is the Taylor series. A Taylor series has some nice properties.

As you've already noted, the denominator of the Taylor coefficient is already a factorial. This comes from the $n$-th derivative of the summand term $x^{n}$ and is simply a result of the properties of the Taylor series. And it is a result of the power rule of differentiation. Higher product rule and higher power rule contain partitions of derivatives of lower degree. This is the combinatorial nature of the factorial in Taylor series and in derivatives.

The numerator of the $n$-th Taylor coefficient, the $n$-th local derivative, can also be interpreted combinatorially. It can be described as Bell polynomial. That is a partition polynomial. It contains all partitions of all derivatives of lower degree, and its coefficients contain the number of integer partitions and the number of set partitions; they are called multinomial coefficients of the third kind. Therefore this coefficients contain factorials in the numerator and in the denominator. Speaking pictorially, partial derivatives and higher derivatives are integer partitions. And this partition structure is maintained in exponential generating functions and in ordinary generating functions.

Factorials are the building blocks of the frequency numbers of combinatorial objects, i.e. of combinatorial numbers. They form binomials, multinomials and other combinatorial numbers which express the frequency of combinatorial structures like combinations, permutations, integer partitions, set partitions and combinatorial objects which are assembled from this basic structures.

The combinatorial interpretation and the interpretation by functions are often two pictures of the same thing. Both are interrelated by generating functions.