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
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 $B\bigg(\dfrac12~,~\dfrac12\bigg)=\pi$.
No comments:
Post a Comment