Series and sequences in physical systems & closed form expressions

Question

I gave a colloquium a while ago about physics inspiring recent developments in mathematics and as is almost borderline cliche in such talks, I mentioned the Fibonacci sequence with closed form expression, $$ F_n = \frac{\varphi^n - (-\varphi)^{-n}}{2 \varphi -1}, \ \varphi = \frac{1+\sqrt 5}{2} $$ and how it shows up in many natural settings because the Fibonacci sequence is the closest integral approximation to the logarithmic spiral series.

Here's my (possibly naive) question: Are there any other series/sequences with closed form expressions which manifest plainly and obviously in nature?

Answer 1

The Casimir effect is a manifestation of $$1+2^3+3^3+\cdots=-\frac{1}{120}.$$

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The vacuum energy $E$ in the space between two metal plates, separated by a distance $a$ equals $$E = \frac{ \hbar c \pi^2 }{6a^3}\sum_{n=1}^\infty n^3.$$ This divergent sum is regularized by analytic continuation of the Riemann zeta function, to give $$1+2^s+3^s+\cdots=\zeta(-s)=-\frac{B_{s+1}}{s+1},$$ with $B_s$ a Bernoulli number, hence $\sum_{n=1}^\infty n^3=-\frac{1}{120}.$

The resulting attractive force $-dE/da$ between the metal plates, the Casimir effect, has been demonstrated in experiments, providing one justification for the zeta-function regularization of the divergent sum.

_{In the physics context, what is going on is that the unobservable vacuum energy $E$ is infinite, while the observable force $-dE/da$ is finite. One way to obtain this finite answer is to add a third metal plate, at a distance $L$ from the two plates at separation $a$. The third plate introduces a cutoff in the infinite sum, and the limit $\lim_{L\rightarrow\infty} dE/da$ gives the same finite answer as the zeta-function regularization.}

Answer 2

The golden ratio is achieved by any sequences of numbers generated by an arbitrary initial seed of two nonzero natural numbers by recursively adding consecutive numbers together, so it is a limiting property of a quite general class of sequences.

The Feigenbaum constants in chaos theory have a similar nature.

Not so simply, the following sequences have important applications in mathematical physics.

The Riemann zeta function evaluated at natural numbers has several physical interpretations. See the responses to the Geometric, physical, probabilistic interpretations of the Riemann Zeta function for zeta(n > 1).

Scattering amplitudes in certain quantum field theories are related to the combinatorics of the associahedra. See references for OEIS A133437. (The generating functions for Fuss-Catalan numbers and their compositional inverses are particular cases.)

The soliton wave solution to the KdV equation for shallow water waves, which also crops up in string theory, is related to the Eulerian numbers. See the MO-Q "Why is there a connection between enumerative geometry and nonlinear waves?"

The coefficients of the Lagurre and Hermite sequences of polynomials have numerous combinatorial interpretations and are related to probabilty wave functions in quantum mechanics connected to orbitals and the harmonic oscillator.

The combinatorics of phylogenetic trees, used in modelling bifurcations in evolution of biological structures, are related to the Ward numbers. (There are other sequences in the OEIS related to genomics and structures of chemicals.)

Compositional inverse pairs of functions or formal series are related to flow equations characterized by autonomous ODEs, and the relations among series reps of these pairs can be couched in terms of the Euler polynomials (with integer coefficients) of the associahedra, the refined Ward numbers A134685, the refined Eulerian numbers A145271, or the noncrossing partitions A134264.

Answer 3

Classical, Fermi-Dirac, and Bose-Einstein statistics involve integrals that can be shown to be equivalent to polylogarithms. Polylogs have a particular simple series form:

$$ \text{Li}_s(x)=\sum_{n=1}^\infty \frac{x^n}{n^s} $$ For $x=\pm1$ and positive integer $s$ these equate to frequently-used constants. However, in many physics problems, $s$ is half-integer, and $x$ is large and negative. For $x$ large and negative, you can't use the series anymore, but an asymptotic formula or analytic continuation will work for calculation.