Euler"s integral for the beta function $B(s,\alpha) = $ (with $x = 1$)
$$ \frac{(s-1)!(\alpha-1)!}{(s+\alpha-1)!} x^{s+\alpha-1} = \int_0^\infty t^{s-1}\; H(x-t) \; (x-t)^{\alpha-1} dt = \int_0^x t^{s-1} \; (x-t)^{\alpha-1} \; dt $$
is one of those iconic equations at the intersection of several domains of physics and mathematical analysis that serve as leaping boards to far-reaching generalizations (see, e.g., Beta integrals by Warnaar and The importance of the Selberg integral by Forrester and Warnaar).
What salient connections to physics, geometry, and geometric probability theory does the Euler beta integral have?
Some associations:
I) Fractional / operational calculus: It can be viewed as a Mellin transform or a Laplace transform convolution integral, or morph into the core Riemann-Liouville fractional integroderivative of fractional calculus analytically continued (e.g., by the Hadamard finite part regularization, cf. "Fractional calculus and interpolation of generalized binomial coefficients")
$$ D_x^{-s} x^{\alpha}= \int_0^\infty \frac{t^{s-1}}{(s-1)!} \; H(x-t) \; (x-t)^\alpha dt = \int_0^x \frac{t^{s-1}}{(s-1)!} \; (x-t)^\alpha \; dt = \frac{\alpha!}{(\alpha+s)!} x^{s+\alpha} \; . $$
II) String theory: The connection between the Veneziano amplitude of nascent string theory and the beta function is well-known. For a very nice overview, see "The Euler Legacy to Modern Physics" by Dattoli.
II) Schwarz triangle, Ehrhart polynimials, and polytopes: Kholodenko in New strings for old Veneziano amplitudes: I Analytical treatment and New models for Veneziano amplitudes: Combinatorial, symplectic, and supersymmetric aspects reviews the string theory connections also and links to a geometric interpretation as lengths of sides of a Schwarz triangle and to Ehrhart polynomials encoding a relation between number of interior points of convex lattice polytopes and the number of their boundary points. See also the Physics Overflow question Relation of Betti numbers to Veneziano's scattering amplitude?.
The relation to the Schwarz triangle is nicely presented in Gauss' hypergeometric function by Beukers.
III Pochhammer contour integration over Riemann surfaces: The integral formulated as a Pochhammer contour integral over Riemann surfaces is beautifully depicted in Exploring visualization methods for complex variables. [Edit 1/3/2021: See also a "A Contour Integral Representation for the Dual Five-Point Function and a Symmetry of the Genus Four Surface in R6" by Hanson and Sha on relations among associahedra, string theory, cross-ratios, the Pochhammer contour, generalized beta integrals, and projective spaces.]
IV: Diff op dualities, symmetric polynomials, and Appell sequences:
There is a dual operator ($D^{s}x^s$, the inverse of $x^{-s}D^{-s}$) and Mellin transform that illuminates the connection to generalizations of symmetric polynomials (and, therefore, the exterior algebra and simplices) by the Weierstrass factorization theorem for meromorphic functions, in this case the generalized rising and falling factorials:
$$ (x D_x x)^{-s} \; x^\alpha = \int_0^\infty \frac{t^{s-1}}{(s-1)!} \; \frac{x^\alpha}{(1+xt)^{\alpha+1}} \; dt = x^{\alpha-s} \frac{(\alpha-s)!}{\alpha!} = x^{-s} D_x ^{-s} x^{-s} \; x^\alpha \; $$
$$= x^{-s} \frac{(xD_x-s)!}{(xD_x)!} \; x^\alpha \; .$$
These factorials specialize to the gamma function and its reciprocal with $\alpha = 0$ and further, with $x = e^z$, to the e.g.f.s for the gamma characteristic classes or genuses, $\frac{e^{sz}}{s!}$ and $e^{sz}s!$ with the periods $\zeta(n>1)$ as coefficients (power functions) for the dual raising operators of the associated dual Appell polynomials (i.e., the Appell polynomials of the two e.g.f.s form an inverse pair under umbral composition). (Cf. Perturbative corrections to Kahler moduli spaces by Halverson, Jockers, Lapan, and Morrison for an application.) These sequences are naturally connected by the cycle index partition polynomials (CIPP, OEIS-A036039) for the symmetric group (Hall-Macdonald scheme and Newton identities). In fact, the operadic infinitesimal generators for the dual operators are these raising operators which are (with the coefficients treated as indeterminates) generators of the CIPP and its umbral compositional inverse.
There is also a connection to Koszul duality, reflecting that, for $z=0$, the e.g.f.s are a multiplicative inverse pair, and to the Kummer confluent hypergeometric functions (cf. this MO-Q).
