We introduce a shifted version of the binomial theorem, and use it to study some remarkable trigonometric integrals and their explicit rewriting in terms of binomial multiple sums. Motivated by the expressions of area generating functions arising in the counting of closed walks on various lattices, we propose similar sums involving fractional values of the area and show that they are closely related to their integer counterparts and lead to rational sequences converging to powers of $\pi$. Our… Expand

Integer compositions with certain colored parts were introduced by Andrews in 2007 to address a number-theoretic problem. Integer compositions allowing zero as some parts were introduced by Ouvry and… Expand

Journal of Physics A: Mathematical and Theoretical

2019

TLDR

A formula for the enumeration of closed lattice random walks of length n enclosing a given algebraic area is proposed, contained in the Kreft coefficients which encode the Hofstadter secular equation for a quantum particle hopping on a lattice coupled to a perpendicular magnetic field.Expand

An effective single-band Hamiltonian representing a crystal electron in a uniform magnetic field is constructed from the tight-binding form of a Bloch band by replacing… Expand