A shifted binomial theorem and trigonometric series

TL;DR

This paper introduces a shifted binomial theorem and applies it to analyze trigonometric integrals, revealing new binomial sums related to lattice walk areas and their connection to powers of pi, with potential implications in statistical mechanics.

Contribution

It presents a novel shifted binomial theorem and demonstrates its application to trigonometric integrals and lattice walk area sums, linking combinatorics with mathematical physics.

Findings

Derived explicit binomial sums for trigonometric integrals.

Established connections between fractional area sums and powers of pi.

Potential applications to models with higher spins or $SU(N)$ symmetries.

Abstract

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 results, other than their mathematical interest, could be relevant to generalizations of statistical mechanical models of the Heisenberg chain type involving higher spins or $SU(N)$ degrees of freedom.