A Rigorous Proof of a Ramanujan Machine Identity for $-\pi/4$ via Exact Recurrence Solving

TL;DR

This paper provides a rigorous proof of a Ramanujan Machine conjecture for -π/4 using exact recurrence solving, deriving a closed-form for the denominator sequence and confirming the identity through analytical methods.

Contribution

It offers the first rigorous proof of a specific Ramanujan Machine identity for -π/4 by solving the associated difference equation and evaluating the limit analytically.

Findings

Derived a closed-form expression for the denominator sequence q_n.

Established absolute convergence of the continued fraction.

Confirmed the identity equals -π/4 through explicit evaluation.

Abstract

We prove a polynomial continued fraction identity for the constant $-\pi/4$, conjectured by the Ramanujan Machine project. The proof proceeds by explicitly solving the underlying second-order linear difference equation. We derive a closed-form expression for the denominator sequence, $q_n = (-1)^n (2n-3)!!\,(n^2+n-1)$, and establish absolute convergence via a Wronskian telescoping argument. The limiting value is reduced by Abel summation to a Beta-function integral, which is evaluated in closed form through an elementary substitution and a single integration by parts, yielding the exact value $-\pi/4$.