A Novel Approach to π Approximation UsingGeneralized Lambert-W Series: ConvergenceAnalysis and Complex Function Connections

We investigate a novel series for approximating π, defined as §(α, β) =∑∞n=1 1nαW(n)β , where W(n) is the principal branch of the Lambert-W func-tion. Numerical computations reveal that §(α, β) can approximate π with acorrection factor C = π§(α,β) , with optimal parameters α ≈ 1.7609582221430886,β ≈ 1.544225750018456. We analyze the series’ convergence behavior, ob-serving that partial sums for different (α, β) pairs exhibit near-parallel tra-jectories on a logarithmic scale, attributed to similar tail decay rates gov-erned by 1nα(ln n)β . Complex analysis techniques, including asymptotic expan-sions and integral approximations, are employed to explore connections tothe Riemann zeta function and polylogarithms. We propose conjectures fora closed-form expression and discuss an unsolved problem of acceleratingconvergence or deriving an exact form. Numerical results, validated to 60-digit precision, and theoretical insights suggest a deep relationship between§(α, β) and special functions, offering a new perspective on π approxima-tio