Alpha Functions: A New Hierarchy with Connections to Elliptic Integrals and Mock Theta Functions

This paper introduces a family of functions αₙ (z) defined by the recursion α₍-₁ (z) = αₙ (z) /αₙ (z+1) with α₀ (z) = 1/z. Closed forms are derived for negative indices. From these functions, we define Lambda functions Λₖ = Σ₍=₁^∞ (1/α-₊ (n) - 1) and obtain exact expressions for Λ₂ = 1 - π²/6, Λ₃ = 25/16 - π²/8 - ζ (3) /2, and Λ₅ involving powers of π up to π¹⁰ and odd zeta values. Numerical values are provided for Λ₄ and Λ₆. Several infinite product conjectures are presented, including ∏ α-₊ (1. 5) /α-₊ (1) = 4/π² and related constants. Connections to mock theta functions and generalized elliptic integrals are explored, with numerical evidence supporting all conjectures. Convergence rates are analyzed, showing that for large k the series become effectively exact within few terms.