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

This paper introduces a family of functions α_n(z) defined by the recursion α_{n-1}(z) = α_n(z)/α_n(z+1) with α_0(z) = 1/z. Closed forms are derived for negative indices. From these functions, we define Lambda functions Λ_k = Σ_{n=1}^∞ (1/α_{-k}(n) - 1) and obtain exact expressions for Λ_2 = 1 - π²/6, Λ_3 = 25/16 - π²/8 - ζ(3)/2, and Λ_5 involving powers of π up to π¹⁰ and odd zeta values. Numerical values are provided for Λ_4 and Λ_6. Several infinite product conjectures are presented, including ∏ α_{-k}(1.5)/α_{-k}(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.