A New Family of Mathematical Constants Arising from a Family of Polynomials Defined by Recurrence Relations

Abstract This paper examines a family of polynomials defined by a recurrence relation. The initial polynomial is P₁⁽ᵇ⁾(t) = tᵇ − tᵇ⁻¹ − 1. Each subsequent polynomial is obtained from the previous one using the recurrence: Pₖ₊₁⁽ᵇ⁾(t) = t⁽ᵇ⁻¹⁾·ᵇᵏ · Pₖ⁽ᵇ⁾(t) − 1 At each step, the exponent of t increases according to a geometric progression: for base 2 this is doubling (1, 2, 4, 8, 16, …), for base 3 — tripling (1, 3, 9, 27, 81, …), and so on for any integer b ≥ 2. Thus, the length of the polynomial grows with each step by adding a new term whose exponent is determined by the recurrence. In explicit form, the polynomials are written as: Pₖ⁽ᵇ⁾(t) = tᵇᵏ − ∑ⱼ₌₀ᵏ⁻¹ tᵇᵏ⁻ᵇʲ − 1 For each integer b ≥ 2, the sequence of dominant positive roots rₖ(b) converges to a limit R(b) satisfying the equation: ∑ⱼ₌₀^∞ R(b)⁻ᵇʲ = 1 High-precision values of R(b) for b = 2…21 are obtained, along with the first 20 terms of the continued fraction expansion for each constant. All 20 sequences are absent from the OEIS database, confirming the novelty of the entire family. Super-exponential convergence of R(b) to the roots of the equation xᵇ − xᵇ⁻¹ − 1 = 0 is observed for b ≥ 5. The constants R(b) are related to lacunary series, Pisot numbers, and may be of interest in number theory, dynamical systems, and fractal geometry.