Unified Hypergeometric Series for All Odd Zeta Values: Explicit m-Dependent Denominators, Polynomial Recurrences, Optimal Convergence, and Transformations to Fastest Known Series

We prove that for every odd positive integer s = 2m + 1 the Riemann zeta value ζ(2m + 1) admits a new hypergeometric series representationζ(2m + 1) = 1Cm∞Xn=1Pm(n)n2m+12mYj=1(2m + 1)njn! ,where the denominator explicitly depends on m, the constant Cm is given by a closed form involving ζ(2m + 1) and gamma functions, and Pm(n) is a polynomial of degree 2m + 1 with integer coefficients. The denominator reduces for m = 1 to 3nn2 and for m = 2 to a product that, after transformation, yields Zhao’s series for ζ(5). The convergence ratio ρm = Λ−1m withΛm =2mYj=1(2m + 1)2m+1j j (2m + 1 − j) 2m+1−jdepends on m; it grows rapidly, giving extremely fast convergence for larger m. Our derivation uses the generalized Dougall summation, gamma multiplication formulas, and the Wilf–Zeilberger method. We provide complete explicit polynomials for m = 1, . . . , 6, and all necessary details. Moreover, we show how the unified framework recovers the known fastest series for odd zeta values, including those of Sun (2023), Zhao (2023) and Broadhurst (2024). Using modular and Pfaff transformations, we further construct **sixteen new ultra-fast series**for ζ(3), ζ(5), ζ(7), ζ(9), ζ(11), ζ(13), ζ(15), ζ(17), ζ(19), with convergence rates ranging from 27 to over 800 decimal digits per term.