An Explicit Closed Form for a Half-Integer 3F2(1) Family with Negative Integral Parameter Differences, via Odd Harmonic Sums

We study the half-integer family F(n,p)=3F2−2n+12,1,2p−12;2n+52,2p+12;1. Although this family belongs to the Karlsson–Minton class of 3F2(1) series with negative integral parameter differences, the most directly relevant symmetric formula of Shpot and Srivastava becomes singular at the free numerator parameter a=1(here a denotes the first numerator parameter of the Shpot–Srivastava reduction formula, the value of which controls the Beta factors in their evaluation). We show that this apparent singularity is removable and identify its finite value with the independently derived closed form. First, Euler’s integral representation and a recurrence for Jm,n(x)=∫01v2m(1−xv2)−ndv yield a closed form for 2F1−2n+12,1;2n+52;x as a polynomial in (1−x) plus arctanh(x)/x. Rainville’s integral formula then reduces F(n,p) to a rational part and a finite rational linear combination of odd harmonic sums Hrodd=∑j=1r(2j−1)−1. For all integers n≥0 and p≥n+3, this proves F(n,p)∈Q. The threshold is sharp: at the adjacent boundary p=n+2 we obtain F(n,n+2)=Rn+Cnπ2 with Rn,Cn∈Q and Cn≠0, so rationality fails. We also prove, by analyticity and the independently derived formula, that the a→1 limit of the Shpot–Srivastava representation equals the present closed form. For computation, the exact assembly uses O(n+p) arithmetic operations under the unit-cost model, while a direct fixed-precision evaluation is ill-conditioned for large n. A stable series/closed-form hybrid for the 2F1 factor combined with adaptive positive-kernel Gauss–Jacobi quadrature gives relative errors at the machine-precision scale over the tested range up to n=64.