A Degenerate Clausenian 3F2(1) Summation and Its Arithmetic Specialization

Clausen’s series 3F2(a,b,c;d,e;1), a generalized hypergeometric series with three numerator and two denominator parameters taken at unit argument, has classical closed forms when its denominator parameters exceed two of its numerator parameters by integers. These closed forms contain gamma or beta factors that become infinite when the remaining numerator parameter a equals 1, so that the sum at a=1 cannot be read off from them, although the series converges there. We give an explicit finite formula for 3F2(1,b,c;b+1+m,c+1+r;1), where m and r are non-negative integers, as a combination of digamma values with coefficients that are rational functions of b and c. For the integer values of b−c at which this formula breaks down we prove a companion formula containing trigamma values. Together the formulas cover the non-terminating domain b,c∈C∖Z≤0. We also show that the same expression is the limit, as a→1, of the formula of Shpot and Srivastava. As an application, we study a two-parameter family of such series indexed by integers n≥2 and p≥1. Each member of the family equals a rational number plus a non-zero rational multiple of πcot(π/n); hence, its sum is rational for n=2 and transcendental for every n≥3. Explicit formulas are given for p=1,2,3, the limit n→∞ is determined, and all results are confirmed by high-precision computation.