Convergence of ray sequences of Pade approximants to 2F1(a,1;c;z), c>a>0

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The Padé table of ₂F₁(a,1;c;z) is normal for c>a>0 (cf. 3). For m ≥ n-1 and c ∉ -, the denominator polynomial Qmn(z) in the [m/n] Padé approximant Pmn(z)/Qmn(z) for ₂F₁(a,1;c;z) and the remainder term Qmn(z)₂F₁(a,1;c;z)-Pmn(z) were explicitly evaluated by Padé (cf. 2, 5 or 7). We show that for c>a>0 and m≥ n-1, the poles of Pmn(z)/Qmn(z) lie on the cut (1,∞). We deduce that the sequence of approximants Pmn(z)/Qmn(z) converges to ₂F₁(a,1;c;z) as m → ∞, n/m → ρ with 0<ρ≤ 1, uniformly on compact subsets of the unit disc |z|<1 for c>a>0