which is the ordinary hypergeometric equation. Provided none of λ1, λ2, λ − μ be zero or integral about x = 0, it has the solutions
F(λ, μ, 1 − λ1, x), xλ1 F(λ + λ1, μ + λ1, 1 + λ1, x);
about x = 1 it has the solutions
F(λ, μ, 1 − λ2, 1 − x), (1 − x)λ2 F(λ + λ2, μ + λ2, 1 + λ2, 1 − x),
where λ + μ + λ1 + λ2 = 1; about x = ∞ it has the solutions
x−λ F(λ, λ + λ1, λ − μ + 1, x−1), x−μ F(μ, μ + λ1, μ − λ + 1, x−1),
where F(α, β, γ, x) is the series
| 1 + | αβx | + | α(α + 1)β(β + 1)x² | ..., |
| γ | 1·2·γ(γ + 1) |
which converges when |x| < 1, whatever α, β, γ may be, converges for all values of x for which |x| = 1 provided the real part of γ − α − β < 0 algebraically, and converges for all these values except x = 1 provided the real part of γ − α − β > −1 algebraically.
In accordance with our general theory, logarithms are to be expected in the solution when one of λ1, λ2, λ − μ is zero or integral. Indeed when λ1 is a negative integer, not zero, the second solution about x = 0 would contain vanishing factors in the denominators of its coefficients; in case λ or μ be one of the positive integers 1, 2, ... (−λ1), vanishing factors occur also in the numerators; and then, in fact, the second solution about x = 0 becomes xλ1 times an integral polynomial of degree (−λ1) − λ or of degree (−λ1) − μ. But when λ1 is a negative integer including zero, and neither λ nor μ is one of the positive integers 1, 2 ... (−λ1), the second solution about x = 0 involves a term having the factor log x. When λ1 is a positive integer, not zero, the second solution about x = 0 persists as a solution, in accordance with the order of arrangement of the roots of the index equation in our theory; the first solution is then replaced by an integral polynomial of degree -λ or −μ1, when λ or μ is one of the negative integers 0, −1, −2, ..., 1 − λ1, but otherwise contains a logarithm. Similarly for the solutions about x = 1 or x = ∞; it will be seen below how the results are deducible from those for x = 0.