Jacobi Function of the Second Kind

$$Q_n^{(\alpha,\beta)}(x)=2^{-n-1}(x-1)^{-\alpha}(x+1)^{-\beta}\int_{-1}^1 (1-t)^{n+\alpha}(1+t)^{n+\beta}(x-t)^{-n-1}\,dt.$$ (1)

In the exceptional case $n=0$, $\alpha+\beta+1=0$, a nonconstant solution is given by

$$Q^{(\alpha)}(x)=\ln(x+1)+\pi^{-1}\sin(\pi\alpha)(x-1)^{-\alp... ...a}\int_{-1}^1 {(1-t)^\alpha(1+t)^\beta\over x-t} \ln(1+t)\,dt.$$ (2)

References

Szegö, G. ``Jacobi Polynomials.'' Ch. 4 in Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 73-79, 1975.