Condon-Shortley Phase

The $(-1)^m$ phase factor in some definitions of the Spherical Harmonics and associated Legendre Polynomials. Using the Condon-Shortley convention gives

$$Y_n^m(\theta,\phi) = (-1)^{m} \sqrt{{2n+1\over 4\pi} {(n-m)!\over (n+m)!}}\,P_n^m(\cos \theta)e^{im\phi}.$$

See also Legendre Polynomial, Spherical Harmonic

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 682 and 692, 1985.

Condon, E. U. and Shortley, G. The Theory of Atomic Spectra. Cambridge, England: Cambridge University Press, 1951.

Shore, B. W. and Menzel, D. H. Principles of Atomic Spectra. New York: Wiley, p. 158, 1968.