Zonal Harmonic
A zonal harmonic is a spherical harmonic of the form 
, i.e., one which reduces to a Legendre polynomial (Whittaker and Watson 1990, p. 302). These harmonics are termed "zonal" since the curves on a unit sphere (with center at the origin) on which 
vanishes are 
parallels of latitude which divide the surface into zones (Whittaker and Watson 1990, p. 392).
Resolving 
into factors linear in 
, multiplied by 
when 
is odd, then replacing 
by 
allows the zonal harmonic 
to be expressed as a product of factors linear in 
, 
, and 
, with the product multiplied by 
when 
is odd (Whittaker and Watson 1990, p. 1990).
REFERENCES:
Byerly, W. E. "Zonal Harmonics." Ch. 5 in An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, pp. 144-194, 1959.
Hashiguchi, H. and Niki, N. "Algebraic Algorithm for Calculating Coefficients of Zonal Polynomials of Order Three." J. Japan. Soc. Comput. Statist. 10, 41-46, 1997.
Kowata, A. and Wada, R. "Zonal Polynomials on the Space of 
Positive Definite Symmetric Matrices." Hiroshima Math. J. 22, 433-443, 1992.
Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.
