Lamé's Differential Equation (Types)

Whittaker and Watson (1990, pp. 539-540) write Lamé's differential equation for Ellipsoidal Harmonics of the four types as

$$4\Delta(\theta) {d\over d\theta}\left[{F(\theta){d\Lambda(\theta)\over d\theta}}\right] = [2m(2m+1)\theta+C]\Lambda(\theta)$$ (1)

$$4\Delta(\theta) {d\over d\theta}\left[{F(\theta){d\Lambda(\theta)\over d\theta}}\right] = [(2m+1)(2m+2)\theta+C]\Lambda(\theta)$$ (2)

$$4\Delta(\theta) {d\over d\theta}\left[{F(\theta){d\Lambda(\theta)\over d\theta}}\right] = [(2m+2)(2m+3)\theta+C]\Lambda(\theta)$$ (3)

$$4\Delta(\theta) {d\over d\theta}\left[{F(\theta){d\Lambda(\theta)\over d\theta}}\right] = [(2m+3)(2m+4)\theta+C]\Lambda(\theta),$$ (4)

where

$$\Delta(\theta) \equiv \sqrt{(a^2+\theta)(b^2+\theta)(c^2+\theta)}$$ (5)

$$\Lambda(\theta) \equiv \prod_{q=1}^m(\theta-\theta_q).$$ (6)

References

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.