§28.31 Equations of Whittaker–Hill and Ince

§28.31(i) Whittaker–Hill Equation

Hill’s equation with three terms

28.31.1

and constant values of , and , is called the Equation of Whittaker–Hill. It has been discussed in detail by Arscott (1967) for , and by Urwin and Arscott (1970) for .

§28.31(ii) Equation of Ince; Ince Polynomials

Formal -periodic solutions can be constructed as Fourier series; compare §28.4:

where the coefficients satisfy

28.31.6
,
28.31.7
,
28.31.8
,
28.31.9
.

When is a nonnegative integer, the parameter can be chosen so that solutions of (28.31.3) are trigonometric polynomials, called Ince polynomials. They are denoted by

28.31.10
28.31.11

and in all cases.

The values of corresponding to , are denoted by , , respectively. They are real and distinct, and can be ordered so that and have precisely zeros, all simple, in . The normalization is given by

ambiguities in sign being resolved by requiring and to be continuous functions of and positive when .

If and in such a way that , then in the notation of §§28.2(v) and 28.2(vi)

28.31.15

For proofs and further information, including convergence of the series (28.31.4), (28.31.5), see Arscott (1967).

§28.31(iii) Paraboloidal Wave Functions

With (28.31.10) and (28.31.11),

28.31.16
28.31.17

are called paraboloidal wave functions. They satisfy the differential equation

with , , respectively.

For ,

More important are the double orthogonality relations for or or both, given by

and

and also for all , given by

where when , and when .

For proofs and further integral equations see Urwin (1964, 1965).

¶ Asymptotic Behavior

For , the functions , behave asymptotically as multiples of as . All other periodic solutions behave as multiples of .

For , the functions , behave asymptotically as multiples of as . All other periodic solutions behave as multiples of .