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29 Lamé FunctionsLamé Polynomials

§29.12 Definitions

Contents

§29.12(i) Elliptic-Function Form

Throughout §§29.1229.16 the order ν in the differential equation (29.2.1) is assumed to be a nonnegative integer.

The Lamé functions Ecνm(z,k2), m=0,1,,ν, and Esνm(z,k2), m=1,2,,ν, are called the Lamé polynomials. There are eight types of Lamé polynomials, defined as follows:

29.12.1 uE2nm(z,k2) =Ec2n2m(z,k2),
29.12.2 sE2n+1m(z,k2) =Ec2n+12m+1(z,k2),
29.12.3 cE2n+1m(z,k2) =Es2n+12m+1(z,k2),
29.12.4 dE2n+1m(z,k2) =Ec2n+12m(z,k2),
29.12.5 scE2n+2m(z,k2) =Es2n+22m+2(z,k2),
29.12.6 sdE2n+2m(z,k2) =Ec2n+22m+1(z,k2),
29.12.7 cdE2n+2m(z,k2) =Es2n+22m+1(z,k2),
29.12.8 scdE2n+3m(z,k2) =Es2n+32m+2(z,k2),

where n=0,1,2,, m=0,1,2,,n. These functions are polynomials in sn(z,k), cn(z,k), and dn(z,k). In consequence they are doubly-periodic meromorphic functions of z.

The superscript m on the left-hand sides of (29.12.1)–(29.12.8) agrees with the number of z-zeros of each Lamé polynomial in the interval (0,K), while n-m is the number of z-zeros in the open line segment from K to K+iK.

The prefixes u, s, c, d, sc, sd, cd, scd indicate the type of the polynomial form of the Lamé polynomial; compare the 3rd and 4th columns in Table 29.12.1. In the fourth column the variable z and modulus k of the Jacobian elliptic functions have been suppressed, and P(sn2) denotes a polynomial of degree n in sn2(z,k) (different for each type). For the determination of the coefficients of the P’s see §29.15(ii).

Table 29.12.1: Lamé polynomials.
ν
eigenvalue
h
eigenfunction
w(z)
polynomial
form
real
period
imag.
period
parity of
w(z)
parity of
w(z-K)
parity of
w(z-K-iK)
2n aν2m(k2) uEνm(z,k2) P(sn2) 2K 2iK even even even
2n+1 aν2m+1(k2) sEνm(z,k2) snP(sn2) 4K 2iK odd even even
2n+1 bν2m+1(k2) cEνm(z,k2) cnP(sn2) 4K 4iK even odd even
2n+1 aν2m(k2) dEνm(z,k2) dnP(sn2) 2K 4iK even even odd
2n+2 bν2m+2(k2) scEνm(z,k2) sncnP(sn2) 2K 4iK odd odd even
2n+2 aν2m+1(k2) sdEνm(z,k2) sndnP(sn2) 4K 4iK odd even odd
2n+2 bν2m+1(k2) cdEνm(z,k2) cndnP(sn2) 4K 2iK even odd odd
2n+3 bν2m+2(k2) scdEνm(z,k2) sncndnP(sn2) 2K 2iK odd odd odd

§29.12(ii) Algebraic Form

With the substitution ξ=sn2(z,k) every Lamé polynomial in Table 29.12.1 can be written in the form

29.12.9 ξρ(ξ-1)σ(ξ-k-2)τP(ξ),

where ρ, σ, τ are either 0 or 12. The polynomial P(ξ) is of degree n and has m zeros (all simple) in (0,1) and n-m zeros (all simple) in (1,k-2). The functions (29.12.9) satisfy (29.2.2).

§29.12(iii) Zeros

Let ξ1,ξ2,,ξn denote the zeros of the polynomial P in (29.12.9) arranged according to

29.12.10 0<ξ1<<ξm<1<ξm+1<<ξn<k-2.

Then the function

29.12.11 g(t1,t2,,tn)=(p=1ntpρ+14|tp-1|σ+14(k-2-tp)τ+14)q<r(tr-tq),

defined for (t1,t2,,tn) with

29.12.12 0t1tm1tm+1tnk-2,

attains its absolute maximum iff tj=ξj, j=1,2,,n. Moreover,

29.12.13 ρ+14ξp+σ+14ξp-1+τ+14ξp-k-2+q=1qpn1ξp-ξq=0,
p=1,2,,n.

This result admits the following electrostatic interpretation: Given three point masses fixed at t=0, t=1, and t=k-2 with positive charges ρ+14, σ+14, and τ+14, respectively, and n movable point masses at t1,t2,,tn arranged according to (29.12.12) with unit positive charges, the equilibrium position is attained when tj=ξj for j=1,2,,n.