SPECIAL FUNCTIONS
where, in the second equality, we have expanded the RHS using the binomial theorem. On
equating coefficients ofhn, we immediately obtain
Lmn(0) =(n+m)!
n!m!.
Recurrence relationsThe various recurrence relations satisfied by the associated Laguerre polynomials
may be derived by differentiating the generating function (18.123) with respect to
either or both ofxandh, or by differentiating with respect toxthe recurrence
relations obeyed by the ordinary Laguerre polynomials, discussed in section 18.7.1.
Of the many recurrence relations satisfied by the associated Laguerre polynomials,
two of the most useful are as follows:
(n+1)Lmn+1(x)=(2n+m+1−x)Lmn(x)−(n+m)Lmn− 1 (x), (18.124)x(Lmn)′(x)=nLmn(x)−(n+m)Lmn− 1 (x). (18.125)For proofs of these relations the reader is referred to exercise 18.7.
18.9 Hermite functionsHermite’s equation has the form
y′′− 2 xy′+2νy=0, (18.126)and has an essential singularity atx=∞. The parameterνis a given real
number, although it nearly always takes an integer value in physical applications.
The Hermite equation appears in the description of the wavefunction of the
harmonic oscillator. Any solution of (18.126) is called aHermite function.
Sincex= 0 is an ordinary point of the equation, we may find two linearlyindependent solutions in the form of a power series (see section 16.2):
y(x)=∑∞m=0amxm. (18.127)Substituting this series into (18.107) yields
∑∞m=0[(m+1)(m+2)am+2+2(ν−m)am]xm=0.Demanding that the coefficient of each power ofxvanishes, we obtain the
recurrence relation
am+2=−2(ν−m)
(m+1)(m+2)am.As mentioned above, in nearly all physical applications, the parameterνtakesinteger values. Therefore, ifν=n,wherenis a non-negative integer, we see that