68 
Proceedings of the Royal Society of Edinburgh. [Sess. 
and it may be noted that M 0 (z) is nothing else than the well-known 
logarithmic-integral function, li(z). 
§ 6. Hermite’s Polynomials ; or the Parabolic-cylinder Function 
when n is an Integer. 
The parabolic-cylinder function,* when n is an integer, degenerates 
into the product of an exponential and a polynomial : 
_z 2 
P >n(z) = e 4 U n (z). 
U n is Hermite’s polynomial, which satisfies the equation 
U n ~ zUn + n\J n = 0. 
A second solution is given by 
V»W = u»W 
• z t . 2 
e 2 dt 
- [Un(t)f 
We shall consider only the case when n is an even number. Then, 
U»(o) = l, B w (o) = 0, and it may be proved that 
n(n - l)U n _ 2 (3)B w (z) - U n (z)B n - 2 (z) -z = 0. 
Now if we proceed with the general method, we shall have to consider 
the decomposition of the rational factor bvd ; and a constant term must 
U n (z) 
be introduced. In that case, we are led to write 
A n + B n zB n — „ 
+ , 
XL 
U. 
where 
r=oo 
The recurrence-formula between B and U may be used to prove that 
1 
F n — . j 
n ! 
and finally we write 
22 
V»(z)= - e 2 B n (,) + ^U n (z)V 0 (z) 
A second solution of the differential equation of the parabolic cylinder 
when n is an even number can then be written 
A n (z) = - B n (z)eA + Id„(2)A 0 (3). 
* Whittaker, Proc. Lond. Math. Soc., xxxv ; Arch. Millie, Proc. Edin. Math. Soc., xxxii. 
