1246 
fr m(#)Pr(a)da = 0 (m == n) 
oe PS GES ys A 
ferences = 1 
0 
In the second place ¢,(x) satisfies the differential equation 
Ln (2) oa (L—a)ypn'(x) + nip;(x) — 0 
which also may be written 
d * 
=, lwe*gn'(a)] + ne-tpa(a) = 0. … … … + 0) 
In the third place we have the following properties, which may 
be easily obtained 
Pale) = Gn (@)—P ati)... . 2 ee 
x 
Pull Pal Gale): ee Oe 
CI) r4ae)—(2n +1 —a) pale) + nga) =O... (6) 
[oeerarn (oye = (— 1)" Cnn lim <n) sve oS ee 
a == (m>>n) 
3. If the expansion (1) is possible, the coefficients a, may be 
expressed by means of the equations (2) 
an = [ryote 
0 
With these values the second member of (1) reduces to 
S= > pula) ELO COR 
0 
In order to determine this sum we introduce g‚(x) in the form 
of a definite integral. This definite integral, which has been given 
by Le Roy, may be found in the following way. 
Denoting by ./,(¢) the Besselian function of order zero, MACLAuRIN’s 
expansion gives easily 
ond. (247 ea = = on os ee 
0 ml! 
eada : ; : 
„— and integrating 
nN: 
Hence, multiplying both members by 
between the limits 0 and oo 
