82 
Proceedings of the Royal Society of Edinburgh. [Sess. 
or 
.m— i 
P(z) ( l) m m\ r(2/t+l)^ 0! ml r ^ 2/t + m ) 1 I (to- 1)! r(2/* + m- 1) 
and the expression between brackets is the polynomial of degree m 
considered by, Sonine * in his researches on the Bessel functions ; it is 
here T m (x), the definition of the polynomial being the expansion 
tx 
e w, ' , ="° 
and its expression, as given by Sonine, being precisely 
afi-* 
T^M- x -i- 
a W p\0\T(a + p) (P-I)l 1 ! r(a + /3- 1) (/5-2)! 2! T(a + P~2) 
We can then write the following expression : 
M m+M+ i, M (^) = (- 1 ) w ^ ! T(2[x+l)x*+h 
a result which can be verified by using the expression of the T polynomial 
in terms of the W k m function, as given by Whittaker, j- 
We then obtain at once the very simple and remarkable expansion 
x+y 
Z/) = ^ + y +i e 2 r(2/»+ l)r(2v+ 1)22(- 
m n 
([x, + v—k+ 1, m + n)T™Jjc) T”(y). 
Some interesting consequences, concerning certain particular cases of 
the M function, can be deduced from this formula. 
If we suppose, in the first place, /jl and v to be of the form 
Z z 
where a and b are integers, we have to consider in the expansion poly- 
nomials of the type 
T r +i (*>- 
for which we readily find the simple expression 
fi a + 1 
r: +i (x)=^xr + Xx). 
But we can observe with Sonine that, if A is an integer, 
Ty*) = lW^) 
where U is Hermite’s polynomial, 
d?j 
* Math. Ann xvi(1880), p. 41. 
t Modern Analysis , 3rd edition (1920), p. 352. 
