A treatise on the theory of Bessel functions, by G. N. Watson.

2-22]

THE BESSEL COEFFICIENTS

23

will be of frequent occurrence in the sequel, enables us to write (1) and (2) in
the compact forms:
co
(5) cos (zsinO)= eJ, (z)cos2nO,
n=O
co
(6) sin (z sin 0) = E e62+1 J2?t+I (Z) sin (2n + 1) 0.
n-=O
If we put 0 = 0 in (5), we find
(7) 1 = 621 J2, (z).
n=0
If we differentiate (5) and (6) any number of times before putting 0 = 0, we
obtain expressions for various polynomials as series of Bessel coefficients. We
shall, however, use a slightly different method subsequently (~ 2'7) to prove
that Zrn is expansible into a series of Bessel coefficients when m is any positive
integer. It is then obvious that any polynomial is thus expansible. This is a
special case of an expansion theorem, due to Neumann, which will be investigated in Chapter xvi.
For the present, we will merely notice that, if (6) be differentiated once
before 0 is put equal to 0, there results
0
(8) z = E2,+1 (2n + 1) J2n+1 (Z),
n=O
while, if 0 be put equal to }rv after two differentiations of (5) and (6), then
(9) zsin z = 2 {22 J2 () - 42 J4 () + 62 Js (z)...},
(10) z cos z = 2 12 J1 (z) - 32 J3 (z) + 52 J, (z) -... }.
These results are due to Lommel*.
NOTE. The expression exp lz (t - l/t)} introduced in ~ 2 1 is not a generating function
in the strict sense. The generating function t associated with E,,JJ (z) is E2 ElttJ. (z).
n=O
If this expression be called S, by using the recurrence formula ~ 2'12 (2), we have
d= t (- +i at + Jo(.
dzh K7 S- t~ 2 )Jo(z).
If we solve this differential equation we get
(11) Sez (t-1t) + ( + 1 e(t-1 ) e (t- /t) J (z) dz.
A result equivalent to this was given by Brenke, Bull. Americcan Jatbh. Soc. xvi. (1910),
pp. 225-230.
* Studien inber die Bessel'schen Functionen (Leipzig, 1868), p. 41.
+ It will be seen in Chapter xvi. that this is a form of "Lommel's function of two variables."