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

11-4, 11-41]

ADDITION THEOREMS

363

But, for an absolutely convergent series,
00 q 00 0
1 Yq UkQ = Y4: Uk, k+n,
q=O =O k=O n=O
and so
n= S I (_)k+n (v +p + 2k) r (v + p + Ic) zp+2k+2n cosp 4 J4vp+2k (Z)
p=0ko =O- 22k+2 p!!n! Fr (v+ p + 2k + n + 1) Z
= S (-)k 2^V+ (v + p + 2k) r (v + p + k) CosP Jv+p+2k (Z) Jv+p+2k (Z)
p=Ok=o p!k! Zv zv
_ GO (_)k 2v+n-2k (v + m) r (v + m - k) COSmm-2k JV+M (Z) JV+M (z)
k=Omn=2k (m - 2k)!k! ZV zv
- <S? ()k 2v+ r-2k (V + m) r (v + m - l) cosm-2k Jv+ (Z) J,+m (z)
m= O k = (n - 2k)! k! Zv!
Now < ()k 2m-2- r (v + m - k) cos- _ - (cos ),
Now E - 7 = (V (cos =),
kN=o (m-2k)! k!r() (C)
where, as in ~ 3'32, C,V (cos 4) denotes the coefficient of am in the expansion
of (1 - 2a cos ) + a2)-V in ascending powers of a. We have therefore obtained
the expansion
JV(p) cc J,+M (Z).,1a(qz) Z)
(2) J- = 2 ( rv (v.n) Z (+ +) - 0 c (cos 4),
which is valid for all values of Z, z, and 4, and for all values of v with the
exception of 0, - 1, -2,....
In the special case in which v = 2, we have
sin c Jm+1(Z) fJM+ (Z)
(3)~ = - "O (m + ) Pm + (Z(cos.
la V=0 2m V' c
This formula is due to Clebsch, Journal fir Math. LXI. (1863), p. 227; it is also
given by Heine, Journal fir Math. LXIX. (1868), p. 133, and Neumann, Leipziger Berichte,
1886, pp. 75-82. The formula in which 2v is a positive integer has been obtained by
Hobson, Proc. London Math. Soc. xxv. (1894), pp. 60-61, from a consideration of solutions
of Laplace's equation for space of 2v +2 dimensions.
An extension of the expansion (2) has been given by Wendt, Monatshefte filu Math. und
Phys. xI. (1900), pp. 125-131; the effect of her generalisation is to express
w-v-P sin2p J,+p (w)
as a series of Bessel functions in which the coefficients are somewhat complicated
determinants.
11'41. The modified form of Gegenbauer's addition theorem.
The formula
J-v (W) 0C J (Z) Jpm ()Z)
(1) _v (~) = 2v r (.); (-) (. + m) ( z) $+ (x) C.^ (cOS )
Va n=o Z- -m=
may be established in the same manner as the Gegenbauer-Sonine formula of
~ 11'4. This formula does not seem to have been given previously explicitly,