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

ASSOCIATED POLYNOMIALS

273

When these inequalities are satisfied, the sum of the moduli of the terms does not
exceed
00 2 ( +2m ).(s -1)! (r x 2) exp( r2)
S=0 Rs1 m=O m (s+'2m)!j ' R-r
Since the expression on the right is independent of z and t, the uniformity of the
convergence follows from the test of Weierstrass.
The function 0, (t) was called by Neumann a Bessel function of the second
kind*; but this term is now used (cf. 3~ 3'53, 3'54) to describe a certain solution
of Bessel's equation, and so it has become obsolete as a description of Neumann's
function. The function 0, (t) is a polynomial of degree n + 1 in 1/t, and it is,
usually called Neumann's polynomial of order n.
If the order of the terms in Neumann's polynomial is reversed by writing
nn- m or 2 (n -1) - for n n (2), according as n is even or odd, it is at
once found that
(5) On (t)=4t i-) 2j!^^; (n even)
1 n2 n2 (-2 2) n2 (n2 - 22) (n2 - 42)
- -+ +...,
t t3 t5 t7
n? (n2-2) n (n2- 12) (n2 32)
2 t4 + t6 t
These results may be combined in the formula
n1 n n ( m) cos21 (m n) r
(7) On (t) lIn-i2 2 L *
o=o x 'l n -.. m +.1)- - (tm+l
The equations (5), (6) and (7) were given by Neumann.
By the methods of ~ 2-11, it is easily proved that
(8) | e On (t) I 2. (n!). ( t )-n- exp ( t 12),
(9) n On (t) = I. (n!). (2 t)- - (1 + 0), (n > 1)
where i0 <O [exp (I t I2)- 1]/(2 - 2).
From these formulae it follows that the series Sa,,^ J, (z) On (t) is convergent
whenever the series S a, (z/t)n is absolutely convergent; and, when z is outside
the circle of convergence of the latter series, an Jn (z) On (t) does not tend to
zero as n -- o, and so the former series does not converge. Again, it is easy
to prove that, as i -- G o,
Cn Jn 0 (z) 0, (t) l - + 0 (n )}
* By analogy with the Legendre function of the second kind, Qn (t), which is such that
1 %
-_ = ( 2n + 1) P, () Q (t).
- Z =0
Cf. Modern Analysis, ~ 15'4.

W. B. F.

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