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

154

THEORY OF BESSEL FUNCTIONS

[CHAP. V

It is not obvious that this assumption is justifiable, though it happens to
be so, and a rigorous proof of the expansion of a quotient of Bessel functions
into an infinite continued fraction will be given in ~ 9'65 with the help of the
theory of " Lommel's polynomials."
[NOTE. The reason why the assumption is not obviously correct is that, even though
the fraction p,/q,, tends to a limit as m -oo, it is not necessarily the case that a"pPn +PM +1
am qm q mn +
tends to that limit; this may be seen by taking
Pn =m+sin m, q, =m, a,= -1.]
The reader will find an elaborate discussion on the representation of J, (z)/J_ 1 (z) as a
continued fraction in a memoir* by Perron, Jiitnchener Sitzungsberichte, xxxvII. (1907),
pp. 483-504; solutions of Riccati's equation, depending on such a representation, have
been considered by Wilton, Quarterly Journal, XLVI. (1915), pp. 320-323. The connexion
between continued fractions of the types considered in this section and the relations connecting contiguous hypergeometric functions has been noticed by Heine, Journal fir
Math. LVII. (1860), pp. 231-247 and Christoffel, Journal fur Math. LVIII. (1861),
pp. 90-92.
5'7. Hacnsen's expression for J, (z) as a limit of a hypergeometric Jfnction.
It was stated by Hansent that
(1) J,(z)= lim ) F",k — ( + 1l) 4 2fA
We shall prove this result for general.(complex) values of v and z when X and, tend to infinity through complex values.
If X = 1/8, = 1/17, the (m + l)th term of the expansion on the right is
_\n (_ I \v+2m m -1
At V[(1 + re) (1 + rm )].
m!F(v+m~1) =i
This is a continuous function of 8 and 7; and, if 30, 70 are arbitrary positive
numbers (less than 2 z 1-1), the series of which it is the (m + l)th term converges uniformly with respect to 8 and V7 whenever both 8 ] < 80 and 1 j < 710.
For the term in question is numerically less than the modulus of the (im + 1)th
term of the (absolutely convergent) expansion of
(^+~) F (1/80, 1/1Vo;v + 1; 4z2 o),
and the uniformity of the convergence follows from the test of Weierstrass.
Since the convergence is uniform, the sum of the terms is a continuous
* This memoir is the subject of a paper by Nielsen, Mlitnchener Sitzungsberichte, xxxVIII.
(1908), pp. 85-88.
f Leipziger Abh. ii. (1855), p. 252; see also a Halberstadt dissertation by F. Neumann, 1909.
[Jahrbuch ilber die Fortschritte der Math. 1909, p. 575.]