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

304

THEORY OF BESSEL FUNCTIONS

[CHAP. IX

These results will be required in the sequel; it will not be necessary to write
down the analogues of all the other formulae of ~~ 9'6-9'64.
The result of eliminating alternate functions from the system (3) is of
some importance. The eliminant is
(a + m) gn+2, (z) = cn (z) gm, (z) - (v + m + 2) Z2gm-, (Z),
where cm (z) = (v + m + 1) t(v + ) (v + m + 2) - 2z}.
We thus obtain the set of equations:
(v + 2) g, (z) = c2 (z) g2, V (z)- (v + 4) Zgo, (z),
(v + 4) g6, v () = C4 (z) g4, (z) - ( + 6) z2g2, (Z),
oooo,.........,,.............................
(8) i ( + 2s) 2+2, v (Z) = C2s (Z) gS, V (Z)- ( + 2s + 2) z2g-2, (z),....................................................................
j ( +- 2m - 2) q,,, (z) c2^ M (Z).2M-2, ) (Z) = ( 2) Z2g 4, - ().)
9'71. The reality of the zeros of g2,,m (z) when v exceeds - 2.
We shall now give Hurwitz' proof of his theorem* that when v > - 2, the
zeros of g,,V (z) are all real; and also that they are all positive, except when
-1 > v > - 2, in which case one of them is negative.
After observing that g2m,v(z) is a polynomial in z of degree m, we shall
shew that the set of functions g2m,, (Z), gm-2, (Z),... g2, (Z),,, (z) form a set
of Sturm's functions. Sufficient conditions for this to be the case are (i) the
existence of the set of relations ~ 9 7 (8), combined with (ii) the theorem that
the real zeros of g2m-2, (z) alternate with those of yg2,, (z).
To prove that the zeros alternate, it is sufficient to prove that the quotient
s9m, v (Z)/g2m-2, V (z) is a monotonic function of the real variable z, except at the
zeros of the denominator, where the quotient is discontinuous.
We have 22-m2, (z) d 92M-, v (Z))= - 2n,2m-2,
dz g2m-2,v (Z))
where Ssr, = 9, v (z) g's, (z) - gs, (z) g'., (z);
and from ~ 9-7 (3) it follows that
U2am, 2m-2 = 22m-2, v (Z) + (a + 2m) M2m-1,2rn-2,
lam — 1,2n?-2 =2 ~22m-3,2rn-4 + (V + 2m - 2) 922n-3, v (z),
so that
m-1l
M2m,2m-2 = g22-2, v (z) +- ( + 2m) E (v + 2r) z2m-2T-2 g2-1 (),
r=l
and therefore, if z > 1, wmm_-2 is expressible as a sum of positive terms
when v >- 2.
* Math. Ann. XXXIII. (1889), pp. 254-256.