HIGHER MONOTONICITY PROPERTIES OF CERTAIN STURM-LIOUVILLE FUNCTIONS
BY LEE L O R C H
University of Alberta, Edmonton, Canada (i) AND
P E T E R S Z E G O
University of Santa Clara, Santa Clara, Calif., U.S.A.(1)
1. Introduction
A n e x a m i n a t i o n of t a b l e s [3; 5; 8; 12; 15] of t h e positive(2) zeros of f a m i l i a r special f u n c t i o n s , such as Bessel functions(a) a n d v a r i o u s o r t h o g o n a l p o l y n o m i a l s , suggests t h a t sequences of differences c o n s t r u c t e d f r o m t h o s e zeros b e h a v e in a r e g u l a r m a n n e r . I n d e e d , c e r t a i n h e u r i s t i c a l l y o b s e r v e d r e g u l a r i t i e s a r e e x p l o i t e d s y s t e m a t i c a l l y b y t a b l e - m a k e r s as checks on t h e i r c o m p u t a t i o n s [1; 5, p. 404; 12, esp. p p . liii-liv].(4)
R i g o r o u s s t u d y of t h i s useful p h e n o m e n o n , h o w e v e r , does n o t a p p e a r t o h a v e p r o - g r e s s e d b e y o n d c o n s i d e r a t i o n of t h e s e c o n d differences of zeros of S t u r m - L i o u v i l l e f u n c t i o n s (solutions of t h e S t u r m - L i o u v i l l e d i f f e r e n t i a l e q u a t i o n y " + ] ( x ) y = O ) . H e r e S t u r m ' s com- p a r i s o n t h e o r e m [13; 14, p p . 19-21] h a s b e e n t h e p r i n c i p a l tool.
F o r i n s t a n c e , d e n o t i n g b y ( c ~ , n = 1,2 .... , t h e a s c e n d i n g sequence of p o s i t i v e zeros of a n a r b i t r a r y Bessel f u n c t i o n C~(x) of o r d e r u, Ch. S t u r m [13, p p . 173-175] u s e d his c o m p a r i s o n t h e o r e m t o show t h a t t h e s e c o n d (forward) differences A~c~n, n = 1,2 ... a r e all p o s i t i v e if Iv] < 89 a n d a r e all n e g a t i v e if [~, 1> 89 I n t h e s a m e m a n n e r , s i m i l a r r e s u l t s h a v e b e e n e s t a b l i s h e d for H e r m i t e , L a g u e r r e a n d L e g e n d r e p o l y n o m i a l s a n d o t h e r S t u r m - L i o u v i l l e
u n c t i o n s .
(1) Some of this work was done a few years ago when the first-named author received partial support from the (U.S.) National Science Foundation through Research Grant NSF G-3663 to Phi- lander Smith College, Little Rock, Arkansas. Its completion was facilitated by a grant from the University of Alberta General Research Fund. Both authors thank Professor Gabor Szeg6 for his interest and encouragement.
(2) All quantities discussed throughout this paper are assumed to be real.
(a) By a Bessel function we mean any real solution of the Bessel differential equation, not merely J~ or Yr.
(4) The regularities now used to check tables are not the ones discussed in this paper. How- ever, the ones established here can also be used conveniently for this purpose.
56 L. LORCH AND P. SZEGO
A n a l o g o u s p r o b l e m s can be f o r m u l a t e d for t h e sequence of t h e a r e a s b o u n d e d b y t h e successive a r c h e s o r w a v e s ( h a v i n g c o n s e c u t i v e zeros as e n d - p o i n t s ) of t h e g r a p h s of such special functions. T h e c o r r e s p o n d i n g results, i n c l u d i n g t h e i n f o r m a t i o n a b o u t t h e s e c o n d differences of t h e zeros, were e s t a b l i s h e d in a u n i f i e d a n d s i m p l e m a n n e r b y E. M a k a i [11].
H i s w o r k was s u b s e q u e n t l y g e n e r a l i z e d in s e v e r a l d i r e c t i o n s b y I . B i h a r i [2].
O u r m a i n p u r p o s e h e r e is t o go b e y o n d t h e s e c o n d differences a n d t o s h o w t h a t all h i g h e r differences of c e r t a i n sequences c o n n e c t e d w i t h t h e zeros a n d a r e a s of C~ (x) h a v e c o n s t a n t sign, a l t e r n a t i n g f r o m one difference t o t h e n e x t , w h e n I v ] > 89 A i r y f u n c t i o n s a n d c e r t a i n g e n e r a l i z a t i o n s of C,(x) a r e s h o w n t o s h a r e t h e s e p r o p e r t i e s . T h e p r e c i s e for- m u l a t i o n s a r e f o u n d in w167 2 - 4 .
I n w 5, t h e s e r e s u l t s a r e e x t e n d e d , in p a r t , t o t h e s t u d y of t h e h i g h e r differences of sequences whose e l e m e n t s a r e t h e (first) differences of t h e r e s p e c t i v e zeros of t h e s o l u t i o n s of d i s t i n c t S t u r m - L i o u v i l l e e q u a t i o n s . T h i s g e n e r a l i z a t i o n is a p p l i e d i n w 6 t o o b t a i n in- f o r m a t i o n c o n c e r n i n g t h e h i g h e r differences of sequences c o m p o s e d of t h e (first) differences of t h e r e s p e c t i v e zeros of a r b i t r a r y Bessel f u n c t i o n s of d i f f e r e n t orders.
T h e g e n e r a l m e t h o d of p r o o f we e m p l o y m a y p e r h a p s b e e x t e n s i b l e t o y i e l d a n a l o g o u s ( a l t h o u g h n o t i d e n t i c a l ) r e s u l t s for C~(x) w i t h Iv] < 1 a n d for o t h e r special f u n c t i o n s , such as H e r m i t e , L a g u e r r e a n d L e g e n d r e p o l y n o m i a l s , t h e B a t e m a n k - f u n c t i o n a n d , i n general, for t h e c o n f l u e n t h y p e r g e o m e t r i c f u n c t i o n for c e r t a i n v a l u e s of t h e parameter.(1) W e list some of o u r c o n j e c t u r e s in w 7.
A l l t h e r e s u l t s b e a r i n g on Bessel f u n c t i o n s w h i c h a r e p r e s e n t e d h e r e d e a l w i t h t h e b e h a v i o r of differences of q u a n t i t i e s c o n n e c t e d w i t h f u n c t i o n s of c o n s t a n t order. E l s e w h e r e [10] we consider some a n a l o g o u s p r o b l e m s i n v o l v i n g i n s t e a d t h e differences of zeros of f i x e d r a n k , b u t of f u n c t i o n s of v a r y i n g order.
2. Definitions, notations and general results ( o n e e q u a t i o n )
I f y ( x ) h a s zeros w h e n x is in a n o p e n i n t e r v a l I a n d is a n o n - t r i v i a l s o l u t i o n of t h e S t u r m - L i o u v i l l e d i f f e r e n t i a l e q u a t i o n
y " + / ( x ) y = 0 , (2.1)
(1) More generally, one may seek a generalization of the Sturm comparison theorem. This theorem shows that the second differences of the sequence of successive zeros of an oscillatory Sturm-Liouville function are all positive if ]'(x)<0, and are all negative if ]'(x)>~0. Perhaps the signs of the first -IV differences of these zeros can be inferred from the signs of ](n)(x), n = 1 . . . . , N . In particular, it would be interesting to determine if the complete monotonicity of /'(x), 0 < x < r implies the complete monotonicity of the sequence composed of the differences of successive zeros of an arbitrary solution of y" +/(x) y = 0, 0 < x < c~. This is the ease for Bessel functions (Theorem 3.1), and for Airy functions (Theorem 4.1).
S T U t t M - - L I O U V I L L E F U I ~ C T I O N S 57 where/(x) is a given function, we designate
any
increasing sequence of consecutive zeros in I b y x,, x 2 ... and define for a n y fixed 2 > - 1f
X k + lM k = ]y(x)]~dx
( k = 1,2 . . . . ). (2.2)d x k
When 2 = 1, the M k are simply the areas, say Ak, of the successive arches of
y(x).
When )~ = 0, the Me become the differences, Axk, of successive zeros of
y(x).
Moreover, Mk exists for a n y 2 > --1, since the zeros ofy(x)
are all of order 1 (the existence of a multiple zero of a solution of (2.1) would imply t h a ty(x)
is identically zero).Where it is convenient to distinguish a n y two (non-trivial) solutions of (2.1), linearly independent or not, we denote t h e m b y
y(x)
and ~(x), with the corresponding zeros and arch-areas, for example, written as xg, s Ak, Ak, k = 1,2 ... respectively. When we designate two such solutions b y yl(x),y2(x)
we mean t h a t t h e y are linearly independent.The symbol A~/~ means, as usual, the n-th (forward) difference of the sequence {/xk}, i.e.,
A~ A/~k=/~k+l--/~, A~/xk=A=-l/xk+l-A~-ll~k ( n = l , 2 ... k = l , 2 .... ).
For typographical convenience, we frequently use the symbol D~0(~) to denote the n-th derivative gc(n)(~).
We prove a general theorem concerning solutions of (2.1) and then specialize in w167 3-4 to general solutions of the Bessel and Airy differential equations, respectively.
Certain preliminary facts are needed in this connection (and again in w 5). Some m a y be known, but, inasmuch as they occupy a central position in this work, we formulate t h e m as lemmas and supply proofs.
L ~ M ~ A 2.1.
Let p(x) be a positive/unction such that p(n~(x) exists in an open interval (a, b). M a p (a, b) onto an interval o / a variable t through the relation x'(t) =p(x). Then,/or any
a > 0and any n=O, 1 ... N, the derivative D~{[x'(t)] ~ is a homogeneous /orm in p(~
p(1)(x) ... p(~(x) /or xE(a,b). It has
(i)order (a+n),
(ii)weight n,
(iii)non-negative coe/- /icients and exponents (both dependent only on n and a), and
(iv)integral exponents with the possible exception o/the exponent o/p(~
Proo/.
The m a p p i n g is possible sincep ( x ) > 0 .
The proof proceeds b y induction.When n = 0 , we have
D~{[x'(t)]~176176
when n = l , we h a v eD~{[x'(t)]~ = a[p(~176
I n b o t h cases, the assertions of the l e m m a are fulfilled.A general t e r m of D; ~ {[x'(t)] ~ would a p p e a r as
S = C [ p ( ~ ' [ p ( i ) ( x ) F ' . . . [ p ( ~ ) ( x ) F ~ ,
58 L . L O R C H A N D P . S Z E G O
w i t h
C, ~o ... an
all n o n - n e g a t i v e a n d ~1 ... ~n all integers. T h e o r d e r w o u l d be a 0 § 9 .. § an a n d t h e w e i g h t a l + 2 ~ 3 . . . + n~n.T h e i n d u c t i o n is c a r r i e d t h r o u g h b y d i f f e r e n t i a t i n g S w i t h r e s p e c t t o t. W e h a v e
dS dS
-dr = p(x) dxx: C~176176176 [P(n)(x)]~n
+ Coq[p(~176176 ~+1 ... [p(n)(x)]~n
J r - . . ,
+ Co~[p(O)(x)]~~
[p(n)(x)]~ ~ i[p(n+l)(x)]"Thus,
S'(t)
is a p o l y n o m i a l i np(~
p(1)(x) ...p(~+l~(x)
w i t h n o n - n e g a t i v e coefficients.N o n e of t h e e x p o n e n t s t h a t a p p e a r b e c o m e n e g a t i v e ; t h e e x p o n e n t of
p(x)
n e v e r d e c r e a s e s while t h e o t h e r s d e c r e a s e b y s t e p s of one u n i t , if a t all. T h e y m a y r e a c h zero, b u t t h i s o c c u r r e n c e w o u l d o n l y p r e s a g e t h e v a n i s h i n g of t h e d e r i v a t i v e of t h e c o r r e s p o n d i n g f a c t o r . T h e order, oJ, a n d w e i g h t , w, of a t e r m inS'(t)
o t h e r t h a n t h e f i r s t a n d l a s t , are, re- s p e c t i v e l y ,~o = (~o + 1) + ~ + ... + ( ~ - 1) + (~+~ + 1) + . . . + a , = 1 + (~0 + ~ + . . . + ~ n ) a n d
w = a l + 2a2 + -.. + i ( a t - 1) + (i + 1) (a~+l + 1) + ... + nan = 1 + (a 1 + 2 a i + ... + nan).
Thus, t h e o r d e r a n d w e i g h t increase b y one u n i t u n d e r t h e process of d i f f e r e n t i a t i o n of t h e s e t e r m s w i t h r e s p e c t t o t. T h e s a m e is c l e a r l y t r u e of t h e f i r s t a n d l a s t t e r m s as well.
F i n a l l y , we o b s e r v e t h a t t h e coefficients a n d e x p o n e n t s d o n o t d e p e n d on
p(x)
b u t o n l y o n n a n d a.Thus,
S'(t)
satisfies t h e a s s e r t i o n of t h e l e m m a a n d t h e i n d u c t i o n (which c a n be con- t i n u e d t o n = N ) is c o m p l e t e .L E M ~ A 2.2.
Let p(x) satis/y the conditions o/Lemma 2.1 and, in addition, the inequalities
( - 1 ) n p ( ~ ) ( x ) > 0 ( n = 0 , 1... N; xe(a,b)).
(2.3)Let the mapping described in Lemma 2.1 be carried out. Then,/or any a >O,
(-1)nDT{[x'(t)] ~
> 0 ( n = 0 , 1 ... N).The lemma remains true i/the/actor ( -
1) nis deleted/tom both
(2.3) a n d (2.4).Proo/.
F r o m L e m m a 2.1, we see t h a t a g e n e r a l t e r m ofD~{[x'(t)] ~}
has t h e f o r m (2.4)S = C[p(~ ~' [p(1)(x)]~' ... [p(n)(X)]~n.
STURM-LIOUVILLE FUNCTIONS 59 B y a s s u m p t i o n (2.3) the sign of S is
( - - 1)a'+2a~+'"+nan.
B u t the e x p o n e n t is simply t h e weight which, according to L e m m a 2.1, has the value n.
Thus, S has sign ( - 1) n a n d (2.4) is established. The final sentence of t h e l e m m a is trivial.
LEMMA 2.3. Let Yl, Y2 be solutions o/(2.1), normalized so that their Wronskian Y l Y ~ - YlY2 = 1. Let p(x) =y~ (x) +y~ (x) and de/ine the trans/ormation y(x) = [p(x)]89 x'(t) =p(x).
This trans/orms (2.1) into the di//erential equation u"(t) +u(t)=0.
Proo]. The f u n c t i o n p(x) is an admissible m a p p i n g f u n c t i o n since it is always positive.
The t r a n s f o r m a t i o n takes ( 2 . 1 ) i n t o u"(t)+q(t)u(t)=0, where
q(t) = 89 p(x) p" (x) - 88 + [p(x) ]2 /(x) (2.5)
so t h a t (with p(x)= y~(x) + ~ y2(x) a n d y" = - / ( x ) y ) ,
' 2 " . ~ ' " 2 " 2 2 " " 2 ' 2
q(t) = p(x) [(y;)~ + (y2) + ylyl y2y2 ) - [Yl(yl) + ylyly2Y2 + y2(y2) ] +
[p(x)] 2/(x)
2 2 t 2 t 2 i P 2
= (Y~+Y2)[(yl) + (y2) ] - [y~(yl)2+2yly~y2y~+ 2 , Y2(Y2) ] = (yiy~ - yly2) = 1.
The m a i n result of this section can be p r o v e d now:
THEOREM 2.1. Suppose that in some closed interval I* there exist two linearly inde- pendent solutions yl(x), y2(x) o/ (2.1) such that
( - 1)~n n { [y~(x)] ~ + [y~(x)] 2 } > O (n = 0, 1 . . . N), (2.6) where the N-th derivative exists in the open interval I, and the lower order derivatives are continuous in its closure I*. Then(i)
( - 1 ) ~ A n M k > 0 ( n = 0 , 1 .... ,N; k = l , 2 , . . . ) , (2.7) so that, in particular,
(--1)n-lAnxk>O ( n = 1,2 ... N + I ; k = 1,2 .... ). (2.8) Moreover, i / x 1 > Xl, then
(--1)nAn(Xk--~k) > 0 ( n = 0 , 1 .... ,N; k = l , 2 , . . . ) . (2.9) I / t h e / a c t o r ( - 1 ) ~ is deleted/rom assumption (2.6), then the conclusions (2.7), (2.8) and (2.9) remain valid provided they are amended by eliminating the/actors ( - 1) ~, ( - 1) ~-1 and ( - 1) n, respectively.
(1) The quantities Mk, xk, ~ are defined in the first paragraph of this section, e.g., by (2.2).
It should be remembered that they refer to any nontrivial solution of (2.1), not merely to yl(x) and y2(x).
60 15. L O R C H A N D P . SZEGO
Proo I.
W i t h o u t loss of generality, we normalize the solutionsyl(x), y2(x)
so t h a t their Wronskian is 1. Then we can a p p l y L e m m a 2.3 and transform the differential equation (2.1) into the formu"(t)+u(t)=0.
Sincep(x)=x'(t)>O
in I , there is a one-to-one corre- spondence between the zeros ofy(x)
and those ofu(t).
B u tu(t) = A
cos ( t - b), A, b constants, so t h a t its zeros are equidistant from one another with Ark=g,k = l , 2
... where t k is the zero ofu(t)
corresponding to x k.Thus, (2.2) becomes
[ ' t k + l t
M k = J t k
[x (t)] 1+89lu(t)l~dt.
Recalling t h a t At~ = ~ (k = 1, 2 . . . . ), and noting t h a t l u ( t + g ) [ = lu(t)], we have
AM~= (*,~+2_ [,,~+, = f~"+* {[x'(t + =)]~+'a[u(t +~)l~-[x'(t)]l+~alu(t)]a}dt
J t k + 1 *It k , I t k
= f'~+l{A~ {[x'(t)]i+'~)} lu(t)l~dt,
where A,{F(t)}
= $'(t + =) - F(t).
I t follows, in the same way, t h a t the higher differences are given b y
• = ~tk +1 { A2 { [x,(t)]l + ~a}} lu(t ) iadt ' (2.10)
d t k
where A", {F(t)} = A. {A~-1F(t)).
:Now, according to a mean-value theorem for higher derivatives and differences [7, p. 73] there exists a 0 such t h a t
A~F(t)
=znF(n)(t+On~)
(0 < 0 < 1), (2.11) providedF(n)(t)
exists in the open interval (t, t + nz) and the lower derivatives are continu- ous in the corresponding closed interval.Applying the extended mean-value theorem (2.11) to the expression (2.10) for AnMk, we obtain
~
t k + lA n i ~ = r ~ n {D'~{[x'(t+Onr~)]l+89
( 0 < 0(t) < 1). (2.12)J t k
I t should be noted t h a t the a r g u m e n t of x' can fall anywhere in
[tk, tk+(n+l)~].
However, this causes no difficulty since the range of t-values occurring in A~Mk encompas- ses the same interval.
L e m m a 2.2 can now be applied, with a = 1 + 89 since condition (2.3) of the l e m m a is fulfilled b y virtue of assumption (2.6) in the theorem. Employing the conclusion of the lemma, (2.4), the result (2.7) of the theorem follows from (2.12).
ST~M--LIOVV~L~ r u N c r I o ~ s 61 T a k i n g 7 = 0 in (2.7), in turn, yields (2.8), since
M k = A x k , A n M k = A n + l x k for ~ = 0 .
The proof of (2.9) is slightly different. L e t tl, t l + ~ , t ~ + 2 g ... be t h e sequence of t-values corresponding to t h e zeros xl,x2, x a .... , a n d tl, il + ~ , tl +2z~ ... be t h e sequence of t-values corresponding t o ~1,~2,~3 ... Then, w i t h x z = x ( t k ) , ~ k = x ( l k ) ,
h ~ ( x k - ~ ) = • x(i~)] = A ~ [ x ( t ~ ) - x ( t ~ - ~ ) ] , where ~ = t k - i k a n d is i n d e p e n d e n t of k.
A p p l y i n g the same m e a n - v a l u e t h e o r e m (2.11) as before, we h a v e An(xk - - ~ k ) =
:~n[x(~)(tk +One) --x~'~)(t~ +On~--7)],
with 0 < 0 < 1. This can be rewritten as
f
t k + O n ~A~(xk-- ~k) = ~ x(~+l)(t) dr.
J t k § ~
Thus, t h e sign of A~(xk--xk) is d e t e r m i n e d b y the sign of x(n+l)(t). However~ L e m m a 2.2 (with a = l ) , applicable in view of hypothesis (2.6), shows this sign t o be ( - 1 ) ~ for n = 0,1 ... N. Hence, t h e q u a n t i t y ( - 1 ) n A n ( x k --xk) is positive or negative according as ~ > 0 or ~ < 0. B u t
x k - ek = x(tk) - x(ik) = x'(t) dr,
k - U
which has t h e same sign as ~ since x ' ( t ) > 0 . I n a s m u c h as x~ > ~ , it is clear t h a t ~] > 0 , so t h a t (2.9) is established. The final sentence of the t h e o r e m follows at once on m a k i n g the obvious changes in the above proof.
R e m a r k . I t should be k e p t in m i n d t h a t t h e definition of x 1, x 2 ... Xl, x2 ... provides a considerable a m o u n t of flexibility. N e i t h e r x I n o r xl need be t h e first zero of its respective function, n o r is t h e r e even a n y requirement t h a t x~ should o c c u p y t h e same relative position a m o n g t h e t o t a l i t y of zeros of y(x) in I as ~1 does relative to ~(x). This is useful to k n o w in connection with applications of T h e o r e m 2.1.
3. Application of w 2 to certain Bessel functions
One o p p o r t u n i t y t o a p p l y T h e o r e m 2.1 to Bessel functions of a p p r o p r i a t e order is facilitated b y Nicholson's integral [15, p. 444(1)].
T h e o r e m 2.1 applies for all non-negative integers N to t h e Bessel differential e q u a t i o n in t h e f o r m
62 L . L O R C H A N D 1 ). S Z E G O
p r o v i d e d [ v[ > 89 The f u n d a m e n t a l
Y.(x)
so t h a ty " ( x ) § 1 8 8
( x > 0), (3.1)solutions Yl, Y2 of (3.1) can be t a k e n to be (89189
J~(x), (89189 p(x) = 89 ~ +
[ Y~(x)]2}.F o r Iv] > 89
p(x)
is analytic, 0 < x < oo, a n d sop(N~(x)
exists for all n o n - n e g a t i v e integers _hr.To satisfy t h e conditions of T h e o r e m 2.1, we need o n l y choose a n i n t e r v a l I e n c o m p a s - sing all zeros u n d e r consideration a n d w i t h its lower e n d - p o i n t a positive n u m b e r . This can obviously be done, since each Bessel function has a least positive zero. Condition (2.6) on the sign of
p(n)(x)
c a n be verified b y using G. N. W a t s o n ' s m a n i p u l a t i o n of Nicholson's integral r e p r e s e n t a t i o n for ourp(x)
[15, p. 446]:W a t s o n ' s calculation gives
p'(x)
= ~ {K o(2x sinh T)} (tanh T) (eosh 2 v T ) . [tanh T - 2v t a n h2vT] dT,
(3.2) where K0(~) is t h e Bessel function of second kind, i m a g i n a r y a r g u m e n t a n d zero order.As W a t s o n points out, t h e b r a c k e t e d factor in t h e a b o v e i n t e g r a n d is n e g a t i v e for Iv I > 89 e v e r y t h i n g else is positive. Hence,
p'(x)<0.
F u r t h e r differentiation yields~
p(~)(x) = ~ {K~n-1)(2x
sinh T)} (2 sinh T) n -~ (tanh T) (eosh 2vT)9 [tanh T - 2v t a n h 2vT]
dT.
(3.3) Now, it is clear f r o m the r e p r e s e n t a t i o n [15, p. 446 footnote]K0(~) = I ~ e
~ oo:h t dt ' Jo
t h a t ( - 1)"K(0 n)(~) > 0 for ~ > 0, n = 0,1,2 ... so t h a t
p(x)
satisfies t h e h y p o t h e s e s of T h e o r e m 2.1, a n d (2.7), (2.8) a n d (2.9) are therefore valid for Bessel functions of order v, Iv] > 1 , for all N = l , 2 ...There is a n a l t e r n a t i v e a r g u m e n t , based on w o r k of P. H a r t m a n [6], available to establish t h e v a l i d i t y of (2.7), (2.8), (2.9) for these Bessel functions.
Applied directly to e q u a t i o n (3.1), H a r t m a n ' s T h e o r e m 18.1n [6, p. 182], w i t h n = + , asserts t h a t
p(x)
is c o m p l e t e l y m o n o t o n i c in (0, ~ ), i.e., it gives (2.6) in t h e slightly w e a k e n e d f o r m in which " > " is replaced b y "~> ". H o w e v e r , as s h o w n in w 8 below, a function com- pletely m o n o t o n i c in (0, c~ ) m u s t be a c o n s t a n t if a n y one of its d e r i v a t i v e s (including theS T U R M - - L I O U V I L L E F U N C T I O N S 63 zeroth, the function itself) vanishes for a n y single positive value. Thus, if e q u a l i t y ever prevailed in (2.6), p(x) would have to be a constant. However, this, in turn, is impossible for Iv[ > 89 since, according to a lemma established in w 9 below, this would require t h e coefficient of y in (3.1) also to be constant, which it is not.
Thus, H a r t m a n ' s T h e o r e m 18.1n, with n = c~, implies (2.6) and so also (2.7), (2.8) a n d (2.9) for Bessel functions of order v, Iv I > 1. Therefore, b y either argument, we obtain the following results for Besse] functions.
THEOREM 3.1. Let c~k, 5~k denote, respectively, the k-th positive zeros, arranged in ascend- ing order, o/ any pair o/ non-trivial solutions (linearly independent or not) o/the Bessel equa- tion (3.1) with [~1 >1. Let ~ be a constant, ,~> - 1 , and
f
Cv, k + lM k = x89 dx. (3.4)
t] C v k
T h e n , / o r Ic = 1, 2 . . .
( -- 1)nA~(C~.m+k -- 5~k) > 0 /or a n y / i x e d m = O, 1 . . . provided c~.m + l > 5~1.
I n particular (still with Ivl > 89
( -- 1)n-lAnj~ > 0, ( -- 1)n-lAny~k > 0
(-- 1)nhnMk> 0 ( n = 0 , 1 . . . . ), (3.5) ( - 1)'-lAnc~k > 0 ( n = 1, 2 . . . . ), (3.6) (n = 0, 1, 2 . . . . ), (3.7)
( n = l , 2 . . . . ), (--1)nAn(?'~k--y~k)>0 ( n = 0 , 1 , 2 . . . . ), where j ~ , y,,~ denote the respective k-th positive zeros o / J ~ ( x ) and Y~(x).
Remark. (3.7) implies (3.9), since ]~l>y~l [15, p. 487 (10)].
(3.8) (3.9)
4. Application of w 2 to Airy functions The Airy functions satisfy the differential equation [14, p. 18]
y"(x) + ~ x y = O. (4.1)
A b r o a d e r class of functions, including the Airy functions, satisfy the differential equa- tion [15, p. 97 (9)]
y"(x) + f12~2x2~-2y = O. (4.2)
These functions are closely related to Bessel functions. Indeed, y = x89 C1,(2~)(yx ~) satisfies (4.2).
64 L . L O R C H A N D P. SZEGO
W h e n 1 < / ~ < ~ , e q u a t i o n (4.2) meets H a r t m a n ' s conditions [6, p. 183 (Theorem 20.1~)] with n = oo, so t h a t t h e special case (4.1), where f l = 3, is covered as well. Accord- ingly, we have a n analogue of T h e o r e m 3.1 for A i r y functions a n d their zeros. I n partic- ular, t h e following result holds:
THEOREM 4.1. I / 1 <fl<~, then
( -- 1)n-lAn([Cl/(2~).k] 1/~} > 0 ( n , k = 1, 2 . . . . ).
I] (a~}, (sk} are the sequences o/positive zeros, arranged in ascending order, o / a n y pair o/
Airy/unctions (solutions o/(4.1)), and a 1 >ax, then
( - - 1 ) n - l A ~ a k > 0 , (--1)~A~(ak--Sk)>0 (n,k = 1,2 .... ). (4.3) This t h e o r e m follows f r o m H a r t m a n ' s T h e o r e m 20.1~, with n = ~ , in precisely the same w a y as T h e o r e m 3.1 was shown to follow f r o m his T h e o r e m 18.1n, with n = ~ , a n d the subordinate results in w167 8 - 9 below.
F o r this theorem, unlike T h e o r e m 3.1, we h a v e no alternative proof to offer.
5. Higher monotonieity of sequences arising from zeros of two Sturm-LiouviHe functions
Previous sections discussed higher m o n o t o n i c i t y of various sequences c o n n e c t e d with, inter alia, the areas u n d e r successive arches of the graphs of certain S t u r m - L i o u v i l l e func- tions and, more particularly, with their zeros. Here we s u p p l y a partial generalization (restricted to zeros) of some of t h e previous results b y relating the zeros of a n a r b i t r a r y solution of one S t u r m - L i o u v i l l e e q u a t i o n to those of an a r b i t r a r y solution of a di//erent S t u r m - L i o u v i l l e equation, u n d e r certain circumstances. This will imply, for example, t h e complete m o n o t o n i c i t y of certain sequences, such as (?',.r+k-Y~. m+k}k%l, for a p p r o p r i a t e /~, ~, r a n d m (cf. T h e o r e m 6.1 for details).
To this end, we need the following extension of our first t w o lemmas:
LEMMA 5.1. Let an open interval I t o/the variable t be mapped onto corresponding open intervals Ix and I x o/ the variables x and X by the mappings x'(t) =p(x). X'(t) =P(X), respec- tively, where p(x), P ( X ) are prescribed positive/unctions such that p(N)(x), P(N~(X) exist in the open interval I x N I x (say Ixx ). Suppose,/urthermore, that p(x), P( X) satis/y the inequalities ( - 1)nD~x (P(X)} >1(- 1)'D~ (p(x)} > 0 (n = 0, 1 . . . N), (5.1) /or x, X E Ixx, where x and X in (5.1) correspond to the same t.
S T U R M - - L I O U V I L L E F U N C T I O N S 65 Then, for a n y a > O,
( - 1)~D~ {[X'(t)] o}/> ( - 1)nD~ {[x'(t)] o} > 0, (5.2)
n = 0 , 1 . . . N , /or t E It.
Again, i / b o t h / a c t o r s ( - 1 ) n are e l i m i n a t e d / t o m azsumption (5.1), then conclusion (5.2) remains valid once the same deletions are made.
Proo/. T h e second inequality of (5.2) is identical w i t h the conclusion of L e m m a 2.2.
T h e first i n e q u a l i t y follows f r o m a t e r m b y t e r m c o m p a r i s o n of t h e h o m o g e n e o u s f o r m s for D~([X'(t)] ~} a n d D'~([x'(t)]~}: According to L e m m a 2.1, a t y p i c a l t e r m of ( - 1 ) n D ~ ( [ x ' ( t ) ] ~}
is
S x = ( - 1)nV[p(x)]~'[p'(x)] ~' ... [p(n)(X)]~n = C[p(x)]~*[( -- 1)p'(x)] ~ ... [( -- 1)np(n)(x)J~n, since a 1 + 2a2 § + nan = n (this being t h e weight), w i t h C positive, a n d w i t h each r e m a i n i n g f a c t o r positive b y (5.1).
T h e corresponding t y p i c a l t e r m of ( - 1)nD.~ ([X'(t)] ~} is
S x = C [ P ( X ) ] ~ ' [ ( - 1) P ' ( X ) ] ~ , . . . [( - 1 ) ~ P ( n ) ( X ) ] ~ n ,
with t h e s a m e values for C, a0 . . . an as for ( - 1)nD~ ([x'(t)]"}, b y L e m m a 2.1.
Thus, t h e proof of (5.2) will be complete once it is established t h a t S ~ > S x . This follows f r o m (5.1), since each f a c t o r in b r a c k e t s in S x is positive a n d n o t m o r e t h a n t h e corresponding f a c t o r of Sx, while C > 0. T h e p r o o f of the final assertion follows on m a k i n g t h e obvious changes in t h e foregoing.
Remark. I f t h e s y m b o l ">~" in (5.1) is replaced b y " > ", t h e n (5.2) can be s t r e n g t h e n e d correspondingly.
F r o m this l e m m a we o b t a i n t h e principal result of the section.
T H ]~ 0 R ~ M 5.1. Suppose that in some closed interval I* there exist two pairs o / l i n e a r l y independent solutions (yl(x), y2(x)}, (Yl(x), Y2(x)} (each pair normalized so that the Wronskian is 1) o / t h e respective di]/erential equations
y"(x) + f i x ) y = 0; Y"(x) + F(x) Y = 0,
(5.3)
(where/, F are given/unctions) such that
( - 1 ) n p ( n ~ ( x ) ~ ( - 1 ) n p ( n ) ( x ) > O ( n = O , 1 ... N), (5.4) where the N - t h derivatives exist in the open interval I, and the lower derivatives are continuous in its closure I*, and where
p(x) = [yl(x)] 2 + [y2(x)]2; P(x) = [YI(x)] 2 + [Y2(x)] 2.
5 - - 6 3 2 9 1 7 . A c t a mathematica 109. I m p r i m 6 le 29 m a r s 1 9 6 3
66 L. LORCH AND P. SZEGO
Then, i / X 1 > X 1 , w e h a v e
( - 1 ) n A n ( x k - X k ) > 0 , (n = 0 , 1 ... N; k = 1,2 .... ), (5.5) /or the zeros xk, X k o / y and Y, respectively, in the intersection 1" N J, where J is an interval de/ined to be either (XI, ~o) or ( X 1, co) according as there does or does not exist a solution ~ ]or the equation
f ( ds (+ ds (5.6)
J ~ , P ( s )
~o being the least such solution.
I / the symbol ">~" in (5.4) is replaced by " > " /or n =O, then the solution o/ (5.6), i/
any, is unique.
Proof. As in w 2, L e m m a 2.3, t h e t r a n s f o r m a t i o n s
y(x) = [x'(t)]89 x'(t) = [yl(x)] 2 + [y~(x)] ~ (5.7) Y ( X ) = [X'(t)]89 V(t); X'(t) = [ YI(X)] ~ + [ Y3(X)] 2
t a k e equations (5.3) into, respectively,
u"(t) + u ( t ) = o; uH(t) + u(t) = o. (5.8)
The m a p p i n g functions (5.7) are n o t unique; a n integration c o n s t a n t is left unspecified.
I n order to particularize (5.7) completely, we t a k e
(x d s = ( X d s (5.9)
t = Jx, p(s) ,]x,P(s)"
The t r a n s f o r m a t i o n s (5.7) n o w establish a one-to-one correspondence between t h e zeros of y(x) a n d u(t) a n d between those of Y ( X ) a n d U(t), respectively, x'(t) a n d X'(t) being positive.
L e t us designate b y tl, t 2 ... t h e zeros of u(t) corresponding, respectively, t o xl, x 3 ....
a n d b y T1, T3 ... t h e zeros of U(t) corresponding to X 1, X2 ... Since all solutions of (5.8) are of the f o r m A cos ( t - b ) , it follows t h a t Ark = A T k =~ for all k.
Now, t l = T l = O , f r o m (5.9), so t h a t t k = T k = ( k - 1 ) ~ , k = l , 2 ... Hence
( - 1 ) n A n ( x k - X k ) = ( - - 1 ) n A n [x(tk) -- X ( t k ) ] , ( 5 . 1 0 ) where the subscript ~ on t h e r i g h t - h a n d difference o p e r a t o r signifies t h a t these differences are to be t a k e n with i n c r e m e n t ~.
According t o a m e a n - v a l u e t h e o r e m for higher differences [7, p. 73], (cf. (2.11)) there exists 0 such t h a t
S T U R M - - L I O U V I L L E F U N C T I O N S 67
AnF(t) = ~ F ( n ) ( t + n ~ O )
(0 < 0 < 1), (5.11) u n d e r conditions onF(t)
which are satisfied here b yx ( t ) - X ( t ) .
A p p l y i n g this to (5.10), we have, for such a 0,(--1)nA~(xk--Xk) =
(--~)n[x(~)(tk+n~O)--X(~)(tk+nnO)]
( 0 < 0 < 1 ;n=O, 1,...,N;
k = l , 2 .... ).(5.12)
T h e a r g u m e n t in (5.12) m a y fall a n y w h e r e in t h e i n t e r v a l [t~, tk + n ~ ] ; this is precisely t h e r a n g e implied inAnF(tk).
Now, (5.5) will follow f r o m (5.12) once it is established t h a t(--1)nx(~)(t) > (--1)~X(n~(t)
(n = 0,1 ... /V), (5.13) for t in the a p p r o p r i a t e interval. T h e case n = 0 requires t h a t we showx(t) > X(t),
(5.14)while t h e o t h e r cases require
( -- 1)nD'~{X'(t)} > ( - 1)nD'~
{x'(t)} (n = O, 1, 2 . . . N - 1). (5.15) I n view of a s s u m p t i o n (5.4), it is clear t h a t L e m m a 5.1 is applicable, so t h a t (5.15) would follow f r o m(-1)~P(n~(X) > (-1)np(~)(x) (n = 0 , 1 ... N - I ) . (5.16) B u t this is e q u i v a l e n t t o
( - 1 ) ~ [ P ( ~ ( X ) - P ( n ) ( x ) ] > (-I)~ip(~)(x)-P(~)(x)] (n
= 0 , 1 ... N - I ) . (5.17) T h e right side of (5.17) is non-positive (from (5.4)). Showing t h a t t h e left side is positive will p r o v e t h e i n e q u a l i t y (5.17). T h a t is, it will suffice to show t h a t=
( -
d s > 0 ( n = O, 1 . . . N -1).
( 1)~*[P(n)(x)
P(~)(x)]H o w e v e r , hypothesis (5.4) states t h a t ( - 1)nP(n+ll(s)<0, so t h a t to p r o v e (5.17), it is suf- ficient to establish t h a t
x(t)>X(t),
i.e., i n e q u a l i t y (5.14). I n o t h e r words, t h e v a l i d i t y of (5.5) is coextensive with t h a t of (5.14).Now, we h a v e a s s u m e d t h a t
X ( 0 ) - - X ( 0 ) : x I - X 1 > 0 .
Hence, if (5.14) is ever violated, t h e n there m u s t exist a t least one value of t, s a y T, for which x(v) =X(~)---} > X r
I f no such ~ exists, t h e n t h e p r o o f is complete, a n d t h e interval J of the t h e o r e m is s i m p l y (X1, ~ ).
68 L . L O R C H A N D P. SZEGO
Otherwise, employing the restriction on t, (5.9), the condition on ~ becomes (5.6).
Accordingly, we define the differentiable function
g(v)
b yg(v) = Jx, P(s)
Obviously,
g(X~)
> O, since x 1 > X1, and_ _ _ f~ .qd~, v>~X r
(5.18)J~,p,s)
from (5.4).
1 1
g'(v)
<0,P(v) p(v)
Thus,
g(v)
is a continuous, non-increasing function which is positive at X1, so t h a t the set of its zeros, i.e., of the values of ~ violating (5.14), would, if non-empty, be a closed inter- val. If there be such ~, then there is a least such, say ~e, and (5.14) is satisfied throughout the open intervalJ----(X1, ~o)"
This completes the proof of the theorem, except for its final sentence. This, in turn, follows immediately on observing t h a t assumption (5.4) becomes, in this case, the in- equality
P(x)
>p(x), so t h a t the functiong(v)
is now strictly decreasing and hence cannot vanish more t h a n once, if at all.Remark 1.
If the functionsp(s), P(s)
are analytic (or even merely members of the same quasi-analytic class), then also the solution of (5.6), if any, is unique. For, otherwise, there would exist an interval of solutions ~, so that, differentiating both sides of (5.6) with respect to ~, we would have p(~)=P(~) for all ~ in the solution interval. Then, from (quasi) analyticity, it follows t h a tp(s) = P(s)
everywhere. Since x 1 > X 1, (5,6) clearly could not be satisfied, a contradiction.Remark 2.
The inequality (2.8) and its consequences, b u t not the whole of Theorem 2.1, follow from Theorem 5.1, on taking the two differential equations (5.3) to be identical, so t h a t (5.4) is satisfied with">~"
reducing to " = ".Remark 3.
As in w 2, if assumption (5.4) is altered by deleting both factors ( - 1) n, then conclusion (5.5) will remain valid provided the factor ( - 1 )n is deleted.Finally, we note the additional corollary below which, in essence, assures us t h a t Theorem 5.1 has non-vacuous content in certain specific circumstances.
COROLLARY 5.1.
Let all the conditions o/Theorem 5.1 be saris]fed and suppose,/urther,
that (i)I*=
[XI, oo ], (if)there are in/initely many xk, Xk in I*, and
(iii)~ { ~ s ) ~ ,
P(s)} ds<c~"
(5.19)S T U R M - L I O U V I L L E F U N C T I O N S 69 Then,/or a n y / i x e d choice o / X 1 , there always exists an x 1 su//iciently large so that (5.5) ho/ds in (X1, co).
Proo/. I t suffices to show t h a t there exists no ~ satisfying (5.6), and for this we need establish only t h a t g(v)>0 for all v where the function g(v) is defined b y (5.18). This func- tion can be p u t in the form
g(v) ,}x,P(s) , p(s) P(s) ds (v>~Xl). (5.20)
Now, g'(v) <<. 0, so t h a t the positivity of g(v), v >~ X 1, can be shown b y proving t h a t g( ~ ) > 0, i.e., b y d e m o n s t r a t i n g t h a t
,P(s) ,]x~ in(s)
for all sufficiently large x 1. B o t h sides of (5.21) are non-negative, since P(s)>~p(s)>0 and x 1 > X 1. However, the left side is an increasing function of Xl, while the right side, which is non-increasing, approaches zero as x l - - > ~ . This proves t h e corollary.
6. An application of w 5 to Bessel functions
The conditions of Corollary 5.1 will be verified for certain Bessel functions of orders /~, ~, respectively.
THEOREM 6.1. Given three numbers 1~, ~, m, with 89 m a positive integer, let c,m, ~,~ denote the m-th positive zero o / a n y solutions o/the Bessel equations o/order i~ and ~, respectively. Then there exists a positive integer r such that c,, > y~,~ and
(--1)nA~{C~.r+~--~v.m+k}>0 ( n , k = 0 , 1 . . . . ). (6.1) I n particular, c and ~ can be chosen to be ] and y in either order.
Proo]. W i t h P(x)= 89 Y~(x)] a n d p ( x ) = 89 Y~(x)], we have to show t h a t (5.4) and (5.19) are satisfied.
To verify (5.4) we obtain from [16, p. 446]
~( - 1)nP(n)(X)a~ ~4_ f / {( _ 1)n[Ko(n_l,(2x sinh t)] (2 sinh t) n-1 t a n h t}
• {(cosh 2~t) ( - 2 t a n h 2rt - 4rt seth ~ 2~t) + (2t sinh 2rt) (tanh t - 2~ t a n h 2~t)} dt.
70 L . L O R C H A N D P . S Z E G O
R e g a r d i n g this integral, we note t h a t t h e expression in t h e first set of braces is ne.
gative, since ( - 1)nK0(~)(T) > 0 for all positive T, a n d t h a t t h e expression in t h e second set of braces is also negative, since v ~> 1 a n d t a n h T ~' 1 as T--> ~ . Hence
8 ( - 1)np(~)(x)>0 for r~> 89 (6.2)
~v
a n d so, since 1 ~/~ < v, we see t h a t ( - 1 ) n P ( n ) ( x ) > ( - 1 ) n p ( n ) ( X ) > 0, which is (5.4).
To establish (5.19), we note t h a t b o t h
p(x)
a n dP(x)
equal 1 +O(x -2)
as x - - > ~ [15, p. 449 (1)], so t h a t1 1
O(s -2)
as s - - > ~ .p(s) P(s)
Thus, (5.19) is satisfied, since
f~ {p(8) P(8)l }ds~O(v_l)
a s v ---~ rThis being t h e case, Corollary 5.1 applies a n d t h e t h e o r e m is proved.
7. Remarks. Open Problems
W e c o m m e n t here on some v a r i a n t s of our results a n d call a t t e n t i o n to some f u r t h e r problems.
(i) The results stated in w 2 a n d w 5 assert strict positivity of certain quantities. I t m a y be useful for other applications to record parallel results in which, instead, non- n e g a t i v i t y is assumed a n d inferred. This involves replacing " > " b y ">~" in t h e l e m m a s a n d theorems of these sections. I t is essential to observe, however, t h a t for the case n = 0 of (2.3) a n d (5.4), i.e., for the m a p p i n g function
p(x)
itself, we m u s t retain strict positivity, so t h a t t h e m a p p i n g exists, even t h o u g h for n = 1,2, ..., N, only n o n - n e g a t i v i t y is assumed.{Suppose
p(xo)=0.
T h e nyl(XO)=y2(xo)=0
a n d the W r o n s k i a n of Yl, Y2 would vanish forx =x o,
c o n t r a r y to t h e a s s u m p t i o n t h a t Yl, Y2 are linearly independent.)(ii) We h a v e dealt explicitly only with those zeros of solutions of (2.1) a n d (5.3) which lie inside t h e interval of definition. This was primarily for ease of statement.
E x c l u d i n g e n d - p o i n t zeros, if any, assured us t h a t the correspondence between t h e zeros of
y(x)
a n d u(t) (as given b y L e m m a 2.3) is one-to-one. A v a r i e t y of other restrictions would do t h e same. F o r instance, ifx'($)
is b o u n d e d a w a y f r o m 0 as t a p p r o a c h e s a n end- point, t h e n t h e requisite one-to-one correspondence would persist even if we include pos- sible end-point zeros. This p a r t i c u l a r condition is satisfied forJr(x),
r > 89 so t h a t our resultsS T U R M - - L I O U V I L L E F U N C T I O : N S 71 for this function can be extended to cover its non-negative zeros rather than, as in Theorem 3.1, only its positive zeros.
(iii) We have been unable to establish an analogue for the case I vl < 89 of Theorem 3.1.
The Sturm comparison theorem shows t h a t the t h e o r e m m u s t .be modified for this instance and numerical calculations suggest the following (which is known for
n = 2
[11; 13; 14]):CO~Z~CTURE.
I / IVl < 89 then (
-1)nAncy, k
> O, k = 1, 2 . . . n = 2, 3 . . .(iv) Many similar conjectures can be advanced; an examination of various tables of zeros will produce them. Appropriate analogues or extensions of Nicholson's integral or H a r t m a n ' s theorems, used in conjunction with an analogue or extension of Theorem 2.1 or Theorem 5.1 m a y settle some of them.
I n some instances, such as the 0-zeros of the Legendre polynomials Pn (cos 0), a n d the H e r m i t e and Laguerre polynomials, the positive zeros appear to form absolutely (rather t h a n c o m p l e t e l y ) m o n o t o n i c sequences; t h a t is, we conjecture for these cases t h a t
all
dif- ferences of the zeros are non-negative.I n this connection, we are h a p p y to t h a n k a n u m b e r of persons who have aided us b y making numerical checks of these conjectures (and of our theorems while t h e y were still conjectures): (1) Messrs. Samuel C. Arthur, Lovell Moore and R o b e r t Robinson for hand calculations involving the differences of zeros of Bessel and Airy functions, done in 1957 while t h e y were students at Philander Smith College; (2) Professor D. H. Lehmer for recomputing certain zeros of Bessel functions which t u r n e d out to have been recorded erroneously in [15] (Corrections noted also in [5, vol. 2, p. 925]); (3) Drs. P. J. Davis a n d P. l~abinowitz [4
(a),
p. 435; (b), p. 619] for verifying, at our request, t h a t all the differences of the 0-zeros of the Legendre polynomials P ~ (cos 0) and P~4 (cos 0) are non-negative, and for noting t h a t the higher differences of the zeros ofP~(x)
do not follow this pattern;(4) the University of Alberta Computing Centre for verifying t h a t all the differences of the positive zeros of the Laguerre polynomials of degrees 6 through 15, inclusive, and of the H e r m i t e polynomials of degrees 6 through 20, inclusive, are non-negative.
The problem of generalizing the Sturm comparison theorem has been mentioned in footnote (1), p. 56.
8. Appendix I: A remark on completely monotonic functions A function
p(x)
is said to becompletely monotonic
in(a, b)
[16, p. 145] if(-1)np(~(x)>~O, a < x < b (n=O, 1,2 .... ).
(8.1) I n connection with the application of H a r t m a n ' s work to the proofs of Theorems 3.1 and 4.1, we used the following result, which we prove now:7 2 L. LORCH AND P. SZEGO
I / p(x) is completely monotonic in
(0,oo),then (-1)'p('~(x) >0,
0 < x < oo, n = 0 , 1 , 2 .... ,unless p(x) is a constant.
Proo/.
Suppose there exists a n o n - n e g a t i v e integer N a n d a positive n u m b e r ~: s u c h t h a t p (N~ (~) = 0. T h e n p (N~(x) = 0 for all x ~> $, since the function ( - 1)Np(N~ (X)
is n o n - n e g a t i v e a n d non-increasing.Now, i0(x) and, with it,
p(~(x)
are a n a l y t i c in (0, oo) [16, p. 146], so t h a tp(~)(x) =0
for 0 < x < oo. Hence,p(x)
is a polynomial for 0 < x < oo. B u tp{x) >~
0, so t h a t t h e highest degree t e r m in this polynomial has a positive coefficient (unlessp(x}
is identically zero, in which case t h e t h e o r e m is obvious).I f
p(x)
be n o t constant, t h e n the highest degree t e r m inp'(x)
also would h a v e a positive coefficient, a n dp'(x)
would assume positive values for all large x, contradicting t h e defini- tion (8.1) when n = l . This proves the result.Remark.
The above result can be shown to be e q u i v a l e n t to t h e following: I f {Pk}~ is a completely m o n o t o n i c sequence (i.e., if ( -1)nAnpk >~ 0, n, k = 0,1,2, ...), t h e n ( -1)nA=/~k > 0 for n, k = 0 , 1 , 2 , ..., unless Pl =P2 = ..- =/x= = ....A direct, e l e m e n t a r y proof of this latter s t a t e m e n t is provided in [9].
9. Appendix II: A remark on Sturm-iJouville equations
I n applying H a r g m a n ' s w o r k to the proofs of Theorems 3.1 a n d 4.1, we m a d e use also of t h e following l e m m a whose proof we s u p p l y now:
Let yl(x), y~(x) be two linearly independent solutions o/the Sturm-Liouville di//erential equation y" +/(x) y = O. I / p ( x ) ~
[yl(x)] 2 + [y2(x)] 2is identically a constant, say
Z,then /(x) is identically a constant also, namely Z -2.
Proo/.
Clearly Z > 0. W i t h o u t loss of generality, we suppose the solutions Yl, Y2 such t h a t their W r o n s k i a n is 1. Then, f r o m L e m m a 2.3, we t r a n s f o r m t h e given differential e q u a t i o n into t h e differential e q u a t i o nu"(t)+u(t)=O,
withx ' ( t ) = p ( x ) = z ,
b y writingy(x) =[x'(t)]89
T h e solution of this l a t t e r e q u a t i o n isu ( t ) = A cos(t-b), A, b
constants.:Now,
x(t)=
Zt + 6, a n d soy(x) = Z89 = Z89
cos ( Z - i x - b - diZ- 1).S u b s t i t u t i n g this in t h e given differential e q u a t i o n t h e n yields t h e desired result.
STURM--LIOUVILLE FUNCTIONS 73
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Received J u n e 24, 1962
