PROCEEDINGS of the
AMERICAN MATHEMATICAL SOCIETY
Volume 83, Number 4, December 1981
A MONOTONIC PROPERTY FOR THE ZEROS
OF ULTRASPHERICAL POLYNOMIALS
ANDREA LAFORGIA1
Abstract.
It is shown that Xx¡$ increases as X increases for 0 < X < 1, k =
I, 2,...,
[|J where xfy is the kth positive zero of ultraspherical polynomial
P?\x).
The aim of this work is to prove the following
Theorem. Let xjfy, k = 1, 2, . . ., [f ], be the zeros of the ultraspherical polynomial
F„(X)(x)in decreasing order on (0, 1), where 0 <X < 1.
Then for every e > 0,
XxfX< (X+ e)x£r\
*=l,2,...,[f].
Remark 1. For our purposes the following form of Sturm comparison theorem
will prove useful. This formulation differs from the usual formulation [2, p. 19] in
that fix) < F(x) is hypothesized for the interval a < x < Xm, rather than for the
larger interval a < x < xm. (See the work [1] for the proof of this formulation of
Sturm theorem.)
Lemma. Let the functions y(x) and Y(x) be nontrivial solutions of the differential
equations
y"(x) + fix)y(x) = 0;
Y"(x) + F(x) Y(x) = 0
and let them have consecutive zeros at xx, x2, .. . , xm and Xx, X2, . . . , Xm respectively on an interval (a, b). Suppose that fix) and F(x) are continuous, that fix) <
F(x), a < x < Xm, and that
(0
lim+ [y'(x)Y(x)
- y(x)Y'(x)]
x—>cr
= 0.
-1
Then
Xk < xk,
Proof of the theorem.
k = 1, 2, . . . , m.
The function
«(x)
= (1 - x2)A/2+1/4P^(x)
Received by the editors March 6, 1981.
1980 Mathematics Subject Classification. Primary 33A45; Secondary 34A50.
Key words and phrases. Zeros of ultraspherical polynomials, Sturm comparison theorem.
'This research was supported by Consiglio Nazionale delle Ricerche.
© 1981 American Mathematical
0002-9939/81/0000-0568/01.50
757
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Society
758
ANDREA LAFORGIA
which has the same zeros on (-1, 1) as P^\x)
equation
v"(x) + px(x)y(x)
satisfies [2, p. 82] the differential
= 0
where
, *
■
(n + X)2
2 + 4X - 4a2 + x2
1 - x1
4(1 - x2)2
The functions u(x/X) and u(x/(X + e)) that have on the interval (0, X)
and (0, X + e) consecutive zeros at Ax*j>,. . . , Xx$„/2] and (X + e)x£x+e\ . . . ,
(X + e)x£[*/e2)]
respectively, satisfy the differential equations
z"(x) + &(*>(*)
= 0,
W"(x) + tpx+e(x)w(x) = 0
where ^„(x) = V^p,^).
It is easy to show that \j/x(x) decreases as X increases for 0 < X < 1 and
0 < x < X, that is if^x(x) > ^\+e(x), e > 0. The limit condition (1) is easily shown
to hold in the present case. Thus the above Lemma is applicable and its conclusion
gives the desired result.
Remark 2. Our result contrasts with x^ > *£*+€) which follows from formula
9x$/8A < 0, k = 1, 2, . . . , [f ]. Putting this result together with the new result
gives
xn,k
1<^)<1+X'
,
e
,
, „
*=1>2'--"
An,k
r/i
2 ■
L
J
Acknowledgement.
We are grateful to Professor Martin E. Muldoon for his
interest and for useful comments.
References
1. S. Ahmed, A. Laf orgia and M. E. Muldoon, On the spacing of the zeros of some classical orthogonal
polynomials, J. London Math. Soc. (to appear).
2. G. Szegö, Orthogonal polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math.
Soc., Providence, R.I., 1975.
Istituto
di Calooli Numerici, Untversita di Torino, Via Carlo Alberto,
Italy
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10-10123 Torino,