Excerpt
Chapter 1
Orthogonal polynomials
1.1
Introduction
Markov and Stieltjes in the late 19th century established the first results on the behaviour of the
zeros of orthogonal polynomials, and this field of research has held the attention of theoreticians,
functional and numerical analysts ever since then. Some of the tools and techniques developed to
analyse different properties of the zeros are Markov's theorem on the monotonicity of zeros ([26]);
Sturm's comparison theorem for the zeros of solutions of second order differential equations
([?]); Obrechkoff's theorem on Descartes' rule of sign ([30]); and the Wall-Wetzel theorem on
eigenvalues of Jacobi matrices ([36]).
There is an extensive literature in the field of orthogonal polynomials which is particularly
concerned with detailed investigation of zeros and related questions (e.g., [29], [34]). Most results
on the zeros of the classical orthogonal polynomials are local in nature, such as inequalities,
asymptotic expansions, and monotonicity properties. For instance, it is well known that if
{p
n
}
1
n=0
is any orthogonal sequence, then the zeros of p
n
are real and simple and each open
interval with endpoints at successive zeros of p
n
contains exactly one zero of p
n-1
; a property
called the interlacing of zeros.
Stieltjes (cf.
[?], Theorem 3.3.3) extended this interlacing
property by proving that if m < n - 1, provided p
m
and p
n
have no common zeros, there
exist m open intervals, with endpoints at successive zeros of p
n
, each of which contains exactly
one zero of p
m
. Beardon (cf. [5], Theorem 5) proved that one can say more, namely, for
each m < n - 1, if p
m
and p
n
are co-prime, there exists a real polynomial S
n-m-1
of degree
n - m - 1 whose real simple zeros together with those of p
m
, interlace with the zeros of p
n
.
The polynomials S
n-m-1
are the dual polynomials introduced by de Boor and Saff in [6] or
equivalently, the associated polynomials analysed by Vinet and Zhedanov in [24].
In recent years, authors including Ismail and Muldoon (cf. [14]), Krasikov (cf. [16]), Gupta
and Muldoon (cf. ), Segura , Ismail and Li (cf. ), and Dimitrov and Nikolov (cf. ) have devel-
oped interesting methods including the use of chain sequences and the derivation of inequalities
for real-root polynomials to refine and improve upper and lower bounds for extreme zeros of
classical orthogonal polynomials.
In this work, we make attempt to gather different results by many mathematicians who
have made landmark achievements on studies on zeros of Laguerre polynomials. We discuss
bounds for extreme zeros of some classical orthogonal polynomials, Bounds for zeros of the
Laguerre polynomials where we quote the works of Ilia Krasikov [25], Limit relation for the
complex zeros of Laguerre polynomials, On the Asymptotic Distribution of the Zeros of Laguerre
Polynomials, Monotonicity of zeros of Laguerre polynomials, Zeros of linear combinations of
2
Laguerre polynomials from different sequences, Linear combinations of Laguerre polynomials of
the same degree, Linear combinations of Laguerre polynomials of different degree and Convexity
of the extreme zeros of Laguerre polynomials
1.2
Preliminaries
Polynomials
A polynomial in a single indeterminate can be written in the form
a
n
x
n
+ a
n-1
x
n-1
+ ... + a
2
x
2
+ a
1
x + a
0
where a
0
, ...a
n
are numbers, or more generally elements of a ring, and x is a symbol which is
called an indeterminate or, for historical reasons, a variable. The symbol x does not represent
any value, although the usual (commutative, distributive) laws valid for arithmetic operations
also apply to it. This can be expressed more concisely by using summation notation:
n
i=0
a
i
x
i
That is, a polynomial can either be zero or can be written as the sum of a finite number
of non-zero terms. Each term consists of the product of a number - called the coefficient of
the term - and a finite number of indeterminates, raised to integer powers. The exponent on
an indeterminate in a term is called the degree of that indeterminate in that term; the degree
of the term is the sum of the degrees of the indeterminates in that term, and the degree of a
polynomial is the largest degree of any one term with nonzero coefficient. Since x = x
1
, the
degree of an indeterminate without a written exponent is one. A term and a polynomial with no
indeterminates are called respectively a constant term and a constant polynomial; the degree of
a constant term and of a nonzero constant polynomial is 0. The degree of the zero polynomial
(which has no term) is not defined.
A polynomial function is a function that can be defined by evaluating a polynomial. A
function f of one argument is called a polynomial function if it satisfies
f (x) = a
n
x
n
+ a
n-1
x
n-1
+ ... + a
2
x
2
+ a
1
x + a
0
for all arguments x, where n is a non-negative integer and a
0
, a
1
, a
2
, ...a
n
, are constant coeffi-
cients. For example, the function f : R R, taking real numbers to real numbers, defined
by
f (x) = x
3
- x
is a polynomial function of one variable. Polynomial functions of multiple variables can also be
defined, using polynomials in multiple indeterminates, as in
f (x, y) = 2x
3
+ 4x
2
y = xy
5
+ y
2
- 7.
An example is also the function
f (x) = cos (2 arccos x)
3
which, although it doesn't look like a polynomial, is a polynomial function on [-1, 1] since for
every x from [-1, 1] it is true that
f (x) = 2x
2
- 1
(see Chebyshev polynomials). Polynomial functions are a class of functions having many im-
portant properties. They are all continuous, smooth, entire, computable, etc
Orthogonality
By using integral calculus. it is common to use the following to define the
inner product of two functions f and g:
f, g
w
=
b
a
f (x)g(x)w(x)dx
Here we introduce a nonnegative weight function w(x) in the definition of this inner product.
In simple cases, w(x) = 1, exactly.
We say that these functions are orthogonal if their inner product is zero:
f, g
w
=
b
a
f (x)g(x)w(x)dx = 0
Hence the members of a set of functions {f
i
, i = 1, 2, 3, ...} are:
· orthogonal on the closed interval [a, b] if
f
i
, f
j w
=
b
a
f
i
(x)f
j
(x)w(x)dx = ||f
i
||
2
i,j
= ||f
j
||
2
i,j
· orthonormal on the interval [a, b] if
f
i
, f
j w
=
b
a
f
i
(x)f
j
(x)w(x)dx =
i,j
where
i,j
=
1
if i = j
0
if i = j
is the "Kronecker delta" function.
1.3
Definitions of Orthogonal Polynomials
A system of polynomials {P
n
} which satisfy the condition of orthogonality
b
a
P
n
(x)P
m
(x)(x)dx = 0,
n = m
whereby the degree of every polynomial P
n
is equal to its index n, and the weight function
(weight) (x) on the interval (a, b) or (when a and b are finite) on [a, b]. Orthogonal polynomials
4
are said to be orthonormalized, and are denoted by { ^
P
n
}, if every polynomial has positive leading
coefficient and if the normalizing condition
b
a
^
P
2
n
(x)(x)dx = 1
is fulfilled.
1.4
Applications
Orthogonal polynomials have very useful properties in the solution of mathematical and phys-
ical problems. Orthogonal polynomials provide a natural way to solve, expand, and interpret
solutions to many types of important differential equations.
Applications of orthogonal polynomials are found in varying degrees in
· Quadrature
· Numerical Interpolation
· Estimation of Error in Pad´
e approximations
· Solutions to ordinary and partial differential equations
· Rational approximations and equilibrium distributions
· Factorization of second order difference equations
· Gauss quadrature for analytic functions
· Electrostatic interpretation of zeros of functins
·
5
Chapter 2
The Laguerre Polynomials
2.1
Introduction
The Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are solutions of La-
guerre's equation:
xy + (1 - x)y + ny = 0
2.2
Recursive relation, closed form, and generating func-
tion
One can also define the Laguerre polynomials recursively, defining the first two polynomials as
L
0
(x) = 1
L
1
(x) = 1 - x
and then using the following recurrence relation for any k 1:
L
k+1
(x) =
1
k + 1
((2k + 1 - x)L
k
(x) - kL
k-1
(x))
The closed form is
The generating function for them likewise follows,
2.3
The first few terms of Laguerre polynomials
n
L
n
(x)
0
1
1
-x + 1
2
1
2
(x
2
- 4x + 2)
3
1
6
(-x
3
+ 9x
2
- 18x + 6)
4
1
24
(x
4
- 16x
3
+ 72x
2
- 96x + 24)
5
1
120
(-x
5
+ 25x
4
- 200x
3
+ 600x
2
- 600x + 120)
6
1
720
(x
6
- 36x
5
+ 450x
4
- 2400x
3
+ 5400x
2
- 4320x + 720)
6
Chapter 3
Zeros of Laguerre Polynomials
3.1
Introduction
3.2
Bounds for extreme zeros of some classical orthogo-
nal polynomials
The following result is from
"Bounds for extreme zeros of some classical orthogonal polynomials" by K Driver, Department
of Mathematics and Applied Mathematics, University of Cape Town, Private Bag X3, Ronde-
bosch 7701, Cape Town, South Africa and K Jordaan, Department of Mathematics and Applied
Mathematics, University of Pretoria, Pretoria, 0002, South Africa
Theorem 3.2.1. Let {p
n
}
n=0
be any sequence of polynomials orthogonal on the (finite or infi-
nite) interval (c, d). Let g
n-k
be any polynomial of degree n - k that satisfies, for each n N ,
and k {1, . . . , n - 1},
f (x)g
n-k
(x) = G
k
(x)p
n
(x) + H(x)p
n+1
(x)
(3.2.1)
where f (x) = 0 for x (c, d) and H(x), G
k
(x) are polynomials with deg(G
k
) = k. Then, for
each fixed n N ,
1. the n real, simple zeros of G
k
g
n-k
interlace with the zeros of p
n+1
if g
n-k
and p
n+1
are
co-prime;
2. if g
n-k
and p
n+1
are not co-prime and have r common zeros, counting multiplicity, then
(a) r min{k, n - k};
(b) these r common zeros are simple zeros of G
k
;
(c) no two successive zeros of p
n+1
, nor its largest or smallest zero, can also be zeros of
g
n-k
;
(d) the n - 2r zeros of G
k
g
n-k
, none of which is also a zero of p
n+1
, together with the r
common zeros of g
n-k
and p
n+1
, interlace with the n+1-r remaining (non-common)
zeros of p
n+1
.
7
Proof. Let w
n+1
< . . . < w
1
denote the zeros of p
n+1
.
1 From (2), provided p
n+1
(x) = 0, we have
f (x)g
n-k
(x)
p
n+1
(x)
= H(x) +
G
k
(x)p
n
(x)
p
n+1
(x)
.
(3.2.2)
Further,
p
n
(x)
p
n+1
(x)
=
n+1
j=1
A
j
x - w
j
where A
j
> 0 for each j {1, . . . , n + 1} (cf. [22, Theorem 3.3.5]). Therefore (3) can be
written as
f (x)g
n-k
(x)
p
n+1
(x)
= H(x) +
n+1
j=1
G
k
(x)A
j
x - w
j
, x = w
j
(4)
(3.2.3)
Since p
n+1
and p
n
are always co-prime while p
n+1
and g
n-k
are co-prime by assumption,
it follows from (2) that G
k
(w
j
) = 0 for any j {1, 2, . . . , n + 1}. Suppose that G
k
does
not change sign in the interval I
j
= (w
j+1
, w
j
) where j {1, 2, . . . , n}. Since A
j
> 0 and
the polynomial H is bounded on I
j
while the right hand side of (4) takes arbitrarily large
positive and negative values on I
j
, it follows that g
n-k
must have an odd number of zeros
in each interval in which G
k
does not change sign. Since G
k
is of degree k, there are at
least n - k intervals (w
j+1
, w
j
), j {1, . . . , n} in which G
k
does not change sign and so
each of these intervals must contain exactly one of the n - k real, simple zeros of g
n-k
.
We deduce that the k zeros of G
k
are real and simple and, together with the zeros of g
n-k
,
interlace with the n + 1 zeros of p
n+1
.
2 If r is the total number of common zeros of p
n+1
and g
n-k
counting multiplicity then
each of these r zeros is a simple zero of p
n+1
and it follows from (2) that any common
zeros of g
n-k
and p
n+1
must also be zeros of G
k
since p
n
and p
n+1
are co-prime. Therefore,
r min{k, n-k} and there must be at least (n-2r) open intervals of the form I
j
= (w
j+1
,
w
j
) with endpoints at successive zeros of p
n+1
where neither w
j+1
nor w
j
is a zero of g
n-k
or G
k
(x).
If G
k
does not change sign in an interval I
j
= (w
j+1
, w
j
), it follows from (4), since A
j
> 0
and H is bounded while the right hand side takes arbitrarily large positive and negative
values for x I
j
, that g
n-k
must have an odd number of zeros in that interval. Therefore,
in at least (n - 2r) intervals I
j
either g
n-k
or G
k
, but not both, must have an odd number
of zeros counting multiplicity. On the other hand,g
n-k
and G
k
have at most (n - k - r)
and (k - r) real zeros respectively that are not zeros of p
n+1
. We deduce that there must
be at most (n - 2r) intervals I
j
= (w
j+1
, w
j
) with endpoints at successive zeros w
j+1
and
w
j
of p
n+1
neither of which is a zero of g
n
- k. It is straightforward to check that if the
number of intervals I
j
= (w
j+1
, w
j
) with endpoints at successive zeros of p
n+1
neither of
which is a zero of g
n-k
is equal to n - 2r, this is only possible if no two consecutive zeros
of p
n+1
, nor the largest or smallest zero of p
n+1
, can be common zero(s) of p
n+1
and g
n-k
.
This proves a) to c) and d) follows from c).
8
3.3
Bounds for zeros of the Laguerre polynomials
The following is extracted from "Ilia Krasikov, Bounds for zeros of the Laguerre polynomials,
Journal of Approximation Theory 121 (2003) 287-291; http://www.elsevier.com/locate/jat
Theorem 3.3.1. Let x
1
and x
k
be the least and the largest zeros of L
()
k
(x), respectively. For
k 7, 8 the following inequalities hold:
x
1
> s - r +
(s - r)
2/3
2r
1/3
,
(3.3.1)
x
k
< s + r +
(s + r)
2/3
2r
1/3
,
(3.3.2)
where
s = 2k + + 1, r =
4k
2
+ (2k - 1)(2k + 2)
More precisely, all the zeros of L
()
k
(x) are confined between the only two real roots of the
following equation:
x
2
- 2sx + b
2
- 1) - 4sx
3
+ 9s
2
x
2
+ (b
2
- 1)(b
2
- 1 - 6sx) = 0
(3.3.3)
where b = + 2
It looks plausible that, up to the factor
1
2
, Theorem 1 gives the correct value of the second
term of the corresponding asymptotics.
Proof. A real entire function (x) is in the Laguerre-Polya class L - P if it has a representation
of the form
(x) = cx
m
e
-x
2
+x
k=1
1 +
x
x
k
e
-x/x
k
( ,
where c, , x
k
are real, 0, m is a nonnegative integer and
x
-2
k
< . Our main tool will
be the following inequality valid for any f L - P [7,9,10],
V
m
(f (x)) =
m
j=-m
(-1)
m+j
f
(m-j)
(x)f
(m+j)
(x)
(m - j)!(m + j)!
0,
m = 0, 1, . . . .
(3.3.4)
We will use m = 2 and set
V = 12V
2
(y) = 3y"
2
- 4y`y"` + yy
(4)
.
Notice that in our case, the positivity (and a plausible connection with the potential theory)
can be seen directly by V =
i=j
(x - x
i
)
-2
(x - x
j
)
-2
, where x
1
, x
2
, . . . are the zeros of y [3].
In the sequel, we deal with the function t = t(x) = y`/y, and set b, r, and s as in Theorem 1
to simplify some expressions. We also assume x > 0. Using differential equation (1) recursively
to express the higher derivatives in V through y and y` we get
2x
3
y
2
V = At
2
+ 2Bt + C,
(3.3.5)
9
where
A = -2x(x
2
- 2sx + (b - 1)(b + 3)),
B = x
3
- (2s + b - 1)x
2
+ (2bs - 3s + (b - 1)(b + 3))x + b - b
3
,
C = (b - s - 1)(x
2
- 2sx - x + b
2
+ b)
Observe that A is positive only for x in the interval (x
a
m
, x
a
m
),
x
a
m,M
= 2k + + 1 ± 2
k
2
+ k + k - - 1.
Let also x
c
m
< x
c
M
be the roots of C.
For the discriminant of the equation At + 2Bt + C = 0; in t we get
(x) = B
2
- AC = (x
2
- 2sx + b
2
- 1)
3
- 4sx
3
+ 9s
2
x
2
+ (b
2
- 1)(b
2
- 1 - 6sx) (3.3.6)
that is exactly expression (6).
We now state two Lemmas that are important for the theorem. theorem Lemma 1.
The equation 9x) = 0 has exactly two real roots x
m
< x
M
, provided k 2 and > -1.
Moreover, x
c
m
< x
a
m
< x
m
< x
M
< x
a
M
< x
c
M
, if k 7, 8. end theorem (See the full
proof of this lemma in "Ilia Krasikov, Bounds for zeros of the Laguerre polynomials, Journal of
Approximation Theory 121 (2003) 287-291; http://www.elsevier.com/locate/jat) begin theorem
Lemma 2.
For k 7, 8, all the zeros of L
()
k
are confined in (x
m
, x
M
) between the only two real roots
of the equation (x) = 0 end theorem
(See the complete proof of this lemma in "Ilia Krasikov, Bounds for zeros of the Laguerre
polynomials, Journal of Approximation Theory 121 (2003) 287-291; http://www.elsevier.com/locate/jat)
Proof of Theorem.
By the previous lemma it is enough to show that inequalities (4) and (5) hold for x
m
and x
M
respectively. Since
s - r +
(s - r)
2/3
2r
1/3
< s < s + r +
(s + r)
2/3
2r
1/3
,
and
(s) = -(s
2
- b
2
+ 1)((s
2
- b
2
)
2
- 3s
2
- b
2
) < 0,
to prove (4) we just check
s ± r +
(s ± r)
2/3
2r
1/3
> 0.
Calculations yield
64r
6
q
4
s - r +
(s - r)
2/3
2r
1/3
= q
8
-32q
5
r-60q
6
r
4/3
-32q
2
r
2
+96q
3
r
4/3
+240q
4
r
8/3
+144r
1
0/3+192qr
1
1/3,
where Since, as it is easy to check, we convince that the above expression is positive. Hence (4)
follows. The proof of (5) is similar using
10
3.4
Limit relation for the complex zeros of Laguerre poly-
nomials
The following result is by Mark V. DeFazio, Dharma P. Gupta, Martin E. Muldoon, on "Limit
relations for the complex zeros of Laguerre and q-Laguerre polynomials", in J. Math. Anal.
Appl. 334 (2007) 977-982; www.elsevier.com/locate/jmaa
Theorem
Let x
1
(), . . . , x
m
() be the m, (2 m n) zeros of L
()
n
(x) in a neighbourhood of x = 0 for
-m. Then
lim -m
m
k=1
1
x
k
()
=
m(m - 2n - 1)
m
2
- 1
.
(3.4.1)
Proof. Let be close to -m. Let us number the zeros of L
(
)
n
(x) so that x
1
, . . . , x
m
are
near zero and x
m+1
, . . . , x
n
are near the positive real zeros of L
(
m)
n-m
(x). The explicit formula
(2) yields
n
k=1
1
x
k
()
=
n
+ 1
,
= -1, . . . , -n.
(3.4.2)
This is readily seen by considering the factorization
(n + n)
-1
L
()
n
(x) =
1 -
x
x
1
()
. . .
1 -
x
x
n
()
= 1 - x
n
k=1
1
x
k
()
+ . . .,
(3.4.3)
and comparing with the expansion (2):
(n + n)
-1
L
()
n
(x) = 1 -
n
+ 1
x + . . . .
(3.4.4)
Letting -m gives
n
1 - m
= lim
-m
n
k=1
1
x
k
()
= lim
m
k=1
1
x
k
()
+ lim
n
k=m+1
1
x
k
()
,
where the zeros in the first sum are the ones in the neighbourhood of 0. But the zeros in the
second sum on the right approach those of L
(
m)
n-m
(x) and hence, using (8), we get
n
1 - m
= lim
-m
m
k=1
1
x
k
()
+ f racn - m1 + m,
(3.4.5)
which gives (7).
3.5
On the Asymptotic Distribution of the Zeros of La-
guerre Polynomials
The following result is by WOLFGANG GAWKONSKI on "On the Asymptotic Distribution of
the Zeros of Hermite, Laguerre, and Jonqui`
ere Polynomials" in JOURNAL OF APPROXIMA-
TION THEORY 50. 2 14-23 i ( 1987)
11
Theorem
If L
()
n
(z) denotes the Laguerre polynomial of degree n, > -1, and N
n
() is the number of
zeros of L
()
n
(z) not exceeding , > 0, then
lim
n
1
n
N
n
(4n) =
2
0
t
-1/2
(1 - t)
1/2
dt,
0 < 1.
(3.5.1)
Theorem 1 gives the asymptotic number of zeros in intervals of the form (4n
1
, 4n
2
] that is
in intervals the length of which tends to infinity with n. In contrast to this result the question
arises: "how many" zeros are located in a "fixed" interval, (
1
,
2
] say? The precise answer is
given by:
THEOREM 2
Under the assumptions and with the notations of Theorem 1 we have
lim
n
1
n
N
n
() =
2
,
> o.
(3.5.2)
3.6
Monotonicity of zeros of Laguerre polynomials
Denote by x
nk
, k = 1, . . . , n, the zeros of the Laguerre polynomial L
n
(x). We establish mono-
tonicity with respect to the parameter of certain functions involving x
nk
(). As a consequence
we obtain sharp upper bounds for the largest zero of L
n
(x)
Theorem 3.6.1. For every n 2 and each k, k = 1, . . . , n, the quantities
x
nk
() - (2n + - 1)
2(n + - 1)
are increasing functions of , for -1/(n - 1). Moreover, when k = 1 the above function
increases in the entire range (-1, ).
For the proof of above, see ([8])
3.7
Zeros of linear combinations of Laguerre polynomials
from different sequences
We consider linear combinations of Laguerre polynomials L
n
of the form R
,t
n
= L
n
+ aL
+t
n
and
S
,t
n
= L
n
+ bL
+t
n-1
where > -1, t > 0 and a, b = 0. We recall that the Laguerre polynomials
(cf. [34]) are orthogonal with respect to the weight function e
-x
x
, > -1 on the interval (0,
)/.
For 0 < t 2, we give proofs ([23]) for the interlacing of the zeros of R
,t
n
and S
,t
n
with the
zeros of L
n
, L
+t
n
, L
n-1
and L
+t
n-1
.
We will make use of two well known identities (cf. [1], 22.7.30 and 22.7.29)
L
n
= L
+1
n
- L
+1
n-1
(3.7.1)
xL
+1
n
(x) = (x - n)L
n
+ ( + n)L
n-1
.
(3.7.2)
12
3.7.1
Linear combinations of Laguerre polynomials of the same de-
gree
Theorem ([23])
Let
R
,t
n
= L
n
+ aL
+t
n
,
> -1.
For 0 < t 2, the zeros of R
,t
n
interlace with the zeros of (i) L
n
, L
+t
n
.
Proof ([23])
We have from [[13], Theorem 2.3] that the zeros of L
n
interlace with the zeros of L
+t
n
for
0 < t 2 which implies that L
n
has a different sign at successive zeros of L
+t
n
and vice versa.
Evaluating (3) at successive zeros x
i
and x
i+1
of L
n
we obtain
R
,t
n
(x
i
)R
,t
n
(x
i+1
= a
2
L
+t
n
(x
i
)L
+t
n
(x
i+1
),
i = 1, 2, . . . , n - 1
< 0f oralla = 0.
Therefore R
,t
n
has a different sign at successive zeros of L
n
and so the zeros interlace. The
same argument shows that the zeros of R
,t
n
interlace with those of L
+t
n
by evaluating (3) at
successive zeros of L
+t
n
.
3.7.2
Linear combinations of Laguerre polynomials of different de-
gree
Next we consider linear combinations of the type
S
,t
n
= L
n
+ bL
+t
n-1
,
b = 0, > -1
(3.7.3)
We will need information on the interlacing properties of the two polynomials L
n
and L
+t
n-1
in
the linear combination. Theorem ([23])
Let > 1 and let
0 < x
1
< x
2
< . . . < x
n
be the zeros of L
n
0 < y
1
< y
2
< . . . < y
n-1
be the zeros of L
n-1
0 < t
1
< t
2
< . . . < t
n-1
be the zeros of L
+t
n-1
and
0 < X
1
< X
2
< . . . < X
n-1
be the zeros of L
+2
n-1
where 0 < t < 2. Then 0 < x
1
< y
1
< t
1
< X
1
< x
2
< . . . < x
n-1
< y
n-1
< t
n-1
< X
n-1
< x
n
Proof ([23])
A simple computation using (1) and (2) leads to
( + 1)L
+1
n
(x) = ( + n + 1)L
n
(x) + xL
+2
n-1
.
(3.7.4)
13
Evaluating (7) at successive zeros x
k
and x
k+1
of L
n
(x), we obtain
x
k
x
k+1
L
+2
n-1
(x
k
)L
+2
n-1
(x
k+1
) = ( + 1)
2
L
+1
n
(x
k
)L
+1
n
(x
k+1
).
The expression on the right is negative since the zeros of L
n
and L
+1
n
interlace (cf. [12],
Theorem 2.3]) and therefore
x
k
< X
k
< x
k+1
for each fixed k, k = 1, . . . , n - 1.
The zeros of L
n-1
increase as increases (cf. [34], p. 122), hence y
k
< t
k
< X
k
for each fixed
k, k = 1, . . . , n - 1
Finally, since the zeros of L
n
and L
n-1
separate each other, we know that
x
k
< y
k
< x
k+1
for each fixed k, k = 1, . . . , n - 1
and this completes the proof
Theorem ([23])
For 0 < t 2, the zeros of S
,t
n
interlace with the zeros of (i) L
n
, (ii) L
+t
n-1
.
Proof ([23])
We know from Theorem 3.1 that the zeros of L
n
interlace with the zeros of L
+t
n-1
for 0 < t 2
which implies that L
+t
n-1
has a different sign at successive zeros of L
n
and vice versa. Evaluating
(6) at successive zeros x
i
and x
i+1
of L
n
we obtain
S
,t
n
(x
i
)S
,t
n
(x
i+1
) = b
2
L
+t
n-1
(x
i
)L
+t
n-1
(x
i+1
,
i = 1, 2, . . . , n - 1
< 0, b = 0.
Therefore S
,t
n
has a different sign at successive zeros of L
n
and so the zeros interlace. The
same argument shows that the zeros of S
,t
n
interlace with those of L
+t
n-1
by evaluating (6) at
successive zeros of L
+t
n-1
.
3.8
Convexity of the extreme zeros of Laguerre polyno-
mials
The convexity theorem as noted by Sturm in ([32] and cited by [7]), can be summarised as
follows.
Theorem
Let y"(t) + F (t)y(t) = 0 be a second-order differential equation in normal form, where F is
continuous in (a, b). Let y(t) be a nontrivial solution in (a, b), and let x
1
< . . . < x
k
< x
k+1
< . . .
denote the consecutive zeros of y(t) in (a, b). Then
1. if F (t) is strictly increasing in (a, b), x
k+2
- x
k+1
< x
k+1
- x
k
,
2. if F (t) is strictly decreasing in (a, b), x
k+2
- x
k+1
> x
k+1
- x
k
.
3. if there exists M > 0 such that F (t) < M in (a, b) then
x
k
x
k+1
- x
k
>
M
,
4. if there exists m > 0 such that F (t) > m in (a, b) then
x
k
<
m
,
14
We say that the zeros of y are concave (convex) on (a, b) for the first (second) case. Hence we
present the following theorem by
Theorem
The zeros of L
n
(t) on (0, 1) are
1. all convex if n > 0 and -1 < 3
2. all convex if > 3 and 0 < n <
+1
-3
3. concave for t < t
0
and convex for t > t
0
when > 3, n >
+1
-3
and t
0
is defined by (3).
Moreover, for the distance between consecutive zeros we have the general estimate
x
k
>
2
2n + + 2n
2
+ 2n + 1
k = 1, . . . , n - 1
(3.8.1)
and also if x
k
> t
0
then
x
k
>
F (x
k
)
k = 1, . . . , n - 1
(3.8.2)
and
x
k
>
F (x
k+1
)
k = 1, . . . , n - 1
(3.8.3)
Conclusion where F is defined by (2).
Proof
For || < 1, t
0
< 0, hence F (t) will be decreasing on (0, ). When 1, F (t) is
increasing on (0, t
0
) and decreasing on (t0, 1). Let x
1
denote the smallest zero of L - n
, then
we know that x
1
>
+1
n
(see [19]). This implies that when t
0
<
+1
n
, F (t) will be decreasing on
the interval (x
1
, ). An easy calculation shows that this condition is equivalent to either 3
or > 3 and n <
+1
-3
. The estimates on the distance x
k
follow from Theorem 2.1(3),(4). The
maximum of F is at t
0
and F (t
0
) > 0, therefore we can take F (t
0
) as M to obtain (4). For (5)
and (6), we use the fact that when x
k
> t
0
, F is monotone decreasing on (x
k
, x
k+1
). In fact,
F is monotone decreasing on (0, 1) and tends to -1/4 as t , so there is exactly one point
t
1
on (t
0
, ), where F crosses the x-axis. The form of the differential equation implies that if
F (t) < 0 and y(t) > 0, the graph will be concave up and similarly, if y(t) < 0, the graph will
be concave down.
Hence there can be at most one zero of the Laguerre polynomial to the right of t
1
. This
means that F (x
n-1
) is positive, but F (x
n
) may be negative and therefore the index in (6) only
runs up to n - 2.
Conclusion
15
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