Volume 8 (2007), Issue 1, Article 24, 7 pp.
INEQUALITIES FOR THE SMALLEST ZEROS OF LAGUERRE POLYNOMIALS
AND THEIR q-ANALOGUES
DHARMA P. GUPTA AND MARTIN E. MULDOON
D EPARTMENT OF M ATHEMATICS AND S TATISTICS
YORK U NIVERSITY, T ORONTO , O NTARIO M3J 1P3
C ANADA
muldoon@yorku.ca
Received 03 August, 2006; accepted 28 February, 2007
Communicated by D. Stefanescu
A BSTRACT. We present bounds and approximations for the smallest positive zero of the La(α)
guerre polynomial Ln (x) which are sharp as α → −1+ . We indicate the applicability of the
results to more general functions including the q-Laguerre polynomials.
Key words and phrases: Laguerre polynomials, Zeros, q-Laguerre polynomials, Inequalities.
2000 Mathematics Subject Classification. 33C45, 33D45.
1. I NTRODUCTION
The Laguerre polynomials are given by the explicit formula [13]
#
"
n n
n
k
k
X
X
(−x)
n
+
α
(−x)
n
+
α
(α)
k
(1.1)
Ln (x) =
=
1+
,
n
−
k
k!
n
(α
+
1)
k
k=0
k=1
valid for all x, α ∈ C (with the understanding that the second sum is interpreted as a limit when
α is a negative integer), where
(α + 1)k = (α + 1)(α + 2) · · · (α + k).
They satisfy the three term recurrence relation
(1.2)
(α)
(α)
(α)
(α)
xLn (x) = −(n + 1)Ln+1 (x) + (α + 2n + 1)Ln (x) − (α + n)Ln−1 (x),
(α)
(α)
with initial conditions L−1 (x) = 0 and L0 (x) = 1 for all complex α and x. When α > −1,
this recurrence relation is positive definite and the Laguerre polynomials are orthogonal with
(α)
respect to the weight function xα e−x on [0, +∞). From this it follows that the zeros of Ln (x)
are positive and simple, that they are increasing functions of α and they interlace with the zeros
(α)
of Ln+1 (x) [13]. When α ≤ −1 we no longer have orthogonality with respect to a positive
weight function and the zeros can be non-real and non-simple.
210-06
2
D HARMA P. G UPTA AND M ARTIN E. M ULDOON
Our purpose here is to present bounds and approximations for the smallest positive zero of
α > −1, which are sharp as α → −1+ . The same kinds of results hold for more
(α)
general functions including the q-Laguerre polynomials Ln (x; q), 0 < q < 1 which satisfy
(α)
(α)
Ln (x(1 − q)−1 ; q) → Ln (x) as q → 1− .
(α)
Ln (x),
2. S MALLEST Z EROS OF L AGUERRE P OLYNOMIALS
In the case α > −1, successively better upper and lower bounds for the zeros of Laguerre
polynomials can be obtained by the method outlined in [7]. They follow from the knowledge
(α)
of the coefficients in the explicit expression for Ln (x). However, they are obtained more
(α)
conveniently by noting that y = Ln (x) satisfies the differential equation
xy 00 + (α + 1 − x)y 0 + ny = 0
(2.1)
and hence that u = y 0 /y satisfies the Riccati type equation
xu02 + (α + 1 − x)u + n = 0.
(2.2)
If we write
y=
(2.3)
n n+α Y
x
,
1−
n
x
i
i=1
where the zeros xi satisfy 0 < x1 < x2 < · · · , then
u=
(2.4)
n
X
i=1
∞
X
1
=−
Sk+1 xk ,
x − xi
k=0
where
Sk =
(2.5)
n
X
x−k
i ,
k = 1, 2, . . . .
i=1
Substituting in (2.2), we get
(2.6)
∞
X
x
k
Sk +
k
X
!
Si Sk−i+1
− (α + k + 1)
i=1
k=1
∞
X
Sk+1 xk + n = 0,
k=0
from which it follows by comparing coefficients that
P
Sk + ki=1 Si Sk−i+1
n
(2.7)
S1 =
, Sk+1 =
, k = 1, 2. . . .
α+1
α+k+1
For the case α > −1, the zeros are all positive and by the method outlined in [7, §3], we have
−1/m
Sm
< x1 < Sm /Sm+1 , m = 1, 2, . . . .
(2.8)
These upper and lower bounds give successively improving [7, §3] upper and lower bounds for
x1 . For example, for α > −1, n ≥ 2, we get, for the smallest zero x1 (α),
1
x1 (α)
(α + 2)
<
<
,
n
α+1
(α + 1 + n)
(2.9)
(2.10)
α+2
n(n + α + 1)
12
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 24, 7 pp.
<
x1 (α)
(α + 3)
<
,
α+1
(α + 1 + 2n)
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Z EROS OF L AGUERRE P OLYNOMIALS
3
where the upper bound recovers that in [13, (6.31.12)], and
13
(α + 2)(α + 3)
x1 (α)
<
n(n + α + 1)(2n + α + 1)
α+1
(α + 2)(α + 4)(α + 2n + 1)
(2.11)
.
< 3
α + 4α2 + 5α + 2 + 5nα2 + 16nα + 11n + 5n2 α + 11n2
Further such bounds may be found but they become successively more complicated. From the
higher estimates we can produce a series expansion valid for −1 < α < 0. The first five terms,
obtained with the help of MAPLE, are:
2
3
α+1 n−1 α+1
n2 + 3n − 4 α + 1
(2.12) x1 (α) =
+
−
n
2
n
12
n
4
7n3 + 6n2 + 23n − 36 α + 1
+
144
n
5
4
3
2
293n + 210n + 235n + 990n − 1728 α + 1
−
+ ··· .
8640
n
It is known [13, Theorem 8.1.3] that
lim n−α L(α)
n
z = z −α/2 Jα (2z 1/2 ),
n
2
and hence that x1 ∼ jα1
/(4n) as n → ∞, with the usual notation for zeros of Bessel functions.
Hence we get
α + 1 (α + 1)2 7(α + 1)3 293(α + 1)4
2
(2.14)
jα1 ∼ 4(α + 1) 1 +
−
+
−
+ ··· ,
2
12
144
8640
(2.13)
n→∞
which agrees with the expansion of [12] for jα1 .
It should be noted that the inequalities obtained here are particularly sharp for α close to −1
but not for large α. Krasikov [10] gives uniform bounds for the extreme zeros of Laguerre and
other polynomials.
The series in (2.12) converges for |α + 1| < 1. This suggests that we consider the case
−2 < α < −1, when the zeros are still real but x1 < 0 < x2 < x3 < · · · [13, Theorem 6.73].
In accordance with [7, Lemma 3.3], the inequalities for x1 are changed, sometimes reversed.
For example, we have, for n ≥ 2,
12
x1 (α)
1
α+2
(2.15)
>
>
,
−2 < α < −1.
n
α+1
n(n + α + 1)
3. q E XTENSIONS
In extending the previous results, it is natural to consider some of the q-extensions of the
Laguerre polynomials. For this purpose we need the standard notations [4, 9] for the basic
hypergeometric functions:
X
∞
(a; q)k (n2 ) (−z)k
a q; z =
q
,
1 φ1
b (b; q)k
(q; q)k
k=0
2 φ1
X
∞
(a; q)k (b; q)k z k
a, b q;
z
=
,
c (c; q)k
(q; q)k
k=0
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 24, 7 pp.
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4
D HARMA P. G UPTA AND M ARTIN E. M ULDOON
where (a; q)n denotes the q-shifted factorial
(a; q)0 = 1, (a; q)n = (1 − a)(1 − aq) · · · (1 − aq n−1 ),
so that (1 − q)−k (q α ; q)k → (α)k as q → 1− .
We seek appropriate q-analogues of the results of Section 2, which will reduce to those results
when q → 1. Different q-analogues are possible; we have found that a good approach is
through what we now call the little q-Jacobi polynomials introduced by W. Hahn [6] (see also
[9, (3.12.1), p.192]):
−n
q , abq n+1 (3.1)
pn (x; a, b; q) = 2 φ1
q; xq .
aq
Hahn proved the discrete orthogonality [4, (7.3.4)]
(3.2)
∞
X
pm (q k ; a, b; q)pn (q k ; a, b; q)
k=0
(bq; q)k
(aq)k
(q; q)k
=
(q; q)n (1 − abq)(bq; q)n (abq 2 ; q)∞ (aq)−n
δm,n ,
(abq; q)n (1 − abq 2n+1 )(aq; q)n (aq; q)∞
where 0 < q, aq < 1 and bq < 1. In this case the orthogonality measure is positive and the
zeros of the polynomials lie in (0, ∞). For a detailed study of the polynomials pn (x; a, b; q), we
refer to the article of Andrews and Askey [2], and the book of Gasper and Rahman [4, §7.3]. In
general, the polynomials give a q-analogue of the Jacobi polynomials but, for b < 0, they give
a q-analogue of the Laguerre polynomials; see (3.6) below.
From (3.1), we get [4, Ex.7.43, p. 210]
−n (1 − q)x α
q
n+α+1
lim pn −
; q , b; q = 1 φ1
q; − x(1 − q)q
q α+1 b→∞
bq
(α)
=
(3.3)
Ln (x; q)
(α)
,
Ln (0; q)
(α)
with the notation of [11, 8, 4] for the q-Laguerre polynomials Ln (x; q). This definition
−n (q α+1 ; q)n
q
(α)
n+α+1
(3.4)
Ln (x; q) =
q; − x(1 − q)q
,
1 φ1
q α+1 (q; q)n
gives [11]
−1
(α)
lim L(α)
n ((1 − q) x; q) = Ln (x).
(3.5)
q→1−
(α)
(We remark that the definition of Ln (x; q) given in [9, p. 108] has x replaced by (1 − q)−1 x.)
On the other hand, again from (3.1), we have
(3.6)
α
β
lim pn (1 − q)x; q , −q ; q =
q→1−
n
X
(−n)k (2x)k
k=0
(1 + α)k k!
(α)
(α)
=
Ln (2x)
(α)
.
Ln (0)
(α)
This is reported in [4, (7.3.9)], but with a small error, Ln (x) rather than Ln (2x) on the righthand side. The relation (3.6) shows that little q-Jacobi polynomials also provide a q-analogue
of the Laguerre polynomials. However, we use the name “q-Laguerre polynomials" only for
(α)
Ln (x; q), as defined in (3.4).
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 24, 7 pp.
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Z EROS OF L AGUERRE P OLYNOMIALS
5
The Wall, or little q-Laguerre, polynomials Wn (x; a; q) ([3], [9, 3.20.1]) are the particular
case b = 0 of pn (x; a, b; q):
−n
q , 0 q ; qx ,
(3.7)
Wn (x; a; q) = pn (x; a, 0; q) = 2 φ1
a, q where 0 < q < 1 and 0 < aq < 1. From the Wall polynomials, we can again obtain the
(α)
q-Laguerre polynomials Ln (x; q) using [9, p. 108] (changed to our notation):
(3.8)
Wn (x; q −α q −1 ) =
(q; q)n
−1
L(α)
n ((1 − q) x; q).
α+1
(q ; q)n
From the relation (3.6), we have
(α)
(3.9)
lim−1 Wn ((1 − q)x; q α ; q) =
q→1
Ln (x)
(α)
.
Ln (0)
Here we present in diagrammatic form the relations between the various polynomials considered:
pn (x; a, b; q)
(3.3)
OOO
OO
b=0OOO
(α)
OO'
o (3.8) Wn (x; a; q)
hhhh
r
rrr
hhhhh
h
h
r(3.5) hh(3.9)
xrrrhhhhhhhh
sh
(3.6) L(α)
n (x; q)
Ln (x)
4. B OUNDS FOR q E XTENSIONS
In finding bounds for the zeros of these polynomials, we no longer have available the differential equations method used in Section 2. However we can still apply the Euler method,
described in [7], based on the explicit expressions for the coefficients in the polynomials to
obtain bounds for the smallest positive zero of the little q-Jacobi polynomials. We consider the
function
∞
X
pn ((1 − q)x; a, b; q) = 1 +
ak x k .
k=1
where
(4.1)
ak =
(q −n ; q)k (abq n+1 ; q)k k
q (1 − q)k
(q; q)k (aq; q)k
We can find S1 , S2 , . . . , defined as in (2.5), in terms of a1 , a2 , . . . . As in Section 2, as long as
0 < q, aq < 1, b < 1, we have 0 < x1 < x2 < · · · . Using [7, (3.4),(3.7)], we have S1 = −a1 ,
and
n−1
X
Sn = −nan −
ai Sn−i .
i=1
Using inequalities (2.8) for m = 1, we obtain the following bounds for the smallest positive
zeros x1 (a, b; q) of pn (x(1 − q); a, b; q), where we assume that 0 < q, aq < 1, b < 1:
(4.2)
1
x1 (a, b; q)
(1 + q)(1 − aq 2 )
<
<
,
(1 − q n )(1 − abq n+1 )
q n−1 (1 − aq)
(1 − q)P
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 24, 7 pp.
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6
D HARMA P. G UPTA AND M ARTIN E. M ULDOON
where
P = 1 + aq 2 + q n − 2aq n+1 − aq n+2 − abq n+1
− 2abq n+2 + abq 2n+1 + a2 bq n+3 + a2 bq 2n+3 .
For m = 2 we get improved lower and upper bounds:
1/2
x1 (a, b; q)
(1 + q)(1 − aq 2 )
< n−1
n
n+1
(1 − q )(1 − q)(1 − abq )P
q (1 − aq)
(1 − q 3 )(1 − aq 3 )P
(4.3)
<
,
σ1 + σ2 + σ3
where
(4.4)
σ1 = 3q 3 (1 − aq)2 (1 − q n−1 )(1 − q n−2 )(1 − abq n+2 )(1 − abq n+3 ),
(4.5)
σ2 = (1 − q 3 )(1 − aq)(1 − aq 3 )(1 − q n )(1 − abq n+1 )P
and
(4.6) σ3 = −q(1 + q + q 2 )(1 − aq)(1 − aq 3 )(1 − q n )(1 − q n−1 )(1 − abq n+1 )(1 − abq n+2 ).
As observed earlier, with the help of (3.6) we should be able to derive corresponding inequalities
(α)
for zeros of Ln (x). If we then make the replacements a → q α , b → −q β in the modified (4.2)
(α)
and (4.3) we recover the inequalities (2.9) and (2.10) for Ln (x) by taking limits q → 1− .
For the case 0 < q, aq < 1, the bounds for the smallest zero x1 (a; q) of the Wall polynomial
(4.7)
Wn ((1 − q)x; a; q) = 2 φ1 (q −n , 0; aq; q(1 − q)x),
are obtained from (4.2) and (4.3) by substituting b = 0.
Finally, we record the bounds for the smallest zero x1 (α; q) for the q-Laguerre polynomial
(α)
Ln (x; q). This can be done either by a direct calculation from the 1 φ1 series in (3.3) or by
obtaining them as a limiting case of little q-Jacobi polynomials, employing (3.7), (4.2) and
(4.3). We obtain, for 0 < q < 1, α > −1:
(4.8)
1
q α+1 x1 (α; q)
(1 + q)(1 − q α+2 )
<
<
,
1 − qn
1 − q α+1
(1 − q)R
where R = 1 + 2q − q n+α+2 − q n − q α+2 , and
1
(1 + q)(1 − q α+2 ) 2
q α+1 x1 (α; q)
(1 − q α+3 )(1 − q)(1 + q + q 2 )R
(4.9)
<
<
(1 − q)(1 − q n )R
1 − q α+1
T
with
(4.10) T = 3q 6 (1 − q n−1 )(1 − q n−2 )(1 − q α+1 )2
+ (1 − q n )(1 − q)(1 − q α+3 )(1 + q + q 2 )R
− q 2 (1 − q n )(1 − q n−1 )(1 − q α+1 )(1 − q α+3 )(1 + q + q 2 ).
From (4.8) and (4.9) we can recover the bounds (2.9) and (2.10) for the smallest zero x1 of
(α)
Laguerre polynomials Ln (x) by taking limits q → 1− .
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 24, 7 pp.
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Z EROS OF L AGUERRE P OLYNOMIALS
7
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J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 24, 7 pp.
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