An upper bound for the Laguerre polynomials
Zdzis law Lewandowski, Jan Szynal∗
Department of Mathematics, M. Curie-Sklodowska University, 20-031 Lublin, Poland
Received 8 January 1998
Abstract
A simple new uniform estimate for the Laguerre polynomials of order , L()n (x); ∈R is given. c 1998 Elsevier Science B.V. All rights reserved.
AMS classication:33C45; 30C50
Keywords:Laguerre polynomials; Laplace integral
The Laguerre polynomials L()
n (x) of order can be dened in many ways [5, 7]. Assuming
∈R; x¿0 and n= 0;1;2; : : : they are dened, for instance, by the generating function
(1−z)−−1
exp
−1xz
−z
=
∞
X
n=0 L()
n (x)z n;
|z|¡1; (1)
or by explicit formula
L(n)(x) = n X
k=0
n+ n−k
! (−x)k
k! : (2)
Two classical global uniform (w.r.t. n; x and ) estimates given by Szego are known (e.g. [1]):
|L(n)(x)|6(+ 1)n n! e
x=2; ¿0; x¿0; n= 0;1;2; : : : ; (S 1)
|L(n)(x)|6
2−(+ 1)n
n!
ex=2; −1¡60; x¿0; n= 0;1;2; : : : : (S2)
∗Corresponding author. [email protected].
The estimate (S2) has been improved in 1985 by Rooney [6], who proved with the aid of Askey formula that
|L(n)(x)|62−
qnex=2; 6−12; x¿0; n= 0;1;2; : : : (R1)
and
|L(n)(x)|6√2qn(+ 1)n (1=2)n
ex=2; ¿−1
2; x¿0; n= 0;1;2; : : : ; (R2)
where qn=√(2n)!=2n+1=2n!; qn∼1=
4 √
4n; n→ ∞.
However, by his method Rooney could not improve the estimate (S1).
Using the less known representation formula given by Koornwinder [2] (¿−1
2; x¿0; n= 0;1;2; : : :),
L()
n (x) =
2(−1)n
√
(+1
2)n!
Z ∞
0
Z
0
(x−r2+ i2√x rcos)ne−r2
r2+1 sin2 ddr; (K)
we nd the simple estimate for |L()
n (x)|which improves (S1) for some values of x and covers wider range of parameter .
More important than the result is the motivation, which comes out from pretty attractive and dicult Krzy˙z conjecture (for the references see, e.g., [3]) in the geometric function theory which says that for any bounded and nonvanishing holomorphic function in the unit disk |z|¡1; which has the form
f(z) = e−t
+a1z+a2z2+· · ·; t¿0; |z|¡1; (3)
we have
sup|an|= 2
e = 0:735: : : ; n= 1;2; : : : (4)
with the equality for the function
Fn(z) =F(1; zn) = exp
−1 +z
n
1−zn
; n= 1;2; : : : ; (5)
where
F(t; z) = exp
−t1 +z
1−z
= e−t
+
∞
X
n=0
An(t)zn; t¿0; |z|¡1: (6)
So far Krzy˙z conjecture is proved only for n= 1;2;3;4 and in general it is known only that
|an|¡0:99918: : : .
From (1) and (6) we easily see that
An(t) = e
−t
L(−1)
n (2t); n= 1;2; : : : : (7)
Therefore, the properties of the Laguerre polynomials are strongly involved in Krzy˙z conjecture [3]. There are many formulae for Laguerre polynomials [5, 7]; however, the direct estimate for |L()
We start with the following simple:
Proof. Using the Laplace’s integral [5] for the Gegenbauer polynomials C()
n ,
and Koornwinder formula (K) we can write the following chain of equalities:
L(n)(x) = 2(−1)
which ends the proof.
Remark. We do not claim the representation (8) is new, however we have diculty to nd refer-ences for the proof.
Compare formula (8) with the denition of the Gamma function
() =
In order to state our result we need the following denition. For the formal series P∞
n=0an and
¿−1 we dene the Cesaro mean ()
n by the formula
Theorem. For ¿−1
Proof. From the Laplace integral (L) it follows directly that
|C()
Corollary 2. Estimate (10) gives by the continuity:
|L(−1=2)
which for large x is better than (R1) and moreover with better constant.
Remark 1. Estimate (10) is better than Szego (S1) for largex, because(n)(expx) is the polynomial. Moreover, it covers for the range ¿−1
The easily checked monotonicity of ()
n (expx) (n(1)(expx)¿n(2)(expx) for 1¡2) shows that the estimate (10) is better than (S1) for all x¿0 if ¿n.
Remark 2. The Askey formula [6]
e−x
L(−)
n (x) = 1 ()
Z ∞
x
(t−x)−1
e−t
L()
n (t) dt; ¿0; ∈R; x¿0; n= 0;1; : : : ; (A)
can be applied to extend estimate (10) for 6−1 2.
Remark 3. The more precise bounds for the Gegenbauer polynomials (see for instance [4]) can be applied to obtain a better estimate than (10). However, they are too complicated to quote them here.
References
[1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1964.
[2] T. Koornwinder, Jacobi polynomials II. An analytic proof of the product formula, SIAM J. Math. Anal. 5 (1974) 125 –137.
[3] Z. Lewandowski, J. Szynal, On the Krzy ˙z conjecture and related problems, in: Laine, Martio (Eds.), XVIth Rolf Nevanlinna Colloquium, Walter de Gruyter, Berlin, 1996, pp. 257– 268.
[4] G. Lohofer, Inequalities for Legendre and Gegenbauer functions, J. Approx. Theory 64 (1991) 226 – 234. [5] E.D. Rainville, Special Functions, Macmillan, New York, 1960.
[6] P.G. Rooney, Further inequalities for generalized Laguerre polynomials, C.R. Math. Rep. Acad. Sci. Canada 7 (1985) 273 – 275.
