Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol99.Issue1-2.1998:
Journal of Computational and Applied Mathematics 99 (1998) 529–533
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
c 1998 Elsevier
A simple new uniform estimate for the Laguerre polynomials of order , L( )
n (x); ∈ R is given.
Science B.V. All rights reserved.
AMS classi cation: 33C45; 30C50
Keywords: Laguerre polynomials; Laplace integral
The Laguerre polynomials Ln( ) (x) of order can be de ned in many ways [5, 7]. Assuming
∈ R; x¿0 and n = 0; 1; 2; : : : they are de ned, for instance, by the generating function
(1 − z)
− −1
xz
exp −
1−z
=
∞
X
Ln( ) (x)z n ;
n=0
|z|¡1;
(1)
or by explicit formula
Ln( ) (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]):
∗
|Ln( ) (x)|6
( + 1)n x=2
e ;
n!
|Ln( ) (x)|6
¿0; x¿0; n = 0; 1; 2; : : : ;
( + 1)n x=2
e ;
2−
n!
−1¡ 60; x¿0; n = 0; 1; 2; : : : :
Corresponding author. jsszynal@golem.umcs.lublin.pl.
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 8 1 - 2
(S1 )
(S2 )
530
Z. Lewandowski, J. Szynal / Journal of Computational and Applied Mathematics 99 (1998) 529–533
The estimate (S2 ) has been improved in 1985 by Rooney [6], who proved with the aid of Askey
formula that
|Ln( ) (x)|62− qn e x=2 ;
6 − 12 ;
x¿0; n = 0; 1; 2; : : :
(R1 )
and
√ qn ( + 1)n x=2
(R2 )
|Ln( ) (x)|6 2
e ;
¿− 21 ; x¿0; n = 0; 1; 2; : : : ;
(1=2)n
√
√
where qn = (2n)!=2n+1=2 n!; 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] ( ¿− 21 ; x¿0;
n = 0; 1; 2; : : :),
Ln( ) (x) =
2(−1) n
√
( + 12 )n!
Z
0
∞
Z
0
√
2
(x − r 2 + i2 x r cos ) n e −r r 2 +1 sin2 d dr;
(K)
we nd the simple estimate for |Ln( ) (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 + a1 z + a2 z 2 + · · · ;
t¿0; |z|¡1;
(3)
we have
2
= 0:735 : : : ; n = 1; 2; : : :
e
with the equality for the function
sup |an | =
1 + zn
Fn (z) = F(1; z ) = exp −
;
1 − zn
n
(4)
n = 1; 2; : : : ;
(5)
where
∞
X
1+z
−t
F(t; z) = exp −t
=e +
An (t)z n ;
1−z
n=0
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 Ln(−1) (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 |Ln( ) (x)|
is not easy to obtain. We are going to apply formula (K) and the estimates for the Gegenbauer
polynomials.
531
Z. Lewandowski, J. Szynal / Journal of Computational and Applied Mathematics 99 (1998) 529–533
We start with the following simple:
Lemma. For ¿− 21 ; x¿0 and n = 0; 1; 2; : : : ; we have
(−1) n
(2 + 1)n ( + 1)
Ln( ) (x) =
Z
∞
Cn( +1=2)
0
x − t −t
e t (t + x) n dt;
x+t
(8)
where Cn() (y); ¿0; y ∈ [−1; 1] denotes the Gegenbauer polynomial of degree n and order .
Proof. Using the Laplace’s integral [5] for the Gegenbauer polynomials Cn() ,
Cn() (y) =
(2)n ( + 21 )
√
()n!
Z
(y +
0
q
y2 − 1 cos ) n sin2−1 d;
(L)
and Koornwinder formula (K) we can write the following chain of equalities:
2(−1) n
Ln( ) (x) = √
( + 12 )n!
Z
∞
(Z
0
0
)
!n
√
2r x
x − r2
2
+i
cos sin d
x + r2
x + r2
2
× (x + r 2 ) n r 2 +1 e−r dr
2(−1) n
=√
( + 12 )n!
×e
−x(1− )=(1+ )
Z
1
−1
Z
( +
√
2
0
1−
x
1+
2x
1+
n
− 1 cos ) sin
n
2
d
x d
(1 + ) 2
1
1−
2(−1) n
Cn( +1=2) ( )e −x(1− )=(1+ ) x
=
(2 + 1)n ( + 1) −1
1+
Z ∞
n
x−t
(−1)
Cn( +1=2)
(x + t) n t e −t dt;
=
(2 + 1)n ( + 1) 0
x+t
Z
2x
1+
n
x d
(1 + ) 2
which ends the proof.
Remark. We do not claim the representation (8) is new, however we have diculty to nd references for the proof.
Compare formula (8) with the de nition of the Gamma function
( ) =
Z
∞
e−t t −1 dt;
¿0:
0
In order to state our result we need the following de nition. For the formal series
¿ − 1 we de ne the Cesaro mean n( ) by the formula
n( )
∞
X
n=1
!
n =
n
X
( + 1)n−k
n!
ak :
( + 1)n k=0 (n − k)!
P∞
n=0
an and
(9)
532
Z. Lewandowski, J. Szynal / Journal of Computational and Applied Mathematics 99 (1998) 529–533
Theorem. For ¿− 21 ; x¿0 and n = 0; 1; 2; : : : we have
|Ln( ) (x)|6
( + 1)n ( )
n (exp x):
n!
(10)
Proof. From the Laplace integral (L) it follows directly that
|Cn() (y)|6
(2)n
;
n!
y ∈ [−1; 1]; ¿0; n = 0; 1; 2 : : : :
(11)
Putting (11) into (8) we nd, for ¿ − 12 ,
|Ln( ) (x)|6
(2 + 1)n
(2 + 1)n ( + 1)n!
Z
∞
e −t t (t + x) n dt =
0
1
(n; ; x):
( + 1)n!
(12)
Moreover, we have
(n; ; x) =
Z
∞
−t
n
e t (t + x) dt =
0
=
k=0
n
X
k=0
=
n
X
n
k
!
xk
Z
∞
e−t t +n−k dt
0
!
n k
x
( + 1 + n − k)
k
n
X
xk
n!
( + 1 + n − k)
( + 1 + n)
( + 1)n
( + 1)
(n − k)!k!
k=0
n
X
n!
( + 1)n−k x k
= ( + 1 + n)
( + 1)n k=0 (n − k)! k!
= ( + 1 + n)n( ) (exp x):
The de nition (9) of n( ) and the well-known formula for Pochhammer symbol: ( + 1)n = ( +
1 + n)= ( + 1), which was also used above, together with (12) complete the proof of (10).
Corollary 1. For = 0 we have
(0)
|L(0)
n (x)|6n (exp x) = 1 +
x2
xn
x
+ + ··· + ;
1! 2!
n!
x¿0; n = 0; 1; 2; : : : :
(13)
Corollary 2. Estimate (10) gives by the continuity:
|Ln(−1=2) (x)|6
( 12 )n (−1=2)
(exp x);
n! n
x¿0; n = 0; 1; 2; : : : ;
(14)
which for large x is better than (R1 ) and moreover with better constant.
Remark 1. Estimate (10) is better than Szego (S1 ) for large x, because n( ) (exp x) is the polynomial.
Moreover, it covers for the range ¿− 12 .
Z. Lewandowski, J. Szynal / Journal of Computational and Applied Mathematics 99 (1998) 529–533
533
The easily checked monotonicity of n( ) (exp x) (n( 1 ) (exp x)¿n( 2 ) (exp x) 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 Ln(−) (x) =
1
()
Z
x
∞
(t − x)−1 e −t Ln() (t) dt;
¿0;
∈ R; x¿0; n = 0; 1; : : : ;
(A)
can be applied to extend estimate (10) for 6− 12 .
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 Krzyz˙ 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.
[7] G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Coll. Publ., vol. 23, American Mathematical Society,
New York, 1939.
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
c 1998 Elsevier
A simple new uniform estimate for the Laguerre polynomials of order , L( )
n (x); ∈ R is given.
Science B.V. All rights reserved.
AMS classi cation: 33C45; 30C50
Keywords: Laguerre polynomials; Laplace integral
The Laguerre polynomials Ln( ) (x) of order can be de ned in many ways [5, 7]. Assuming
∈ R; x¿0 and n = 0; 1; 2; : : : they are de ned, for instance, by the generating function
(1 − z)
− −1
xz
exp −
1−z
=
∞
X
Ln( ) (x)z n ;
n=0
|z|¡1;
(1)
or by explicit formula
Ln( ) (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]):
∗
|Ln( ) (x)|6
( + 1)n x=2
e ;
n!
|Ln( ) (x)|6
¿0; x¿0; n = 0; 1; 2; : : : ;
( + 1)n x=2
e ;
2−
n!
−1¡ 60; x¿0; n = 0; 1; 2; : : : :
Corresponding author. jsszynal@golem.umcs.lublin.pl.
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 8 1 - 2
(S1 )
(S2 )
530
Z. Lewandowski, J. Szynal / Journal of Computational and Applied Mathematics 99 (1998) 529–533
The estimate (S2 ) has been improved in 1985 by Rooney [6], who proved with the aid of Askey
formula that
|Ln( ) (x)|62− qn e x=2 ;
6 − 12 ;
x¿0; n = 0; 1; 2; : : :
(R1 )
and
√ qn ( + 1)n x=2
(R2 )
|Ln( ) (x)|6 2
e ;
¿− 21 ; x¿0; n = 0; 1; 2; : : : ;
(1=2)n
√
√
where qn = (2n)!=2n+1=2 n!; 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] ( ¿− 21 ; x¿0;
n = 0; 1; 2; : : :),
Ln( ) (x) =
2(−1) n
√
( + 12 )n!
Z
0
∞
Z
0
√
2
(x − r 2 + i2 x r cos ) n e −r r 2 +1 sin2 d dr;
(K)
we nd the simple estimate for |Ln( ) (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 + a1 z + a2 z 2 + · · · ;
t¿0; |z|¡1;
(3)
we have
2
= 0:735 : : : ; n = 1; 2; : : :
e
with the equality for the function
sup |an | =
1 + zn
Fn (z) = F(1; z ) = exp −
;
1 − zn
n
(4)
n = 1; 2; : : : ;
(5)
where
∞
X
1+z
−t
F(t; z) = exp −t
=e +
An (t)z n ;
1−z
n=0
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 Ln(−1) (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 |Ln( ) (x)|
is not easy to obtain. We are going to apply formula (K) and the estimates for the Gegenbauer
polynomials.
531
Z. Lewandowski, J. Szynal / Journal of Computational and Applied Mathematics 99 (1998) 529–533
We start with the following simple:
Lemma. For ¿− 21 ; x¿0 and n = 0; 1; 2; : : : ; we have
(−1) n
(2 + 1)n ( + 1)
Ln( ) (x) =
Z
∞
Cn( +1=2)
0
x − t −t
e t (t + x) n dt;
x+t
(8)
where Cn() (y); ¿0; y ∈ [−1; 1] denotes the Gegenbauer polynomial of degree n and order .
Proof. Using the Laplace’s integral [5] for the Gegenbauer polynomials Cn() ,
Cn() (y) =
(2)n ( + 21 )
√
()n!
Z
(y +
0
q
y2 − 1 cos ) n sin2−1 d;
(L)
and Koornwinder formula (K) we can write the following chain of equalities:
2(−1) n
Ln( ) (x) = √
( + 12 )n!
Z
∞
(Z
0
0
)
!n
√
2r x
x − r2
2
+i
cos sin d
x + r2
x + r2
2
× (x + r 2 ) n r 2 +1 e−r dr
2(−1) n
=√
( + 12 )n!
×e
−x(1− )=(1+ )
Z
1
−1
Z
( +
√
2
0
1−
x
1+
2x
1+
n
− 1 cos ) sin
n
2
d
x d
(1 + ) 2
1
1−
2(−1) n
Cn( +1=2) ( )e −x(1− )=(1+ ) x
=
(2 + 1)n ( + 1) −1
1+
Z ∞
n
x−t
(−1)
Cn( +1=2)
(x + t) n t e −t dt;
=
(2 + 1)n ( + 1) 0
x+t
Z
2x
1+
n
x d
(1 + ) 2
which ends the proof.
Remark. We do not claim the representation (8) is new, however we have diculty to nd references for the proof.
Compare formula (8) with the de nition of the Gamma function
( ) =
Z
∞
e−t t −1 dt;
¿0:
0
In order to state our result we need the following de nition. For the formal series
¿ − 1 we de ne the Cesaro mean n( ) by the formula
n( )
∞
X
n=1
!
n =
n
X
( + 1)n−k
n!
ak :
( + 1)n k=0 (n − k)!
P∞
n=0
an and
(9)
532
Z. Lewandowski, J. Szynal / Journal of Computational and Applied Mathematics 99 (1998) 529–533
Theorem. For ¿− 21 ; x¿0 and n = 0; 1; 2; : : : we have
|Ln( ) (x)|6
( + 1)n ( )
n (exp x):
n!
(10)
Proof. From the Laplace integral (L) it follows directly that
|Cn() (y)|6
(2)n
;
n!
y ∈ [−1; 1]; ¿0; n = 0; 1; 2 : : : :
(11)
Putting (11) into (8) we nd, for ¿ − 12 ,
|Ln( ) (x)|6
(2 + 1)n
(2 + 1)n ( + 1)n!
Z
∞
e −t t (t + x) n dt =
0
1
(n; ; x):
( + 1)n!
(12)
Moreover, we have
(n; ; x) =
Z
∞
−t
n
e t (t + x) dt =
0
=
k=0
n
X
k=0
=
n
X
n
k
!
xk
Z
∞
e−t t +n−k dt
0
!
n k
x
( + 1 + n − k)
k
n
X
xk
n!
( + 1 + n − k)
( + 1 + n)
( + 1)n
( + 1)
(n − k)!k!
k=0
n
X
n!
( + 1)n−k x k
= ( + 1 + n)
( + 1)n k=0 (n − k)! k!
= ( + 1 + n)n( ) (exp x):
The de nition (9) of n( ) and the well-known formula for Pochhammer symbol: ( + 1)n = ( +
1 + n)= ( + 1), which was also used above, together with (12) complete the proof of (10).
Corollary 1. For = 0 we have
(0)
|L(0)
n (x)|6n (exp x) = 1 +
x2
xn
x
+ + ··· + ;
1! 2!
n!
x¿0; n = 0; 1; 2; : : : :
(13)
Corollary 2. Estimate (10) gives by the continuity:
|Ln(−1=2) (x)|6
( 12 )n (−1=2)
(exp x);
n! n
x¿0; n = 0; 1; 2; : : : ;
(14)
which for large x is better than (R1 ) and moreover with better constant.
Remark 1. Estimate (10) is better than Szego (S1 ) for large x, because n( ) (exp x) is the polynomial.
Moreover, it covers for the range ¿− 12 .
Z. Lewandowski, J. Szynal / Journal of Computational and Applied Mathematics 99 (1998) 529–533
533
The easily checked monotonicity of n( ) (exp x) (n( 1 ) (exp x)¿n( 2 ) (exp x) 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 Ln(−) (x) =
1
()
Z
x
∞
(t − x)−1 e −t Ln() (t) dt;
¿0;
∈ R; x¿0; n = 0; 1; : : : ;
(A)
can be applied to extend estimate (10) for 6− 12 .
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 Krzyz˙ 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.
[7] G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Coll. Publ., vol. 23, American Mathematical Society,
New York, 1939.