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An upper bound for the Laguerre polynomials
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
ELSEVIER
Journal of Computational and Applied Mathematics 99 (1998) 529-533
An upper bound for the Laguerre polynomials Z d z i s ~ a w L e w a n d o w s k i , 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 ~, Ll,~)(x), ~ C ~ is given. (~) 1998 Elsevier Science B.V. All rights reserved. A M S classification." 33C45; 30C50 Keywords: Laguerre polynomials; Laplace integral
The Laguerre polynomials Ll,,~)(x) of order ~ can be defined in many ways [5, 7]. Assuming E N, x/> 0 and n = 0, 1,2,... they are defined, for instance, by the generating function
xz 1---z
(1 - z ) ..... lexp
= ~,,=0 Ll/~(x)z ",
Izl<
1,
(1)
or by explicit formula
Two classical global uniform (w.r.t. n , x and ~) estimates given by Szeg6 are known (e.g. [1]):
ILl/'(x)l~< (~ +n ~1), e~-..~ ~ ' ILl'~'(x)'~< ( 2
7~>0, x~>0, n = 0 , 1 , 2 . . . . .
( ° ~e+~1;)2"')n ,
-1<0~<0,
x~>0, n - - 0 , 1 , 2 . . . . .
* Corresponding author, [email protected]. 0377-0427/98/S-see front matter (~) 1998 Elsevier Science B.V. All rights reserved. PII: S0377-0427(98)00 1 81-2
(S,) (S:)
Z. LewandowskL J. Szynal/Journal of Computational and Applied Mathematics 99 (1998) 529 533
530
The estimate ($2) has been improved in 1985 by Rooney [6], who proved with the aid of Askey formula that
[L~)(x)[<~2-~q, eX;2'
~<<_l,
x>>-O, n - - 0 , 1 , 2 , . . .
(R~)
and
IL )(x)l ~ x/2 q"(~ + 1), eX/2' (1/2),
~>-½,
x~>0, n = 0 , 1 , 2 . . . . .
where q, = ~ / 2 n + J / 2 n ! , q,, ~ 1/ ~ 4x~, n --* oo. However, by his method Rooney could not improve the estimate (S~). Using the less known representation formula given by Koornwinder [2] ( ~ > - ½ , n = 0 , 1,2,...),
L(~)(x) =
2(-1)"
(x - r 2 + i2v/xr cos ~)"e
1 v~r(~ + ~)n!
r2r2~+! sin 2~ 4~d~dr,
(R2)
x~>0,
(K)
we find the simple estimate for [L~)(x)l which improves ($1) 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 difficult Krzy2 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+alz+a2z2+
...,
t>O,
[z[
(3)
we have sup lan [ = _2 = 0.735 .... n = 1,2,... e with the equality for the function
F,(z)=F(1,z")=exp
1-
,
(4)
n=l,2,...,
(5)
where OG
F ( t , z ) = e x p ~/ - t ] -1- +z'x ~_zJ=e
' + ~-~A,(t)z",
t > O, [z[< I.
(6)
n=0
So far Krzy2 conjecture is proved only for n = 1,2,3,4 and in general it is known only that la, I <0.99918 . . . . From (1) and (6) we easily see that
A,(t)=e-'L~-l)(2t),
n = 1,2 . . . . .
(7)
Therefore, the properties of the Laguerre polynomials are strongly involved in Krzy2 conjecture [3]. There are many formulae for Laguerre polynomials [5, 7]; however, the direct estimate for IL~)(x)] is not easy to obtain. We are going to apply formula (K) and the estimates for the Gegenbauer polynomials.
Z. Lewandowski, J. Szynal/Journal of Computational and Applied Mathematics 99 (1998) 529 533
531
We start with the following simple: 1 Lemma. For ~ > - ~., x >t 0 and n = O, 1,2,..., we have
L}y(x)= ( 2 ~ +
(-1) n 1 ) , r ( ~ + 1)
/o
C~+l,2) _,
x-t ~
()
e-'t~(t+
x) n
dt,
(8)
where C~"~(y), v > 0 , y E [ - 1 , 1] denotes the Gegenbauer polynomial of degree n and order v. Proof. Using the Laplace's integral [5] for the Gegenbauer polynomials C,~,''), 1
v/~F(v) n!
C},'~(y)=
(y+
y 2 _ lcos4~),,sin 2,' 14~d45'
(L)
and Koornwinder formula (K) we can write the following chain of equalities: 2 ( - 1 )n
[~
~ (x - r2
2rv/x
× (x + r2)nr2~+le-"-~ dr =
(~ + V ~ - 1 cos 4)" sin 2~ @ dcb ,
\ l+rJ =(2~+ _ -
2(-1)" 1),F(~+I)
(l+r) 2
.f_' . (xl-Z'~( 2x "~" xdz __1Q~+ll21(v)e-X/~-°/l~+'~ \ ~+--~r/ \ l - - - ~ r / ( 1 + ~ ) 2
(-1)" f~ (2a + 1),F(~ + 1)
which ends the proof.
~
~+,/2)(x-t) C~ ~
(x + t)nt~e -' dt,
[]
Remark. We do not claim the representation (8) is new, however we have difficulty to find references for the proof. Compare formula (8) with the definition of the Gamma function e - ' t ~-I dt,
F(e)=
~>0.
In order to state our result we need the following definition. For the formal series ~n~__0a, and ~ > - 1 we define the Ces/iro mean a~) by the formula c~) a,
n! an -- (a + 1 ), \ n= I
/
(~ + 1)n_k (-n -- ~.i ak. k=0
(9)
532
Z. L e w a n d o w s k i , J. S z y n a l / J o u r n a l o f C o m p u t a t i o n a l a n d A p p l i e d M a t h e m a t i c s 99 (1998) 5 2 9 - 5 3 3
Theorem. For ~ ~> - 5,1 x ~ 0 and n = 0, 1,2,... we have ILL~)(x)I ~< (~ + 1)n a},~)(expx).
(lO)
n!
Proof. From the Laplace integral (L) it follows directly that (2v),
IC~V)(Y)l~ n-~-'
yE[-1,1],
(11)
v>O, n = 0 , 1 , 2 . . . .
Putting (11) into (8) we find, for ~ > (2~ + 1 )n [LT)(x)[ ~< (2~ + 1 ).F(~ + 1 )n!
' 2' e-'t~(t + x ) ~ dt =
1 F(~ + 1)ni 7(n'~'x)"
(12)
Moreover, we have 7(n,~,x) =
e tt~(t + x)" d t =
e-~t~+"-k dt
xk k=O
= }--~r(~ + 1 + n - k)
xk
k=0
n! (-~ +- l ) n F ( ~ + l + n ) n!
=r(~+l+n)(~l)
@F(~+l+n-k)
2-" k=0 ~
xk
r(~+~
(n - k ) ! k !
(~ + 1 )n-~ x k
o ~,:0 ( ~ - '
1,!
= F(~ + 1 + n ) a ~ ) ( e x p x ) .
The definition (9) of a~,~) and the well-known formula for Pochhammer symbol: (~ + 1 )n = F(~ + []
1 + n ) / F ( ~ + 1), which was also used above, together with (12) complete the proof of (10).
Corollary 1. For ~ = 0 we have X
X2
Xn
IL~°)(x)[ ~
x~O, n = 0 , 1 , 2 . . . . .
(13)
Corollary 2. Estimate (10) gives by the continuity: ~ )n O"n(_ l / 2 ),( e x nv x'~,, ],Ln-l/2)(X),'-~( ] .< -( -n!
x~O,
n----O, 1,2,.. .
,
(14)
which f o r large x is better than (R~) and moreover with better constant.
Remark 1. Estimate (10) is better than Szeg6 (S~) for large x, because a(,~) (expx) is the polynomial. Moreover, it covers for ~ the range ~ >~- 5"
Z. Lewandowski, J. Szynal/Journal of Computational and Applied Mathematics 99 (1998) 529 533
533
The easily checked monotonicity of o-~)(expx) (trY.~' )(exp x) > o-~.~2~(expx) for ~ < ,~_~) shows that the estimate (10) is better than (S1) for all x>~0 if 7~>n.
Remark 2. The Askey formula [6] e -~ L.~,_~,)
1
(t_
x)~,_,e_,L},,)(t)dt '
can be applied to extend estimate (10) for ~ <
/~>0,
ve~,
x~>0, n = 0 , 1 .... ,
(A)
-),
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, 1.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 Krzy2 conjecture and related problems, in: Laine, Martio (Eds.), XVIth Rolf Nevanlinna Colloquium, Walter de Gruyter, Berlin, 1996, pp. 257-268. [4] G. Loh6fer, 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. Szeg6, Orthogonal Polynomials, Amer. Math. Soc. Coll. Publ., vol. 23, American Mathematical Society, New York, 1939.
ELSEVIER
Journal of Computational and Applied Mathematics 99 (1998) 529-533
An upper bound for the Laguerre polynomials Z d z i s ~ a w L e w a n d o w s k i , 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 ~, Ll,~)(x), ~ C ~ is given. (~) 1998 Elsevier Science B.V. All rights reserved. A M S classification." 33C45; 30C50 Keywords: Laguerre polynomials; Laplace integral
The Laguerre polynomials Ll,,~)(x) of order ~ can be defined in many ways [5, 7]. Assuming E N, x/> 0 and n = 0, 1,2,... they are defined, for instance, by the generating function
xz 1---z
(1 - z ) ..... lexp
= ~,,=0 Ll/~(x)z ",
Izl<
1,
(1)
or by explicit formula
Two classical global uniform (w.r.t. n , x and ~) estimates given by Szeg6 are known (e.g. [1]):
ILl/'(x)l~< (~ +n ~1), e~-..~ ~ ' ILl'~'(x)'~< ( 2
7~>0, x~>0, n = 0 , 1 , 2 . . . . .
( ° ~e+~1;)2"')n ,
-1<0~<0,
x~>0, n - - 0 , 1 , 2 . . . . .
* Corresponding author, [email protected]. 0377-0427/98/S-see front matter (~) 1998 Elsevier Science B.V. All rights reserved. PII: S0377-0427(98)00 1 81-2
(S,) (S:)
Z. LewandowskL J. Szynal/Journal of Computational and Applied Mathematics 99 (1998) 529 533
530
The estimate ($2) has been improved in 1985 by Rooney [6], who proved with the aid of Askey formula that
[L~)(x)[<~2-~q, eX;2'
~<<_l,
x>>-O, n - - 0 , 1 , 2 , . . .
(R~)
and
IL )(x)l ~ x/2 q"(~ + 1), eX/2' (1/2),
~>-½,
x~>0, n = 0 , 1 , 2 . . . . .
where q, = ~ / 2 n + J / 2 n ! , q,, ~ 1/ ~ 4x~, n --* oo. However, by his method Rooney could not improve the estimate (S~). Using the less known representation formula given by Koornwinder [2] ( ~ > - ½ , n = 0 , 1,2,...),
L(~)(x) =
2(-1)"
(x - r 2 + i2v/xr cos ~)"e
1 v~r(~ + ~)n!
r2r2~+! sin 2~ 4~d~dr,
(R2)
x~>0,
(K)
we find the simple estimate for [L~)(x)l which improves ($1) 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 difficult Krzy2 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+alz+a2z2+
...,
t>O,
[z[
(3)
we have sup lan [ = _2 = 0.735 .... n = 1,2,... e with the equality for the function
F,(z)=F(1,z")=exp
1-
,
(4)
n=l,2,...,
(5)
where OG
F ( t , z ) = e x p ~/ - t ] -1- +z'x ~_zJ=e
' + ~-~A,(t)z",
t > O, [z[< I.
(6)
n=0
So far Krzy2 conjecture is proved only for n = 1,2,3,4 and in general it is known only that la, I <0.99918 . . . . From (1) and (6) we easily see that
A,(t)=e-'L~-l)(2t),
n = 1,2 . . . . .
(7)
Therefore, the properties of the Laguerre polynomials are strongly involved in Krzy2 conjecture [3]. There are many formulae for Laguerre polynomials [5, 7]; however, the direct estimate for IL~)(x)] is not easy to obtain. We are going to apply formula (K) and the estimates for the Gegenbauer polynomials.
Z. Lewandowski, J. Szynal/Journal of Computational and Applied Mathematics 99 (1998) 529 533
531
We start with the following simple: 1 Lemma. For ~ > - ~., x >t 0 and n = O, 1,2,..., we have
L}y(x)= ( 2 ~ +
(-1) n 1 ) , r ( ~ + 1)
/o
C~+l,2) _,
x-t ~
()
e-'t~(t+
x) n
dt,
(8)
where C~"~(y), v > 0 , y E [ - 1 , 1] denotes the Gegenbauer polynomial of degree n and order v. Proof. Using the Laplace's integral [5] for the Gegenbauer polynomials C,~,''), 1
v/~F(v) n!
C},'~(y)=
(y+
y 2 _ lcos4~),,sin 2,' 14~d45'
(L)
and Koornwinder formula (K) we can write the following chain of equalities: 2 ( - 1 )n
[~
~ (x - r2
2rv/x
× (x + r2)nr2~+le-"-~ dr =
(~ + V ~ - 1 cos 4)" sin 2~ @ dcb ,
\ l+rJ =(2~+ _ -
2(-1)" 1),F(~+I)
(l+r) 2
.f_' . (xl-Z'~( 2x "~" xdz __1Q~+ll21(v)e-X/~-°/l~+'~ \ ~+--~r/ \ l - - - ~ r / ( 1 + ~ ) 2
(-1)" f~ (2a + 1),F(~ + 1)
which ends the proof.
~
~+,/2)(x-t) C~ ~
(x + t)nt~e -' dt,
[]
Remark. We do not claim the representation (8) is new, however we have difficulty to find references for the proof. Compare formula (8) with the definition of the Gamma function e - ' t ~-I dt,
F(e)=
~>0.
In order to state our result we need the following definition. For the formal series ~n~__0a, and ~ > - 1 we define the Ces/iro mean a~) by the formula c~) a,
n! an -- (a + 1 ), \ n= I
/
(~ + 1)n_k (-n -- ~.i ak. k=0
(9)
532
Z. L e w a n d o w s k i , J. S z y n a l / J o u r n a l o f C o m p u t a t i o n a l a n d A p p l i e d M a t h e m a t i c s 99 (1998) 5 2 9 - 5 3 3
Theorem. For ~ ~> - 5,1 x ~ 0 and n = 0, 1,2,... we have ILL~)(x)I ~< (~ + 1)n a},~)(expx).
(lO)
n!
Proof. From the Laplace integral (L) it follows directly that (2v),
IC~V)(Y)l~ n-~-'
yE[-1,1],
(11)
v>O, n = 0 , 1 , 2 . . . .
Putting (11) into (8) we find, for ~ > (2~ + 1 )n [LT)(x)[ ~< (2~ + 1 ).F(~ + 1 )n!
' 2' e-'t~(t + x ) ~ dt =
1 F(~ + 1)ni 7(n'~'x)"
(12)
Moreover, we have 7(n,~,x) =
e tt~(t + x)" d t =
e-~t~+"-k dt
xk k=O
= }--~r(~ + 1 + n - k)
xk
k=0
n! (-~ +- l ) n F ( ~ + l + n ) n!
=r(~+l+n)(~l)
@F(~+l+n-k)
2-" k=0 ~
xk
r(~+~
(n - k ) ! k !
(~ + 1 )n-~ x k
o ~,:0 ( ~ - '
1,!
= F(~ + 1 + n ) a ~ ) ( e x p x ) .
The definition (9) of a~,~) and the well-known formula for Pochhammer symbol: (~ + 1 )n = F(~ + []
1 + n ) / F ( ~ + 1), which was also used above, together with (12) complete the proof of (10).
Corollary 1. For ~ = 0 we have X
X2
Xn
IL~°)(x)[ ~
x~O, n = 0 , 1 , 2 . . . . .
(13)
Corollary 2. Estimate (10) gives by the continuity: ~ )n O"n(_ l / 2 ),( e x nv x'~,, ],Ln-l/2)(X),'-~( ] .< -( -n!
x~O,
n----O, 1,2,.. .
,
(14)
which f o r large x is better than (R~) and moreover with better constant.
Remark 1. Estimate (10) is better than Szeg6 (S~) for large x, because a(,~) (expx) is the polynomial. Moreover, it covers for ~ the range ~ >~- 5"
Z. Lewandowski, J. Szynal/Journal of Computational and Applied Mathematics 99 (1998) 529 533
533
The easily checked monotonicity of o-~)(expx) (trY.~' )(exp x) > o-~.~2~(expx) for ~ < ,~_~) shows that the estimate (10) is better than (S1) for all x>~0 if 7~>n.
Remark 2. The Askey formula [6] e -~ L.~,_~,)
1
(t_
x)~,_,e_,L},,)(t)dt '
can be applied to extend estimate (10) for ~ <
/~>0,
ve~,
x~>0, n = 0 , 1 .... ,
(A)
-),
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, 1.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 Krzy2 conjecture and related problems, in: Laine, Martio (Eds.), XVIth Rolf Nevanlinna Colloquium, Walter de Gruyter, Berlin, 1996, pp. 257-268. [4] G. Loh6fer, 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. Szeg6, Orthogonal Polynomials, Amer. Math. Soc. Coll. Publ., vol. 23, American Mathematical Society, New York, 1939.