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Hermite Polynomials in Asymptotic Representations of Generalized Bernoulli, Euler, Bessel, and Buchholz Polynomials
Journal of Mathematical Analysis and Applications 239, 457᎐477 Ž1999. Article ID jmaa.1999.6584, available online at http:rrwww.idealibrary.com on
Hermite Polynomials in Asymptotic Representations of Generalized Bernoulli, Euler, Bessel, and Buchholz Polynomials Jose ´ L. Lopez ´ Departamento de Matematica Aplicada, Facultad de Ciencias, Uni¨ ersidad de Zaragoza, ´ 50013-Zaragoza, Spain
and Nico M. Temme1 CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands Submitted by L. Debnath Received April 8, 1999
This is the second paper on finite exact representations of certain polynomials in terms of Hermite polynomials. The representations have asymptotic properties and include new limits of the polynomials, again in terms of Hermite polynomials. This time we consider the generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. The asymptotic approximations of these polynomials are valid for large values of a certain parameter. The representations and limits include information on the zero distribution of the polynomials. Graphs are given that indicate the accuracy of the first term approximations. 䊚 1999 Academic Press
1. INTRODUCTION Generalized Bernoulli, Euler, Bessel, and Buchholz polynomials of degree n, complex order and complex argument z, denoted respectively by Bn Ž z ., En Ž z ., Yn Ž z . and Pn Ž z ., can be defined by their generating 1
E-mail: [email protected]. 457 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
LOPEZ AND TEMME ´
458 functions ww11x, chap. 6x, w ew z
Ž e w y 1.
s
Bn Ž z . n!
ns0
2 ew z
Ž e w q 1.
⬁
Ý
s
⬁
Ý
En Ž z . n!
ns0
wn,
< w < - 2 ,
Ž 1.
w n,
Ž 2.
ww4x, page 181x, 1
ž
'1 y 2 zw s
1 q '1 y 2 zw Yn Ž z .
⬁
Ý
2
n!
ns0
/
e 2 w rŽ1q '1y2 z w .
<2 zw < - 1,
w n,
Ž 3.
and ww2x, sec. 3x, in a slightly different notation, e
zŽcot wy1r w .r2
sin w
ž / w
s
⬁
Ý
Pn Ž z . w n ,
Ž 4.
ns0
The generalized Bernoulli and Euler polynomials play an important role in the calculus of finite differences. In fact, the coefficients in all the usual central-difference formulae for interpolation, numerical differentiation and integration, and differences in terms of derivatives can be expressed in terms of these polynomials w11x. Many properties of these polynomials can be found in ww3x, chap. 6x, ww5x, vol. 1, chap. 1x, w10x, and w11x. An explicit formula for the generalized Bernoulli polynomials can be found in w12x. Asymptotic expansions in terms of elementary functions and in terms of gamma and polygamma functions are obtained in w16x. Properties and explicit formulas for the generalized Bernoulli and Euler numbers can be found in w9x, w14x, w15x and references therein. The generalized Bessel polynomials form a set of orthogonal polynomials on the unit circle in the complex plane. They are important in certain problems of mathematical physics; for example, they arise in the study of electrical networks and when the wave equation is considered in spherical coordinates. For a historical survey and discussion of many interesting properties, we refer to w6x. New asymptotic expansions of Yn Ž x . Žand its zeros. for large values of n are given in w17x. Buchholz polynomials are used for the representation of the Whittaker functions as convergent series expansions of Bessel functions w2x. They appear also in the convergent expansions of the Whittaker functions in ascending powers of their order and in the asymptotic expansions of the
HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
459
Whittaker functions in descending powers of their order w7x. Explicit formulas for obtaining these polynomials may be found in w1x. In our first paper w8x it has been shown that Jacobi, Gegenbauer, Laguerre, and Tricomi-Carlitz polynomials have asymptotic representations in terms of the Hermite polynomials ? nr2 @
Hn Ž x . s n!
Ž y1.
Ý
k
k! Ž n y 2 k . !
ks0
Ž2 x.
ny 2 k
.
These asymptotic representations include well-known limits of the polynomials in terms of the Hermite polynomials, and provide a powerful tool for approximating the zeros of these polynomials in terms of the zeros of the Hermite polynomials in the asymptotic limit w8x. The polynomials of the previous paper are all orthogonal on a set of the real line. The present group is quite different. Only the Bessel polynomials are orthogonal, but not in the standard sense: they are orthogonal on the unit circle. In a certain sense the polynomials of the present group become orthogonal if the parameter becomes large. We give similar asymptotic representations for the new group as in the previous paper for the asymptotic limit < < ª ⬁. From these representations we can derive
lim
ª⬁
lim
ª⬁
ž / ž / 8
nr2
En
nr2
ª⬁
ª⬁
Bn
lim i n Ž 2 .
lim
nr2
24
6
ž /
ž ( / ž ( /
Yn y
2
2
2
q q
6
2
z s Hn Ž z . ,
z s Hn Ž z . , 2
ž ( / 1qi
nr2
Pn Ž y2 6 z . s
'
z
1 n!
s Hn Ž z . ,
Hn Ž z . .
From these limits we can obtain approximations for the zeros of these polynomials in the asymptotic regime. In the following section we give the principles of the Hermite-type asymptotic approximations used in this paper. In later sections we apply the method to obtain expansions for the generalized Bernoulli and Euler polynomials, and the Bessel and Buchholz polynomials. We also obtain estimates of their zeros for large .
LOPEZ AND TEMME ´
460
2. EXPANSIONS IN TERMS OF HERMITE POLYNOMIALS The Hermite polynomials have the generating function ⬁
2
e 2 z wyw s
Hn Ž z .
Ý
n!
ns0
z, w g ⺓,
wn,
which gives the Cauchy-type integral Hn Ž z . s
n!
He 2 i C
2 z wyw 2
dw
, Ž 5. w nq1 where C is a circle around the origin and the integration is in positive direction. The polynomials defined in Ž1. ᎐ Ž4., as well as many other well-known polynomials, may be defined by a generating function F Ž z, w . in the form F Ž z, w . s
⬁
Ý
pn Ž z . w n ,
Ž 6.
ns0
where F: ⺓ = ⺓ ª ⺓ is analytic with respect to w in a domain that contains the origin. We assume that F Ž z, 0. s p 0 Ž z . s 1 and that the polynomials pnŽ z . are independent of w. We have the Cauchy-type integral representation pn Ž z . s
dw
1
H F Ž z, w . w 2 i C
nq1
,
where C is a circle around the origin inside the domain where F is analytic Žas a function of w .. We write the generating function F Ž z, w . of pnŽ z . in the form 2
F Ž z, w . s e AŽ z .wyBŽ z .w f Ž z, w . ,
Ž 7.
where AŽ z . and B Ž z . are independent of w, and it follows that pn Ž z . s
1
He 2 i C
AŽ z .wyBŽ z .w 2
dw
f Ž z, w .
. Ž 8. w nq1 The function f is also analytic around the origin w s 0. Therefore, we can expand f Ž z, w . s 1 q p1 Ž z . y A Ž z . w q p 2 Ž z . y A Ž z . p1 Ž z . q B Ž z . q s
1 2
2
A Ž z . w 2 q ⭈⭈⭈
⬁
Ý ck Ž z . w k . ks0
Ž 9.
461
HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
Substituting this in Ž8. and using Ž5., we obtain the finite expansion pn Ž z . s Ž B Ž z . .
nr2
n
Ý ks0
Hnyk Ž .
ck Ž z .
Ž BŽ z. .
kr2
Ž n y k. !
s
,
AŽ z .
'
2 BŽ z.
, Ž 10 .
because terms with k ) n do not contribute to the integral in Ž8.. If B Ž z . happens to be zero for a special z-value, say z 0 , we write pn Ž z 0 . s A Ž z 0 .
n
n
Ý ks0
ck Ž z0 . AŽ z0 .
k
Ž n y k. !
.
Ž 11 .
In the examples considered in the following sections, the choice of AŽ z . and B Ž z . is based on our requirement that c1Ž z . s c 2 Ž z . s 0, in order to make the function f Ž z, w . close to 1 near the origin wnote that f Ž z, 0. s 1x. Then, the generating function F Ž z, w . is close to the generating function of the Hermite polynomials. Using c 0 Ž z . s 1 and requiring c1Ž z . s c 2 Ž z . s 0, we have, from Ž9., A Ž z . s p1 Ž z . ,
B Ž z . s 12 p12 Ž z . y p 2 Ž z . .
Ž 12 .
We can summarize the above discussion in the following. PROPOSITION 2.1. Consider the polynomials pnŽ z . defined in Ž6. by a generating function F Ž z, w . analytic in w s 0 and normalized in the form F Ž z, 0. s 1. Then, they may be represented as the finite sum Ž10. if p12 Ž z . y 2 p 2 Ž z . / 0, and as the finite sum Ž11. if p12 Ž z 0 . y 2 p 2 Ž z 0 . s 0. The functions c k Ž z . are the coefficients of the Taylor expansion of F Ž z, w . exp
½Ž
1 2
2
p1 Ž z . y p 2 Ž z . . w 2 y p1 Ž z . w
5
at w s 0, c 0 s 1, c1 s c 2 s 0 and Hn are the Hermite polynomials. In the following sections we verify if the finite sum in Ž10. yields asymptotic representations for the generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. The special choice of AŽ z . and B Ž z . given in Ž12. is crucial for obtaining asymptotic properties. To prove these properties we will use the following lemma. LEMMA 2.1. Let Ž w . be analytic at w s 0, with Maclaurin expansion of the form Ž w . s w n Ž a0 q a1w q a2 w 2 q ⭈⭈⭈ . q b1w q b 2 w 2 q ⭈⭈⭈ , where n is a positi¨ e integer, a k , bk are complex numbers that do not depend on the complex number , a0 / 0; let c k denote the coefficients of the power series of e Ž w ., that is, e Ž w . s Ý⬁ks 0 c k w k . Then c k s O Ž < < ? k r n@ . ,
ª ⬁.
LOPEZ AND TEMME ´
462
Proof. The proof follows from expanding e Ž w . s Ý⬁ks 0 w Ž w .x krk!, substituting the power series of and collecting equal powers of w.
3. GENERALIZED BERNOULLI POLYNOMIALS From Ž1. we obtain the following Cauchy-type integral for the generalized Bernoulli polynomials Bn Ž z . s
w ew z
n!
H 2 i C Ž e
w
y 1.
dw
w nq1
,
where C is a circle around the origin with radius less than 2 . We assume that F Ž z, w . s w e w zrŽ e w y 1. assumes real values for real values of z, w and . We have B0 Ž z . s 1,
B1 Ž z . s z y
2
B2 Ž z . s z 2 y z q
,
Ž 3 y 1. 12
.
Hence, by Ž12. and pnŽ z . s Bn Ž z .rn!, AŽ z . s z y
2
,
BŽ z. s
24
,
and by Ž10.,
Bn Ž z . s n!
ž / 24
nr2
n
ck Ž .
Ý ks0 Ž n y k . !
24
ž /
kr2
Hnyk
ž
'6 Ž z y r2.
'
/
.
Ž 13 . Observe that this representation shows the symmetry of Bn Ž z . with respect to the point z s 12 . The coefficients c k Ž . of the expansion are given in the following lemma. LEMMA 3.1. The odd coefficients c nŽ . in the expansion Ž13. ¨ anish, c 2 nq1 Ž . s 0
᭙ n G 0,
Ž 14 .
HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
463
and the e¨ en ones are independent of z; they are gi¨ en for n G 2 by the recurrence c2 n Ž . s
n
12 n y
Ý ks2 n
1
Ž 2 k q 1. Ž k y 3. q 6 c 2Ž nyk . Ž . Ž 2 k q 1. ! Ž n y k.
c Ž . , Ý n ks1 Ž 2 k q 1 . ! 2Ž nyk .
Ž 15 .
with c 0 Ž . s 1, c 2 Ž . s 0 and satisfy < < ª ⬁.
c 2 n Ž . s O Ž ? n r2 @ . ,
Ž 16 .
Proof. Using equation Ž7., the function f Ž z, w . of the generalized Bernoulli polynomials reads w e Ž1qw r12.w r2
f Ž z, w . s
Ž e w y 1.
s
⬁
Ý ck Ž . w k .
Ž 17 .
ks0
This is an even function in the variable w and Ž14. follows. It is independent of z and so are the coefficients c k Žwhich only depend on .. Moreover, it satisfies the differential equation
1q
ž
1 2
y
1 w
y
w 12
/Ž
e w y 1 . f Ž z, w . q Ž e w y 1 .
d dw
f Ž z, w . s 0,
and introducing the expansion Ž17. wwith c 2 nq1Ž . s 0x in this differential equation we obtain Ž15.. The function f Ž z, w . can be written in this case in the form f Ž z, w . s e w
4
1Ž w .
,
1Ž w . s
1 2880
q O Ž w2 . ,
w ª 0,
and 1Ž w . does not depend on and z. Hence, the proof of Ž16. follows from Lemma 2.1. PROPOSITION 3.1. The generalized Bernoulli polynomials Bn Ž z . ha¨ e the finite expansion in terms of Hermite polynomials Bn
Ž z. s
ž / 24
nr2
Hn Ž . q n!
ž / 24
nr2 ? nr2 @
Ý ks2
c2 k Ž .
24
ž /
k
Hny 2 k Ž .
,
Ž n y 2k. ! Ž 18 .
LOPEZ AND TEMME ´
464 where
s
'6 Ž z y r2.
Ž 19 .
'
and c 2 k Ž . are gi¨ en in Ž15.. This is actually an asymptotic expansion of Bn Ž z . for < < ª ⬁ with respect to the sequence ? k r2 @yk , uniformly with respect to . Proof. Ž18. follows trivially by using Ž13. and Lemma 3.1. The asymptotic property of Ž18. follows from Ž16.. If < < is bounded, the combination c 2 k Ž . yk in Ž18. gives the asymptotic nature for large values of < <; if < < is not bounded, then the property HnŽ . s O Ž n . gives extra asymptotic convergence in the sum in Ž18.. Figure 1 shows the accuracy of the approximation Bn Ž z . ,
nr2
ž /
Hn
24
ž
'6 Ž z y r2.
'
/
Ž 20 .
for n s 10, real z and several values of . 3.1. Approximating the Zeros When computing approximations of the zeros of the generalized Bernoulli polynomials for large values of we start with the zeros of the Hermite polynomial HnŽ . in Ž20.. Let bn, m and h n, m be the mth zero of Bn Ž z . and HnŽ z ., respectively, m s 1, 2, . . . , n. Then, for given and n we take the relation for given in Ž19. to compute a first approximation of bn, m by writing bn , m ;
2
q
(
6
hn, m .
For s 10, 20, 30 and n s 10, the best relative accuracy in the zeros is ; 10y3 and the worst result Žfor the largest zero. is ; 10y2 . For s 40, 50 it oscillates between 10y3 and 10y4 , whereas for s 100 it oscillates between 10y4 and 10y5 .
4. GENERALIZED EULER POLYNOMIALS From Ž2. we obtain the Cauchy-type integral for the generalized Euler polynomials n! 2 ew z dw En Ž z . s , w 2 i C Ž e q 1 . w nq1
H
465
HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
Ž . FIG. 1. Solid lines represent B10 x for several values of , whereas dashed lines represent the right-hand side of Ž20..
where C is a circle around the origin with radius less than . We assume that F Ž z, w . s 2 e w zrŽ e w q 1. assumes real values for real values of z, w, and . We have E0 Ž z . s 1,
E1 Ž z . s z y
2
E2 Ž z . s z 2 y z q
,
Ž y 1. 4
.
Hence, by Ž12. and pnŽ z . s En Ž z .rn!, AŽ z . s z y
2
BŽ z. s
,
8
.
It follows from Ž10. that En
Ž z . s n!
ž / 8
nr2
n
ck Ž .
Ý ks0 Ž n y k . !
8
ž /
kr2
Hnyk
ž
'2 Ž z y r2.
'
/
, Ž 21 .
466
LOPEZ AND TEMME ´
FIG. 1. Continued.
where the coefficients c k Ž . of the expansion are given below. This representation shows the symmetry of En Ž z . with respect to the point z s 12 . We have the following results. The proofs are as in the case of the Bernoulli polynomials. LEMMA 4.1. The odd coefficients c nŽ . in the expansion Ž21. ¨ anish, c 2 nq1 Ž . s 0
᭙ n G 0,
Ž 22 .
467
HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
and the e¨ en ones are independent of z; they are gi¨ en for n G 2 by the recurrence c2 n Ž . s
2k y 3
n
c Ž . Ý 16 n ks2 Ž 2 k y 1 . ! 2Ž nyk . q
Ž k y n y 1.
n
1
Ý 2 n ks2 Ž 2 k y 2 . !
c2Ž nykq1. Ž . ,
Ž 23 .
where c 0 Ž . s 1, c 2 Ž . s 0, and satisfy < < ª ⬁.
c 2 n Ž . s O Ž ? n r2 @ . ,
Ž 24 .
PROPOSITION 4.1. The generalized Euler polynomials En Ž z . ha¨ e the finite expansion in terms of Hermite polynomials En Ž z . s
ž / 8
nr2
Hn Ž . q n!
nr2 ? nr2 @
ž /
c2 k Ž .
Ý
8
ks2
8
ž /
k
Hny 2 k Ž .
,
Ž n y 2k. ! Ž 25 .
where
s
'2 Ž z y r2.
Ž 26 .
'
and c k Ž . are gi¨ en in Ž23.. This is actually an asymptotic expansion of En Ž z . for < < ª ⬁ with respect to the sequence ? k r2 @yk , uniformly with respect to . Figure 2 shows the accuracy of the approximation En Ž z . ,
ž / 8
nr2
Hn
ž
'2 Ž z y r2.
'
/
Ž 27 .
for n s 10, real z and several values of . 4.1. Approximating the Zeros Let e n, m and h n, m be the mth zero of En Ž z . and HnŽ z ., respectively, m s 1, 2, . . . , n. Then, for given and n we take the relation for given
LOPEZ AND TEMME ´
468
Ž . FIG. 2. Solid lines represent E10 x for several values of , whereas dashed lines represent the right-hand side of Ž27..
in Ž26. to compute a first approximation of e n, m by writing en , m ;
2
q
(
2
hn, m .
The accuracy is as in the case of the generalized Bernoulli polynomials.
5. GENERALIZED BESSEL POLYNOMIALS From Ž3. we obtain the following Cauchy-type integral for the generalized Bessel polynomials Yn
Ž z. s
n!
1
H 2 i C '1 y 2 zw
ž
2 1 q '1 y 2 zw
/
e 2 w rŽ1q '1y2 z w .
dw w nq1
where C is a circle around the origin with radius less than <1rŽ2 z .<.
,
HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
FIG. 2. Continued.
We have Y0 Ž z . s 1, Y2 Ž z . s
1 4
Y1 Ž z . s
1 2
2 q Ž q 2. z ,
4 q 4 Ž q 3 . z q Ž q 7 . q 12 4 z 2 .
Hence, by Ž12. and pnŽ z . s Yn Ž z .rn!,
AŽ z . s
1 2
2 q Ž q 2. z ,
BŽ z. s y
z 8
4 q Ž 3 q 8. z .
469
LOPEZ AND TEMME ´
470 It follows from Ž10. that Yn Ž z . s n! B Ž z .
nr2
c k Ž z, .
n
Ý ks0
Hnyk Ž .
kr2
BŽ z.
Ž n y k. !
s
,
AŽ z .
'
2 BŽ z.
Ž 28 . where the coefficients c k Ž z, . of the expansion satisfy the properties given in the following lemma. We introduce the notation y Ž w . s '1 y 2 zw s 1 y
⬁
Ý
bk s
z k bk w k ,
ks1
2k
1
ž /
2 k! 2
. ky 1
LEMMA 5.1. The coefficients c k Ž z, . in the expansion Ž28. are gi¨ en by the recursion relation 16 Ž k q 1 . c kq 1 s 48 kzck y 28 Ž k y 1 . z 2 c ky1 q 4 6 q Ž 18 y k q 5 . z z 2 c ky2 y 32 q Ž 64 q 3 Ž k y 3 . q 25 . z z 3 c ky 3 ky4
q2
Ý Ž 4 jbkq 1yj js0
y 8 jbky j y bky1yj . z kq 1yj y2 Ž 4 q 3 q 8 z . z ky j bky 2yj c j ,
Ž 29 .
where c j s 0 if j - 0 and empty sums are zero with c 0 Ž z, . s 1, c1Ž z, . s c 2 Ž z, . s 0, and they satisfy the asymptotic estimate < < ª ⬁.
c k Ž z, . s O Ž ? k r3@ . ,
Ž 30 .
Proof. The function f Ž z, w . s
eyA wqB w
2
q2 w rŽ1q 1y2 z w .
'
ž
'1 y 2 zw
2 1 q '1 y 2 zw
/
satisfies the differential equation 2
2
y 2 Ž 1 q y . f ⬘ s Ž 2 Bw y A . y 2 Ž 1 q y . q 2 y 2 Ž 1 q y . q 2 zwy 2
qz Ž 1 q y . q zy Ž 1 q y . f .
471
HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
Then, writing f Ž z, w . s Ý⬁ks 0 c k w k we obtain the recursion Ž29. upon substitution. The function f Ž z, w . can be written in the form f Ž z, w . s e 1Ž z , w .q 2 Ž z , w . , where 1 , 2 do not depend on , with
1 Ž z, w . s ln
2
y
1 q '1 y 2 zw 5
s w3
12
z3 q O Ž w. ,
1 2
zw y
3 8
z2 w2
wª0
and
2 Ž z, w . s w 3
1 6
Ž 8 z q 3. z 2 q O Ž w . ,
w ª 0.
Hence, Ž30. follows from Lemma 2.1. The first few terms are c 0 Ž z, . s 1, c1Ž z, . s c 2 Ž z, . s 0, and c 3 Ž z, . s c 4 Ž z, . s c5 Ž z, . s
z2 12 z3 64 z4 80
Ž 5 q 16 . z q 6 , Ž 35 q 128. z q 40 , Ž 63 q 256. z q 70 .
PROPOSITION 5.1. The generalized Bessel polynomials Yn Ž z . ha¨ e the finite expansion in terms of Hermite polynomials Yn Ž z . s B Ž z .
nr2
n
Hn Ž . q n!
Ý ks3
c k Ž z, . BŽ z.
kr2
Hnyk Ž .
Ž n y k. !
,
Ž 31 .
where is gi¨ en in Ž28.. This is actually an asymptotic expansion of Yn Ž z . for < < ª ⬁ and holds for fixed ¨ alues of z and n. Proof. Ž31. follows trivially from Ž28. and using c 0 s 1, c1 s c 2 s 0. The asymptotic property follows from Ž30. and by using HnŽ z . s O Ž z n .. 5.1. Approximating the Zeros Let yn, m and h n, m be the mth zero of Yn Ž z . and HnŽ z ., respectively, m s 1, 2, . . . , n. Then, for given and n we can compute a first approxi-
LOPEZ AND TEMME ´
472
mation of yn, m . We obtain, inverting the relation for given in Ž28., pz 2 q qz q 1 s 0, p s
1 4
2 q 4 q 4 q 2 2 Ž 3 q 8 . , q s q 2 q 2 2 .
This gives the relation zŽ . s
yq q i 2 Ž q 4 y 2
'
2
.
.
2p
Using this with s h n, m we obtain a first approximation of z s yn, m . The zeros of Yn Ž z . are complex, in contrast with those of the classical orthogonal polynomials, where the zeros are real and inside the domain of orthogonality. Information on the zeros distribution of Yn Ž z . for large values of seems not to be available in the literature. In Fig. 3 we show the curves z Ž . for g wy'2 n q 1 , '2 n q 1 x, in which interval the zeros h n, m of the Hermite polynomial HnŽ . occur w13x. 6. BUCHHOLZ POLYNOMIALS From Ž4. we obtain the following Cauchy-type integral for the Buchholz polynomials Pn
Ž z. s
1
He 2 i C
zŽcot wy1r w .r2
sin w
ž / w
dw w nq1
,
where C is a circle around the origin with radius less than .
FIG. 3. The curves in the z-plane under the mapping ª z Ž . are the images of the intervals wy'2 n q 1 , '2 n q 1 x where the zeros of the Hermite polynomial HnŽ . occur. We take n s 10 and show the curves Žfrom left to right. for s 100, 200, . . . , 500.
473
HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
We have P0 Ž z . s 1,
P1 Ž z . s y
z 6
P2 Ž z . s y
,
1 72
Ž 12 y z 2 . .
Hence, by Ž12. and pnŽ z . s Pn Ž z ., AŽ z . s y
z 6
BŽ z. s
,
6
.
It follows that Pn
Ž z. s
nr2
ž / 6
n
c k Ž z, .
Ý ks0 Ž n y k . !
kr2
6
ž /
Hnyk
yz
ž' / 2 6
,
Ž 32 .
where the coefficients c k Ž z, . of the expansion satisfy the properties given in the following lemma. LEMMA 6.1. The first six coefficients c nŽ z, . in the expansion Ž32. are c 0 Ž z, . s 1, c 4 Ž z, . s y
c1 Ž z, . s c 2 Ž z, . s 0,
180
,
c5 Ž z, . s y
z 945
c 3 Ž z, . s y c6 Ž z, . s
,
z 90
,
7z 2 y 40 113400
,
Ž 33 . and the remaining ones satisfy, for k G 1, < < q < z < ª ⬁.
c k Ž z, . s O Ž ? k r4 @ q z ? k r3@ . ,
Ž 34 .
Proof. Using equation Ž7., the function f Ž z, w . of the Buchholz polynomials can be written in the form f Ž z, w . s e 1Ž z , w .q 2 Ž z , w . , where 1 , 2 do not depend on , with
1 Ž z, w . s ln
sin w w
q BŽ z. w2 s w4 y
1 180
q O Ž w2 . ,
wª0
and
2 Ž z, w . s z Ž cot w y 1rw . r2 y A Ž z . w s zw 3 y 901 q O Ž w 2 . , w ª 0. Hence, Ž34. follows from Lemma 2.1.
LOPEZ AND TEMME ´
474
Ž . FIG. 4. Solid lines represent P10 x for several values of , whereas dashed lines represent the right-hand side of Ž37. with z s x.
Figure 4 shows the accuracy of the approximation
Pn Ž z . ,
1 n!
nr2
ž / 6
Hn
yz
ž' /
Ž 35 .
2 6
for n s 10, z s x and several values of . PROPOSITION 6.1. The Buchholz polynomials Pn Ž z . ha¨ e the finite expansion in terms of Hermite polynomials
Pn
Ž z. s
1 n!
ž / 6
nr2
Hn Ž . q
ž / 6
nr2
n
Ý c k Ž z, . ks3
6
ž /
kr2
Hny k Ž .
,
Ž n y k. ! Ž 36 .
HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
475
FIG. 4. Continued.
where
sy
z 2 6
'
,
Ž 37 .
the first six coefficients c k Ž z, . are gi¨ en in Ž33.. This is actually an asymptotic expansion of Pn Ž z . for < < q < z < ª ⬁. Proof. Ž36. follows trivially from Ž32. and using c 0 s 1, c1 s c 2 s 0. The asymptotic property follows from Lemma 6.1 and by using HnŽ z . s O Ž z n ..
476
LOPEZ AND TEMME ´
6.1. Approximating the Zeros We proceed in a similar way as in the previous cases. Let pn, m and h n, m be the mth zero of Pn Ž z . and HnŽ z ., respectively, m s 1, 2, . . . , n. Then, for given and n we take the relation for given in Ž37. to compute a first approximation of pn, m by writing pn , m ; y2 6 h n , m .
'
The accuracy of this approximation increases for increasing . For example, for s 20 or 40 and n s 10, the relative accuracy in the zeros is ; 10y2 . For s 100 or 200, the relative accuracy oscillates between 10y2 and 10y3 .
7. CONCLUSIONS Finite approximations of the generalized Bernoulli, Euler, Bessel, and Buchholz polynomials in terms of Hermite polynomials have been given. These are also asymptotic expansions of these polynomials with respect to certain sequences of the order parameter for < < ª ⬁. For large < <, the nth order polynomials become, up to a factor, the nth Hermite polynomial of a certain variable. From these approximations in terms of Hermite polynomials we have obtained asymptotic estimates of the zeros of these polynomials.
ACKNOWLEDGMENTS J. L. Lopez wants to thank C.W.I. of Amsterdam for its scientific support and hospitality ´ during the realization of this work. The financial support of Comision ´ Interministerial de Ciencia y Tecnologıa ´ is acknowledged. N. M. Temme wants to acknowledge the hospitality and financial support of the Facultad de Ciencias of the Universidad de Zaragoza.
REFERENCES 1. J. Abad and J. Sesma, Computation of Coulomb wave functions at low energies, Comp. Phys. Commun. 71 Ž1992., 110᎐124. 2. H. Buchholz, ‘‘The Confluent Hypergeometric Function,’’ Springer-Verlag, Berlin, 1969. 3. R. A. Buckingham, ‘‘Numerical Methods,’’ Pitman, London, 1957. 4. T. S. Chihara, ‘‘An Introduction to Orthogonal Polynomials,’’ Gordon and Breach, New York, 1978. 5. A. Erdelyi ´ et al., ‘‘Higher Transcendental Functions,’’ McGraw-Hill, London, 1953. 6. E. Grosswald, ‘‘Bessel Polynomials,’’ Lecture Notes in Mathematics 698, Springer, New York, 1978.
HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
477
7. J. L. Lopez and J. Sesma, The Whittaker function Mk, as a function of k, Const. ´ Approx. 15 Ž1999., 83᎐95. 8. J. L. Lopez and N. M. Temme, Approximations of orthogonal polynomials in terms of ´ Hermite polynomials, Meth. Appl. Anal. Žin press.. 9. A. T. Lundell, On the denominator of generalized Bernoulli numbers, J. Number Theory 26 Ž1987., 79᎐88. 10. L. M. Milne-Thomson, ‘‘The Calculus of Finite Differences Ž2nd ed..,’’ Macmillan, New York, 1965. 11. N. E. Norlund, ‘‘Vorlesungen ¨ uber Differenzenrechnung,’’ Springer, Berlin, 1924. ¨ 12. H. M. Srivastava and P. G. Todorov, An explicit formula for the generalized Bernoulli polynomials, J. Math. Anal. Appl. 130 Ž1988., 509᎐513. 13. N. M. Temme, ‘‘Special Functions: An Introduction to the Classical Functions of Mathematical Physics,’’ Wiley, New York, 1996. 14. P. G. Todorov, Une formule simple explicite des nombres de Bernoulli generalises, ´ ´ ´ C. R. Acad. Sci. Paris Ser. ´ I. Math. 301 Ž1985., 665᎐666. 15. P. G. Todorov, Explicit and recurrence formulas for generalized Euler numbers, Funct. Approx. 22 Ž1993., 113᎐117. 16. A. Weinmann, Asymptotic expansions of generalized Bernoulli polynomials, Proc. Camb. Phil. Soc. 59 Ž1963., 73᎐80. 17. R. Wong and J.-M. Zhang, Asymptotic expansions of the generalized Bessel polynomials, J. Comput. Appl. Math. 85 Ž1997., 87᎐112.
Hermite Polynomials in Asymptotic Representations of Generalized Bernoulli, Euler, Bessel, and Buchholz Polynomials Jose ´ L. Lopez ´ Departamento de Matematica Aplicada, Facultad de Ciencias, Uni¨ ersidad de Zaragoza, ´ 50013-Zaragoza, Spain
and Nico M. Temme1 CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands Submitted by L. Debnath Received April 8, 1999
This is the second paper on finite exact representations of certain polynomials in terms of Hermite polynomials. The representations have asymptotic properties and include new limits of the polynomials, again in terms of Hermite polynomials. This time we consider the generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. The asymptotic approximations of these polynomials are valid for large values of a certain parameter. The representations and limits include information on the zero distribution of the polynomials. Graphs are given that indicate the accuracy of the first term approximations. 䊚 1999 Academic Press
1. INTRODUCTION Generalized Bernoulli, Euler, Bessel, and Buchholz polynomials of degree n, complex order and complex argument z, denoted respectively by Bn Ž z ., En Ž z ., Yn Ž z . and Pn Ž z ., can be defined by their generating 1
E-mail: [email protected]. 457 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
LOPEZ AND TEMME ´
458 functions ww11x, chap. 6x, w ew z
Ž e w y 1.
s
Bn Ž z . n!
ns0
2 ew z
Ž e w q 1.
⬁
Ý
s
⬁
Ý
En Ž z . n!
ns0
wn,
< w < - 2 ,
Ž 1.
w n,
Ž 2.
ww4x, page 181x, 1
ž
'1 y 2 zw s
1 q '1 y 2 zw Yn Ž z .
⬁
Ý
2
n!
ns0
/
e 2 w rŽ1q '1y2 z w .
<2 zw < - 1,
w n,
Ž 3.
and ww2x, sec. 3x, in a slightly different notation, e
zŽcot wy1r w .r2
sin w
ž / w
s
⬁
Ý
Pn Ž z . w n ,
Ž 4.
ns0
The generalized Bernoulli and Euler polynomials play an important role in the calculus of finite differences. In fact, the coefficients in all the usual central-difference formulae for interpolation, numerical differentiation and integration, and differences in terms of derivatives can be expressed in terms of these polynomials w11x. Many properties of these polynomials can be found in ww3x, chap. 6x, ww5x, vol. 1, chap. 1x, w10x, and w11x. An explicit formula for the generalized Bernoulli polynomials can be found in w12x. Asymptotic expansions in terms of elementary functions and in terms of gamma and polygamma functions are obtained in w16x. Properties and explicit formulas for the generalized Bernoulli and Euler numbers can be found in w9x, w14x, w15x and references therein. The generalized Bessel polynomials form a set of orthogonal polynomials on the unit circle in the complex plane. They are important in certain problems of mathematical physics; for example, they arise in the study of electrical networks and when the wave equation is considered in spherical coordinates. For a historical survey and discussion of many interesting properties, we refer to w6x. New asymptotic expansions of Yn Ž x . Žand its zeros. for large values of n are given in w17x. Buchholz polynomials are used for the representation of the Whittaker functions as convergent series expansions of Bessel functions w2x. They appear also in the convergent expansions of the Whittaker functions in ascending powers of their order and in the asymptotic expansions of the
HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
459
Whittaker functions in descending powers of their order w7x. Explicit formulas for obtaining these polynomials may be found in w1x. In our first paper w8x it has been shown that Jacobi, Gegenbauer, Laguerre, and Tricomi-Carlitz polynomials have asymptotic representations in terms of the Hermite polynomials ? nr2 @
Hn Ž x . s n!
Ž y1.
Ý
k
k! Ž n y 2 k . !
ks0
Ž2 x.
ny 2 k
.
These asymptotic representations include well-known limits of the polynomials in terms of the Hermite polynomials, and provide a powerful tool for approximating the zeros of these polynomials in terms of the zeros of the Hermite polynomials in the asymptotic limit w8x. The polynomials of the previous paper are all orthogonal on a set of the real line. The present group is quite different. Only the Bessel polynomials are orthogonal, but not in the standard sense: they are orthogonal on the unit circle. In a certain sense the polynomials of the present group become orthogonal if the parameter becomes large. We give similar asymptotic representations for the new group as in the previous paper for the asymptotic limit < < ª ⬁. From these representations we can derive
lim
ª⬁
lim
ª⬁
ž / ž / 8
nr2
En
nr2
ª⬁
ª⬁
Bn
lim i n Ž 2 .
lim
nr2
24
6
ž /
ž ( / ž ( /
Yn y
2
2
2
q q
6
2
z s Hn Ž z . ,
z s Hn Ž z . , 2
ž ( / 1qi
nr2
Pn Ž y2 6 z . s
'
z
1 n!
s Hn Ž z . ,
Hn Ž z . .
From these limits we can obtain approximations for the zeros of these polynomials in the asymptotic regime. In the following section we give the principles of the Hermite-type asymptotic approximations used in this paper. In later sections we apply the method to obtain expansions for the generalized Bernoulli and Euler polynomials, and the Bessel and Buchholz polynomials. We also obtain estimates of their zeros for large .
LOPEZ AND TEMME ´
460
2. EXPANSIONS IN TERMS OF HERMITE POLYNOMIALS The Hermite polynomials have the generating function ⬁
2
e 2 z wyw s
Hn Ž z .
Ý
n!
ns0
z, w g ⺓,
wn,
which gives the Cauchy-type integral Hn Ž z . s
n!
He 2 i C
2 z wyw 2
dw
, Ž 5. w nq1 where C is a circle around the origin and the integration is in positive direction. The polynomials defined in Ž1. ᎐ Ž4., as well as many other well-known polynomials, may be defined by a generating function F Ž z, w . in the form F Ž z, w . s
⬁
Ý
pn Ž z . w n ,
Ž 6.
ns0
where F: ⺓ = ⺓ ª ⺓ is analytic with respect to w in a domain that contains the origin. We assume that F Ž z, 0. s p 0 Ž z . s 1 and that the polynomials pnŽ z . are independent of w. We have the Cauchy-type integral representation pn Ž z . s
dw
1
H F Ž z, w . w 2 i C
nq1
,
where C is a circle around the origin inside the domain where F is analytic Žas a function of w .. We write the generating function F Ž z, w . of pnŽ z . in the form 2
F Ž z, w . s e AŽ z .wyBŽ z .w f Ž z, w . ,
Ž 7.
where AŽ z . and B Ž z . are independent of w, and it follows that pn Ž z . s
1
He 2 i C
AŽ z .wyBŽ z .w 2
dw
f Ž z, w .
. Ž 8. w nq1 The function f is also analytic around the origin w s 0. Therefore, we can expand f Ž z, w . s 1 q p1 Ž z . y A Ž z . w q p 2 Ž z . y A Ž z . p1 Ž z . q B Ž z . q s
1 2
2
A Ž z . w 2 q ⭈⭈⭈
⬁
Ý ck Ž z . w k . ks0
Ž 9.
461
HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
Substituting this in Ž8. and using Ž5., we obtain the finite expansion pn Ž z . s Ž B Ž z . .
nr2
n
Ý ks0
Hnyk Ž .
ck Ž z .
Ž BŽ z. .
kr2
Ž n y k. !
s
,
AŽ z .
'
2 BŽ z.
, Ž 10 .
because terms with k ) n do not contribute to the integral in Ž8.. If B Ž z . happens to be zero for a special z-value, say z 0 , we write pn Ž z 0 . s A Ž z 0 .
n
n
Ý ks0
ck Ž z0 . AŽ z0 .
k
Ž n y k. !
.
Ž 11 .
In the examples considered in the following sections, the choice of AŽ z . and B Ž z . is based on our requirement that c1Ž z . s c 2 Ž z . s 0, in order to make the function f Ž z, w . close to 1 near the origin wnote that f Ž z, 0. s 1x. Then, the generating function F Ž z, w . is close to the generating function of the Hermite polynomials. Using c 0 Ž z . s 1 and requiring c1Ž z . s c 2 Ž z . s 0, we have, from Ž9., A Ž z . s p1 Ž z . ,
B Ž z . s 12 p12 Ž z . y p 2 Ž z . .
Ž 12 .
We can summarize the above discussion in the following. PROPOSITION 2.1. Consider the polynomials pnŽ z . defined in Ž6. by a generating function F Ž z, w . analytic in w s 0 and normalized in the form F Ž z, 0. s 1. Then, they may be represented as the finite sum Ž10. if p12 Ž z . y 2 p 2 Ž z . / 0, and as the finite sum Ž11. if p12 Ž z 0 . y 2 p 2 Ž z 0 . s 0. The functions c k Ž z . are the coefficients of the Taylor expansion of F Ž z, w . exp
½Ž
1 2
2
p1 Ž z . y p 2 Ž z . . w 2 y p1 Ž z . w
5
at w s 0, c 0 s 1, c1 s c 2 s 0 and Hn are the Hermite polynomials. In the following sections we verify if the finite sum in Ž10. yields asymptotic representations for the generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. The special choice of AŽ z . and B Ž z . given in Ž12. is crucial for obtaining asymptotic properties. To prove these properties we will use the following lemma. LEMMA 2.1. Let Ž w . be analytic at w s 0, with Maclaurin expansion of the form Ž w . s w n Ž a0 q a1w q a2 w 2 q ⭈⭈⭈ . q b1w q b 2 w 2 q ⭈⭈⭈ , where n is a positi¨ e integer, a k , bk are complex numbers that do not depend on the complex number , a0 / 0; let c k denote the coefficients of the power series of e Ž w ., that is, e Ž w . s Ý⬁ks 0 c k w k . Then c k s O Ž < < ? k r n@ . ,
ª ⬁.
LOPEZ AND TEMME ´
462
Proof. The proof follows from expanding e Ž w . s Ý⬁ks 0 w Ž w .x krk!, substituting the power series of and collecting equal powers of w.
3. GENERALIZED BERNOULLI POLYNOMIALS From Ž1. we obtain the following Cauchy-type integral for the generalized Bernoulli polynomials Bn Ž z . s
w ew z
n!
H 2 i C Ž e
w
y 1.
dw
w nq1
,
where C is a circle around the origin with radius less than 2 . We assume that F Ž z, w . s w e w zrŽ e w y 1. assumes real values for real values of z, w and . We have B0 Ž z . s 1,
B1 Ž z . s z y
2
B2 Ž z . s z 2 y z q
,
Ž 3 y 1. 12
.
Hence, by Ž12. and pnŽ z . s Bn Ž z .rn!, AŽ z . s z y
2
,
BŽ z. s
24
,
and by Ž10.,
Bn Ž z . s n!
ž / 24
nr2
n
ck Ž .
Ý ks0 Ž n y k . !
24
ž /
kr2
Hnyk
ž
'6 Ž z y r2.
'
/
.
Ž 13 . Observe that this representation shows the symmetry of Bn Ž z . with respect to the point z s 12 . The coefficients c k Ž . of the expansion are given in the following lemma. LEMMA 3.1. The odd coefficients c nŽ . in the expansion Ž13. ¨ anish, c 2 nq1 Ž . s 0
᭙ n G 0,
Ž 14 .
HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
463
and the e¨ en ones are independent of z; they are gi¨ en for n G 2 by the recurrence c2 n Ž . s
n
12 n y
Ý ks2 n
1
Ž 2 k q 1. Ž k y 3. q 6 c 2Ž nyk . Ž . Ž 2 k q 1. ! Ž n y k.
c Ž . , Ý n ks1 Ž 2 k q 1 . ! 2Ž nyk .
Ž 15 .
with c 0 Ž . s 1, c 2 Ž . s 0 and satisfy < < ª ⬁.
c 2 n Ž . s O Ž ? n r2 @ . ,
Ž 16 .
Proof. Using equation Ž7., the function f Ž z, w . of the generalized Bernoulli polynomials reads w e Ž1qw r12.w r2
f Ž z, w . s
Ž e w y 1.
s
⬁
Ý ck Ž . w k .
Ž 17 .
ks0
This is an even function in the variable w and Ž14. follows. It is independent of z and so are the coefficients c k Žwhich only depend on .. Moreover, it satisfies the differential equation
1q
ž
1 2
y
1 w
y
w 12
/Ž
e w y 1 . f Ž z, w . q Ž e w y 1 .
d dw
f Ž z, w . s 0,
and introducing the expansion Ž17. wwith c 2 nq1Ž . s 0x in this differential equation we obtain Ž15.. The function f Ž z, w . can be written in this case in the form f Ž z, w . s e w
4
1Ž w .
,
1Ž w . s
1 2880
q O Ž w2 . ,
w ª 0,
and 1Ž w . does not depend on and z. Hence, the proof of Ž16. follows from Lemma 2.1. PROPOSITION 3.1. The generalized Bernoulli polynomials Bn Ž z . ha¨ e the finite expansion in terms of Hermite polynomials Bn
Ž z. s
ž / 24
nr2
Hn Ž . q n!
ž / 24
nr2 ? nr2 @
Ý ks2
c2 k Ž .
24
ž /
k
Hny 2 k Ž .
,
Ž n y 2k. ! Ž 18 .
LOPEZ AND TEMME ´
464 where
s
'6 Ž z y r2.
Ž 19 .
'
and c 2 k Ž . are gi¨ en in Ž15.. This is actually an asymptotic expansion of Bn Ž z . for < < ª ⬁ with respect to the sequence ? k r2 @yk , uniformly with respect to . Proof. Ž18. follows trivially by using Ž13. and Lemma 3.1. The asymptotic property of Ž18. follows from Ž16.. If < < is bounded, the combination c 2 k Ž . yk in Ž18. gives the asymptotic nature for large values of < <; if < < is not bounded, then the property HnŽ . s O Ž n . gives extra asymptotic convergence in the sum in Ž18.. Figure 1 shows the accuracy of the approximation Bn Ž z . ,
nr2
ž /
Hn
24
ž
'6 Ž z y r2.
'
/
Ž 20 .
for n s 10, real z and several values of . 3.1. Approximating the Zeros When computing approximations of the zeros of the generalized Bernoulli polynomials for large values of we start with the zeros of the Hermite polynomial HnŽ . in Ž20.. Let bn, m and h n, m be the mth zero of Bn Ž z . and HnŽ z ., respectively, m s 1, 2, . . . , n. Then, for given and n we take the relation for given in Ž19. to compute a first approximation of bn, m by writing bn , m ;
2
q
(
6
hn, m .
For s 10, 20, 30 and n s 10, the best relative accuracy in the zeros is ; 10y3 and the worst result Žfor the largest zero. is ; 10y2 . For s 40, 50 it oscillates between 10y3 and 10y4 , whereas for s 100 it oscillates between 10y4 and 10y5 .
4. GENERALIZED EULER POLYNOMIALS From Ž2. we obtain the Cauchy-type integral for the generalized Euler polynomials n! 2 ew z dw En Ž z . s , w 2 i C Ž e q 1 . w nq1
H
465
HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
Ž . FIG. 1. Solid lines represent B10 x for several values of , whereas dashed lines represent the right-hand side of Ž20..
where C is a circle around the origin with radius less than . We assume that F Ž z, w . s 2 e w zrŽ e w q 1. assumes real values for real values of z, w, and . We have E0 Ž z . s 1,
E1 Ž z . s z y
2
E2 Ž z . s z 2 y z q
,
Ž y 1. 4
.
Hence, by Ž12. and pnŽ z . s En Ž z .rn!, AŽ z . s z y
2
BŽ z. s
,
8
.
It follows from Ž10. that En
Ž z . s n!
ž / 8
nr2
n
ck Ž .
Ý ks0 Ž n y k . !
8
ž /
kr2
Hnyk
ž
'2 Ž z y r2.
'
/
, Ž 21 .
466
LOPEZ AND TEMME ´
FIG. 1. Continued.
where the coefficients c k Ž . of the expansion are given below. This representation shows the symmetry of En Ž z . with respect to the point z s 12 . We have the following results. The proofs are as in the case of the Bernoulli polynomials. LEMMA 4.1. The odd coefficients c nŽ . in the expansion Ž21. ¨ anish, c 2 nq1 Ž . s 0
᭙ n G 0,
Ž 22 .
467
HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
and the e¨ en ones are independent of z; they are gi¨ en for n G 2 by the recurrence c2 n Ž . s
2k y 3
n
c Ž . Ý 16 n ks2 Ž 2 k y 1 . ! 2Ž nyk . q
Ž k y n y 1.
n
1
Ý 2 n ks2 Ž 2 k y 2 . !
c2Ž nykq1. Ž . ,
Ž 23 .
where c 0 Ž . s 1, c 2 Ž . s 0, and satisfy < < ª ⬁.
c 2 n Ž . s O Ž ? n r2 @ . ,
Ž 24 .
PROPOSITION 4.1. The generalized Euler polynomials En Ž z . ha¨ e the finite expansion in terms of Hermite polynomials En Ž z . s
ž / 8
nr2
Hn Ž . q n!
nr2 ? nr2 @
ž /
c2 k Ž .
Ý
8
ks2
8
ž /
k
Hny 2 k Ž .
,
Ž n y 2k. ! Ž 25 .
where
s
'2 Ž z y r2.
Ž 26 .
'
and c k Ž . are gi¨ en in Ž23.. This is actually an asymptotic expansion of En Ž z . for < < ª ⬁ with respect to the sequence ? k r2 @yk , uniformly with respect to . Figure 2 shows the accuracy of the approximation En Ž z . ,
ž / 8
nr2
Hn
ž
'2 Ž z y r2.
'
/
Ž 27 .
for n s 10, real z and several values of . 4.1. Approximating the Zeros Let e n, m and h n, m be the mth zero of En Ž z . and HnŽ z ., respectively, m s 1, 2, . . . , n. Then, for given and n we take the relation for given
LOPEZ AND TEMME ´
468
Ž . FIG. 2. Solid lines represent E10 x for several values of , whereas dashed lines represent the right-hand side of Ž27..
in Ž26. to compute a first approximation of e n, m by writing en , m ;
2
q
(
2
hn, m .
The accuracy is as in the case of the generalized Bernoulli polynomials.
5. GENERALIZED BESSEL POLYNOMIALS From Ž3. we obtain the following Cauchy-type integral for the generalized Bessel polynomials Yn
Ž z. s
n!
1
H 2 i C '1 y 2 zw
ž
2 1 q '1 y 2 zw
/
e 2 w rŽ1q '1y2 z w .
dw w nq1
where C is a circle around the origin with radius less than <1rŽ2 z .<.
,
HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
FIG. 2. Continued.
We have Y0 Ž z . s 1, Y2 Ž z . s
1 4
Y1 Ž z . s
1 2
2 q Ž q 2. z ,
4 q 4 Ž q 3 . z q Ž q 7 . q 12 4 z 2 .
Hence, by Ž12. and pnŽ z . s Yn Ž z .rn!,
AŽ z . s
1 2
2 q Ž q 2. z ,
BŽ z. s y
z 8
4 q Ž 3 q 8. z .
469
LOPEZ AND TEMME ´
470 It follows from Ž10. that Yn Ž z . s n! B Ž z .
nr2
c k Ž z, .
n
Ý ks0
Hnyk Ž .
kr2
BŽ z.
Ž n y k. !
s
,
AŽ z .
'
2 BŽ z.
Ž 28 . where the coefficients c k Ž z, . of the expansion satisfy the properties given in the following lemma. We introduce the notation y Ž w . s '1 y 2 zw s 1 y
⬁
Ý
bk s
z k bk w k ,
ks1
2k
1
ž /
2 k! 2
. ky 1
LEMMA 5.1. The coefficients c k Ž z, . in the expansion Ž28. are gi¨ en by the recursion relation 16 Ž k q 1 . c kq 1 s 48 kzck y 28 Ž k y 1 . z 2 c ky1 q 4 6 q Ž 18 y k q 5 . z z 2 c ky2 y 32 q Ž 64 q 3 Ž k y 3 . q 25 . z z 3 c ky 3 ky4
q2
Ý Ž 4 jbkq 1yj js0
y 8 jbky j y bky1yj . z kq 1yj y2 Ž 4 q 3 q 8 z . z ky j bky 2yj c j ,
Ž 29 .
where c j s 0 if j - 0 and empty sums are zero with c 0 Ž z, . s 1, c1Ž z, . s c 2 Ž z, . s 0, and they satisfy the asymptotic estimate < < ª ⬁.
c k Ž z, . s O Ž ? k r3@ . ,
Ž 30 .
Proof. The function f Ž z, w . s
eyA wqB w
2
q2 w rŽ1q 1y2 z w .
'
ž
'1 y 2 zw
2 1 q '1 y 2 zw
/
satisfies the differential equation 2
2
y 2 Ž 1 q y . f ⬘ s Ž 2 Bw y A . y 2 Ž 1 q y . q 2 y 2 Ž 1 q y . q 2 zwy 2
qz Ž 1 q y . q zy Ž 1 q y . f .
471
HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
Then, writing f Ž z, w . s Ý⬁ks 0 c k w k we obtain the recursion Ž29. upon substitution. The function f Ž z, w . can be written in the form f Ž z, w . s e 1Ž z , w .q 2 Ž z , w . , where 1 , 2 do not depend on , with
1 Ž z, w . s ln
2
y
1 q '1 y 2 zw 5
s w3
12
z3 q O Ž w. ,
1 2
zw y
3 8
z2 w2
wª0
and
2 Ž z, w . s w 3
1 6
Ž 8 z q 3. z 2 q O Ž w . ,
w ª 0.
Hence, Ž30. follows from Lemma 2.1. The first few terms are c 0 Ž z, . s 1, c1Ž z, . s c 2 Ž z, . s 0, and c 3 Ž z, . s c 4 Ž z, . s c5 Ž z, . s
z2 12 z3 64 z4 80
Ž 5 q 16 . z q 6 , Ž 35 q 128. z q 40 , Ž 63 q 256. z q 70 .
PROPOSITION 5.1. The generalized Bessel polynomials Yn Ž z . ha¨ e the finite expansion in terms of Hermite polynomials Yn Ž z . s B Ž z .
nr2
n
Hn Ž . q n!
Ý ks3
c k Ž z, . BŽ z.
kr2
Hnyk Ž .
Ž n y k. !
,
Ž 31 .
where is gi¨ en in Ž28.. This is actually an asymptotic expansion of Yn Ž z . for < < ª ⬁ and holds for fixed ¨ alues of z and n. Proof. Ž31. follows trivially from Ž28. and using c 0 s 1, c1 s c 2 s 0. The asymptotic property follows from Ž30. and by using HnŽ z . s O Ž z n .. 5.1. Approximating the Zeros Let yn, m and h n, m be the mth zero of Yn Ž z . and HnŽ z ., respectively, m s 1, 2, . . . , n. Then, for given and n we can compute a first approxi-
LOPEZ AND TEMME ´
472
mation of yn, m . We obtain, inverting the relation for given in Ž28., pz 2 q qz q 1 s 0, p s
1 4
2 q 4 q 4 q 2 2 Ž 3 q 8 . , q s q 2 q 2 2 .
This gives the relation zŽ . s
yq q i 2 Ž q 4 y 2
'
2
.
.
2p
Using this with s h n, m we obtain a first approximation of z s yn, m . The zeros of Yn Ž z . are complex, in contrast with those of the classical orthogonal polynomials, where the zeros are real and inside the domain of orthogonality. Information on the zeros distribution of Yn Ž z . for large values of seems not to be available in the literature. In Fig. 3 we show the curves z Ž . for g wy'2 n q 1 , '2 n q 1 x, in which interval the zeros h n, m of the Hermite polynomial HnŽ . occur w13x. 6. BUCHHOLZ POLYNOMIALS From Ž4. we obtain the following Cauchy-type integral for the Buchholz polynomials Pn
Ž z. s
1
He 2 i C
zŽcot wy1r w .r2
sin w
ž / w
dw w nq1
,
where C is a circle around the origin with radius less than .
FIG. 3. The curves in the z-plane under the mapping ª z Ž . are the images of the intervals wy'2 n q 1 , '2 n q 1 x where the zeros of the Hermite polynomial HnŽ . occur. We take n s 10 and show the curves Žfrom left to right. for s 100, 200, . . . , 500.
473
HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
We have P0 Ž z . s 1,
P1 Ž z . s y
z 6
P2 Ž z . s y
,
1 72
Ž 12 y z 2 . .
Hence, by Ž12. and pnŽ z . s Pn Ž z ., AŽ z . s y
z 6
BŽ z. s
,
6
.
It follows that Pn
Ž z. s
nr2
ž / 6
n
c k Ž z, .
Ý ks0 Ž n y k . !
kr2
6
ž /
Hnyk
yz
ž' / 2 6
,
Ž 32 .
where the coefficients c k Ž z, . of the expansion satisfy the properties given in the following lemma. LEMMA 6.1. The first six coefficients c nŽ z, . in the expansion Ž32. are c 0 Ž z, . s 1, c 4 Ž z, . s y
c1 Ž z, . s c 2 Ž z, . s 0,
180
,
c5 Ž z, . s y
z 945
c 3 Ž z, . s y c6 Ž z, . s
,
z 90
,
7z 2 y 40 113400
,
Ž 33 . and the remaining ones satisfy, for k G 1, < < q < z < ª ⬁.
c k Ž z, . s O Ž ? k r4 @ q z ? k r3@ . ,
Ž 34 .
Proof. Using equation Ž7., the function f Ž z, w . of the Buchholz polynomials can be written in the form f Ž z, w . s e 1Ž z , w .q 2 Ž z , w . , where 1 , 2 do not depend on , with
1 Ž z, w . s ln
sin w w
q BŽ z. w2 s w4 y
1 180
q O Ž w2 . ,
wª0
and
2 Ž z, w . s z Ž cot w y 1rw . r2 y A Ž z . w s zw 3 y 901 q O Ž w 2 . , w ª 0. Hence, Ž34. follows from Lemma 2.1.
LOPEZ AND TEMME ´
474
Ž . FIG. 4. Solid lines represent P10 x for several values of , whereas dashed lines represent the right-hand side of Ž37. with z s x.
Figure 4 shows the accuracy of the approximation
Pn Ž z . ,
1 n!
nr2
ž / 6
Hn
yz
ž' /
Ž 35 .
2 6
for n s 10, z s x and several values of . PROPOSITION 6.1. The Buchholz polynomials Pn Ž z . ha¨ e the finite expansion in terms of Hermite polynomials
Pn
Ž z. s
1 n!
ž / 6
nr2
Hn Ž . q
ž / 6
nr2
n
Ý c k Ž z, . ks3
6
ž /
kr2
Hny k Ž .
,
Ž n y k. ! Ž 36 .
HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
475
FIG. 4. Continued.
where
sy
z 2 6
'
,
Ž 37 .
the first six coefficients c k Ž z, . are gi¨ en in Ž33.. This is actually an asymptotic expansion of Pn Ž z . for < < q < z < ª ⬁. Proof. Ž36. follows trivially from Ž32. and using c 0 s 1, c1 s c 2 s 0. The asymptotic property follows from Lemma 6.1 and by using HnŽ z . s O Ž z n ..
476
LOPEZ AND TEMME ´
6.1. Approximating the Zeros We proceed in a similar way as in the previous cases. Let pn, m and h n, m be the mth zero of Pn Ž z . and HnŽ z ., respectively, m s 1, 2, . . . , n. Then, for given and n we take the relation for given in Ž37. to compute a first approximation of pn, m by writing pn , m ; y2 6 h n , m .
'
The accuracy of this approximation increases for increasing . For example, for s 20 or 40 and n s 10, the relative accuracy in the zeros is ; 10y2 . For s 100 or 200, the relative accuracy oscillates between 10y2 and 10y3 .
7. CONCLUSIONS Finite approximations of the generalized Bernoulli, Euler, Bessel, and Buchholz polynomials in terms of Hermite polynomials have been given. These are also asymptotic expansions of these polynomials with respect to certain sequences of the order parameter for < < ª ⬁. For large < <, the nth order polynomials become, up to a factor, the nth Hermite polynomial of a certain variable. From these approximations in terms of Hermite polynomials we have obtained asymptotic estimates of the zeros of these polynomials.
ACKNOWLEDGMENTS J. L. Lopez wants to thank C.W.I. of Amsterdam for its scientific support and hospitality ´ during the realization of this work. The financial support of Comision ´ Interministerial de Ciencia y Tecnologıa ´ is acknowledged. N. M. Temme wants to acknowledge the hospitality and financial support of the Facultad de Ciencias of the Universidad de Zaragoza.
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HERMITE POLYNOMIALS IN ASYMPTOTIC REPRESENTATIONS
477
7. J. L. Lopez and J. Sesma, The Whittaker function Mk, as a function of k, Const. ´ Approx. 15 Ž1999., 83᎐95. 8. J. L. Lopez and N. M. Temme, Approximations of orthogonal polynomials in terms of ´ Hermite polynomials, Meth. Appl. Anal. Žin press.. 9. A. T. Lundell, On the denominator of generalized Bernoulli numbers, J. Number Theory 26 Ž1987., 79᎐88. 10. L. M. Milne-Thomson, ‘‘The Calculus of Finite Differences Ž2nd ed..,’’ Macmillan, New York, 1965. 11. N. E. Norlund, ‘‘Vorlesungen ¨ uber Differenzenrechnung,’’ Springer, Berlin, 1924. ¨ 12. H. M. Srivastava and P. G. Todorov, An explicit formula for the generalized Bernoulli polynomials, J. Math. Anal. Appl. 130 Ž1988., 509᎐513. 13. N. M. Temme, ‘‘Special Functions: An Introduction to the Classical Functions of Mathematical Physics,’’ Wiley, New York, 1996. 14. P. G. Todorov, Une formule simple explicite des nombres de Bernoulli generalises, ´ ´ ´ C. R. Acad. Sci. Paris Ser. ´ I. Math. 301 Ž1985., 665᎐666. 15. P. G. Todorov, Explicit and recurrence formulas for generalized Euler numbers, Funct. Approx. 22 Ž1993., 113᎐117. 16. A. Weinmann, Asymptotic expansions of generalized Bernoulli polynomials, Proc. Camb. Phil. Soc. 59 Ž1963., 73᎐80. 17. R. Wong and J.-M. Zhang, Asymptotic expansions of the generalized Bessel polynomials, J. Comput. Appl. Math. 85 Ž1997., 87᎐112.