Characterization of generalized Bessel polynomials in terms of polynomial inequalities

Characterization of Generalized Bessel Polynomials in

Terms of Polynomial Inequalities*

Eliana X. L. de Andrade, Dimitar K. Dimitrov,†_{and A. Sri R anga}

Departamento de Ciencias de Computacˆ ¸ao e Estatıstica, IBILCE, Uni˜ ´ ¨ersidade Estadual Paulista, Sao Jose do Rio Preto, 15054-000 SP, Brazil˜ ´

Submitted by Robert A. Gustafson

R eceived July 12, 1996

Ž .

Generalized Bessel polynomials GBPs are characterized as the extremal poly-nomials in certain inequalities in L2 _{norm of Markov type. Q1998 A ca dem ic P re ss}

1. INTR ODU CTION AND STATEMENT OF R ESU LTS

Ž .

Generalized Bessel polynomials y x_n ;a,b are defined by

k

n

_Ž

_yxrb

_.

yn

Ž

x;a,b

.

[

_Ý

Ž

yn

. Ž

k nqay1

.

k k!

ks0

s2F0

Ž

yn,nqay1;y;yxrb

.

,

Ž .

1

Ž . Ž . Ž . Ž .

where a _k denotes the Pochhammer symbol, a₀[1, a_k[a aq1

Ž .

??? aqky1 , kG1. For asbs2 they reduce to the simple Bessel

Ž . Ž . w x

polynomials y x_n [y x_n ; 2, 2 . A recent review of Srivastava 5 contains comprehensive historical information and new results about GBPs. H ere

w x

we mention only the following orthogonality relation given in 5 :

`

ay2 ybrx

x e y x

Ž

;a,b

.

y

Ž

x;a,b

.

dx

H

j m

0

j! ay1

sb G

Ž

2yayj

.

djm, 1yay2j

2

Ž .

4

R e

Ž .

a -1ymyj, R e

Ž

b

.

)0, m,jgN0[Nj 0 .

* R esearch supported by the Brazilian foundations FAPESP and CNPq and the Bulgarian Science Foundation under Grant MM-414.

†

On leave from the U niversity of R ousse, Bulgaria. E-mail: dimitrov@nimitz.dcce. ibilce.unesp.br.

538

0022-247Xr98 $25.00

CopyrightQ_{1998 by A ca de m ic P re ss}

Even though the parameter b does not have essential influence in our investigation, we keep it in order to maintain the common notation

Ž .

y xn ;a,b .

Observe that, for a,bgR, b)0, ngN0, and 2n-1ya,

1r2 `

ay2 ybrx 2

5 5p a,PP_ns

ž

H

x e p

Ž .

x dx

/

Ž .

3

0

is a well-defined norm in the space PP _{of real algebraic polynomials of} n

degree not exceeding n. Then under the same restrictions on a,b, and n, 5p95aq2,PP_ny1 is well defined because the inequalities 2n-1ya and

Ž . Ž .

2 ny1 -1y aq2 are equivalent.

Ž .

First we prove an inequality of Markov type for the norm 3 for which GBPs are the extremal polynomials:

TH EOR EM1. Leta,bgR,b)0, ngN0,and2n-1ya. For e¨ery

pgPP_{n and any positi¨e integer k, kFn,}

1r2 k

n!

Žk.

5p 5aq2k,PP_nykF

ž

_Ł

Ž

4yayny3j

.

/

5 5p a,PP_n.

Ž .

4

nyk !

Ž

.

js1

Moreo¨er, for each k, ks1, . . . ,n, equality is attained if and only if p is a

Ž .

constant multiple of y x_n ;a,b .

Ž . k _{Ž .}

It is clear that the expression n!r nyk !Ł_js14yayny3j , which appears in the best constant

1r2 k

n!

Mn

Ž

k,a

.

s

ž

_Ł

Ž

4yayny3j

.

/

nyk !

Ž

.

js1

Ž .

in 4 , is positive because 2n-1ya. Ž .

Inequalities similar to 4 which characterize the classical Jacobi and

w x

general Laguerre polynomials are obtained by the second author in 1 and

w x w x

independently by Guessab and Milovanovic 2 and Min 4 . Chapter 6 of

_´

w x

Milovanovic, Mitrinovic, and R assias’s book 3 provides complete informa-

´

´

tion about various inequalities for polynomials.

Next we shall prove an inequality of Markov type for a special type of ‘‘quasi-polynomials.’’ Let a,bgR, b)0, ngN0, and 2n-y1ya.

Ž .

The set of ‘‘quasi-polynomials QQ_{n a is defined by}

Q

Q _{a [ q x sx}a_eybrx_{p x : p} gPP _.

Ž .

Ž .

Ž .

4

n n

Ž .

Then QQ_n a can be equipped with the norm

1r2 `

ya brx 2

5 5q a,QQn_Ža.s

ž

H

x e q

Ž .

x dx

/

.

Ž .

5

Ž . Ž .

Since for any qgQQ _{a its derivative belongs to} QQ _{ay2 and the}

n nq1

Ž . Ž .

restriction 2n-y1ya is equivalent to 2 nq1 -y1y ay2 , then 5q95a,QQnq1Žay2. is well defined.

TH EOR EM 2. Let a,bgR, b)0, ngN0, and 2n-y1ya. Then

the inequality

1r2

5q95ay2 ,QQnq1_Žay2.F

Ž

Ž

nq1

. Ž

yayn

.

.

5 5q a,QQn_Ža.

Ž .

6 Ž .

holds for e¨ery qgQQ_{n a . Moreo¨er, equality is attained if and only if q is a}

a ybrx X _{Ž .}

constant multiple of x e ynq1 x;a,b .

Ž .Ž .

Note that the expression nq1 yayn which appears on the right-Ž .

hand side of 6 is positive because 2n-y1ya.

2. PR OOFS

It is well known that the generalized Bessel polynomial of degree n

satisfies the second-order differential equation

x2y0 q

Ž

axqb

.

y9yn n

Ž

qay1

.

ys0.

Ž .

7

Obviously, this can be rewritten in the self-adjoint form:

xa_eybrx_{y9 9}

sn n

_Ž

qay1

_.

xay2_eybrx_y.

_{Ž .}

₈

Ž

.

Ž . It follows immediately from 1 that

n n

Ž

qay1

.

X

yn

Ž

x;a,b

.

s yny1

Ž

x;aq2,b

.

.

Ž .

9

b

Ž .

Proof Theorem1. First we prove 4 for ks1. Let p be an arbitrary polynomial of degree n. U nder the restrictions on a, b, and n, the

Ž . Ž .

polynomials y x₀ ; a,b , . . . ,y x_n ;a,b are orthogonal polynomials. H ence they form a basis in PP _{and p can be uniquely represented in the}

n

Ž . n _{Ž .}

form p x sÝjs0a y xj

˜

j ;a,b , where

1r2

1yay2j

y x

Ž

;a,b

.

[ y x

Ž

;a,b

.

, js0, . . . ,n,

˜

j

ž

_bay1_j!

/

j

G

Ž

2yayj

.

are the orthonormalized GBPs. Therefore

n

2 2

5 5p a,PP_ns

_Ý

a_j.

Ž . n X_{Ž . Ž .}

Since p9 x sÝjs1a y xj

˜

j ;a,b then 9 yields

1r2

n _1yay2j

p9

Ž .

x s

_Ý

aj

ž

_bay1_{j!G 2}

/

yayj

Ž

.

js1

j j

Ž

qay1

.

= yjy1

Ž

x;aq2,b

.

.

b

Ž . Ž .

Note that the polynomials y x0 ;aq2,b , . . . ,yny1 x;aq2,b are

or-Ž . a ybrx

thogonal on 0,` with respect to the weight function x e . The Ž .

requirement a-1y2n is equivalent to aq2-1y2 ny1 . There-Ž .

fore we can apply 2 for asaq2 and m,jFny1. Thus

2 2

n _{1yay2j j}

_Ž

_jqay1

_.

2 aq1

5p95aq2 ,PP_ny1s

_Ý

_{ay1 2} b b j!G

Ž

2yayj

.

b

js1

jy1 !

Ž

.

₂

= G

Ž

1yayj a

.

j

1yay2j

n

2

s

_Ý

j

Ž

1yayj a

.

j.

js1

The statement of the theorem for ks1 will be proved if we show that the solution of the extremal problem

n n n

2 2 2

max

½

_Ý

j

Ž

1yayj a

.

j

_Ý

aj: a0, . . . ,angR,

_Ý

aj/0

5

Ž .

10

js0 js0 js0

2_{Ž .}

is M_n 1,a and it is attained for a₀s ??? sa_ny1s0. It is well known Ž .

and easy to see that 10 is equivalent to the problem of determining

max j

Ž

1yayj

.

.

1FjFn

Ž . Ž .

Since f j [j1yayj is a concave binomial with zeros at 0 and 1ya, Ž Ž . .

then f is an increasing function of j for jg 0, 1ya r2 . On the other Ž .

hand, the requirement a-1y2nis equivalent to n- 1ya r2. H ence Ž . Ž .

f j -f n for js0, . . . ,ny1. Therefore the solution of the extremal Ž . Ž . 2_{Ž .}

problem 10 is n1yayn sM_n 1,a and the maximal value is at-tained only for a₀s ??? sa_ny1s0 and an arbitrary nonzero a_n. This

Ž .

completes the proof for ks1. The inequalities 4 for ks2, . . . ,n follow by repeated application of that for ks1.

Ž .

Proof of Theorem2. Let qgQQ_{n a be arbitrary. Since the polynomials}

X_{Ž .} X _{Ž . Ž .}

y x₁ ;a,b , . . . ,y_n x;a,b form a basis in QQ _{a , then qcan be uniquely}

q1 n

Ž . a ybrx nq1 X_{Ž .}

mind that under the restrictions on a, b, and n, the polynomials

X_{Ž . Ž .}

y xj ;a,b , js1, . . . ,nq1, are orthogonal on 0,` with respect to the a ybrx _{Ž . Ž .}

weight function x e , and using 9 and 2 for asaq2, we get

2 nq1

`

2 a ybrx X

5 5q a,QQ_ns

H

x e

ž

_Ý

a y_{j j}

Ž

x;a,b

.

/

dx

0 _js1

nq1

2

X

2

s

_Ý

aj yj

Ž

x;a,b

.

aq2 , PP_n

y1

js1

nq1

_Ž

_{jy1 !}

_.

2 2 ay1 2

s

_Ý

ajb j

Ž

jqay1

.

G

Ž

1yayj

.

.

Ž .

11 1yay2j

js1

Ž . On the other hand, 8 yields

nq1

ay2 ybrx

q9

Ž .

x sx e

_Ý

a j jj

Ž

qay1

. Ž

y xj ;a,b

.

.

js1

Then

2 nq1

`

2 ay2 ybrx

5q95ay2 ,QQ_nq1s

H

x e

ž

_Ý

a j j_j

Ž

qay1

. Ž

y x_j ;a,b

.

/

dx

0 _js1

nq1 _j!

2 2 ay1 2

s

_Ý

ajb j

Ž

jqay1

.

G

Ž

2yayj

.

. 1yay2j

js1

12

Ž .

Introducing the notation

jy1 !

Ž

.

2

2 2 ay1 2

bj [ajb j

Ž

jqay1

.

G

Ž

1yayj

.

, 1yay2j

Ž . Ž .

we can rewrite 11 and 12 in the form

nq1

2 2

5 5q a,QQ_ns

_Ý

b_j js1

and

nq1

2 2

5q95ay2 ,QQ_nq1s

_Ý

j

Ž

1yayj b

.

_j,

respectively. Note that the requirements of the theorem are equivalent to Ž .

nq1- 1ya r2. Then the same reasoning as in the proof of Theorem 1 completes the proof.

R EFER ENCES

Ž . Ž .

1. D. K. Dimitrov, Bernstein inequality in L₂ norm, Math. Balkanica N.S. 7 1993 , 131]136.

2. A. Guessab and G. V. Milovanovic, Weighted_´ L2_{-analogues of Bernstein’s inequality and}

Ž .

classical orthogonal polynomials,J.Math. Anal. Appl.182 1994 , 244]249.

3. G. V. Milovanovic, D. S. Mitrinovic, and Th. M. R assias, ‘‘Topics in Polynomials: Extremal´ ´

Problems, Inequalities, Z eros,’’ World Scientific, Singapore, 1994.

4. G. H . Min, Bernstein]Markov inequalities in L2 _{spaces and their optimal constants II,}

Ž .

J. Math.Res.Exposition14 1994 , 135]138.

5. H . M. Srivastava, Orthogonality relations and generating functions for the generalized

Ž .