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Orthogonality with respect to the sum of two semiclassical regular linear forms
Journal of Computational North-Holland
and Applied
Mathematics
265
37 (1991) 265-272
Orthogonality with respect to the sum of two semiclassical regular linear forms A. Ronveaux Physique MathPmatique, Belgium
S. Belmehdi
Facultb
Universitaires
F. Marcellan
Received
61, B-5000 Namur,
*
Laboratoire d’Analyse Numkique, Paris Cedex 05, France
Departamento Spain
Notre Dame de la Paix, rue de Bruxelles
UniversitP Pierre et Marie Curie, Tour 55-65, 5e &age, 4 Place Jussieu, F-75252
**
de Matemciticas,
30 December
ETS de Ingenieros Industriales,
c/JosP
Gutikrrez Abascal 2, E-28006 Madrid,
1990
Abstract Ronveaux, A., S. Belmehdi and F. Marcellkr, Orthogonality with respect to the sum of two semiclassical linear forms, Journal of Computational and Applied Mathematics 37 (1991) 2655272.
regular
The linear form sum of two semiclassical regular linear forms verifies in general a second-order differential equation and we examine some situations where this new form remains semiclassical. The sequences of polynomials orthogonal with respect to this new form, called of second category, have associated polynomials which do not belong to the Laguerre-Hahn class. Keywords: Orthogonal
polynomials,
semiclassical
form, Stieltjes
functions.
1. Introduction
Let { P,(X)} be a family of polynomials P,(x), denoted orthogonal with respect to a positive linear form 2 [1,13]:
P,,, of degree exactly equal to n,
(L-i?, Pn’) =d,Z. (9, P,P,) =O, n#m, In some situations described below it will be convenient to assume that 2 positive Bore1 measure dp of support I:
(2,
.‘=l.
* This author’s research * * This author’s research 0377-0427/91/$03.50
(1)
is represented by a
dl-L.
(2)
has been partially supported by Action Integrado Hispano-Francesa has been supported by CECYT PB89/01981/C02/01.
0 1991 - Elsevier Science Publishers
B.V. AlJ rights reserved
31/90.
A. Ronveaux et al. / Orthogonality and linear forms
266
Modifications polynomials, l
of t_he form 2, and study of the corresponding new family of orthogonal say { P,, }, have already been examined since a long time in the following cases.
S-type modifications
(classical
type, for instance).
~=~+XS(x-cc),
h>O,
(3)
where 6(x - c) is the usual Dirac distribution and h is assumed to be positive that 2 is still positive definite (A > 0 can of course be weakened). l
Polynomial-type P=
modification.
n(x)9,
(4)
where l-I(x) is a rational l
in order to insure
function.
Truncated weights p(x). ,ii=
where H(x)
zH(d,-x)
~H(x-c;)+ ,[ i
j
is the Heaviside
H(x)=O,
p,
dp=p
dx,
(5)
I
step function:
x
H(x)=l,
x20.
(6)
Modifications of the first type appear in [16] (see also [4,5,8-11,191); polynomial-type modifications were already examined by Christoffel [2,6,17,18] when II(x) is a polynomial and more recently by Ouvarov [16] when IJ( x) is a rational function; truncated weights were met by physicists [12,20-223 in several situations and were treated in [l]. These three new forms 2 can also be interpreted as a sum of two linear forms L?=.S’i +LF* where -Ep, is singular in the first case, and equal to (n(x) - l)P in the second case. The third case merits some special attention and some comments. The relative positions of ci and di give several configurations for the new weight p, so let us assume first that there exist only two cuts at x = c and x = d with c < d, c and d inside the support (a, b) of p(x). Two situations appear: &=
[H(x-c)+H(d-x)]p,
(7)
,&=
[H(c-x)+H(x-d)]p.
(8)
p2 for instance can now be interpreted as the sum of two disjoint weights represented by the same function p(x) where the value of p(x) between c and d is reduced to zero [l]. If the starting weight p (or the corresponding linear form 9) is semiclassical [7,13], L? satisfies the functional equation D(+.Y)
+ +Z=
where I$( x) and q(x) ($2, (D(W% Equation
0,
are polynomial
in x and where (10)
PA = (=K Ic,P”)? P,,) = -(G’,
(9) can also be written cpD@‘)
(9)
+ ($’ + #)z=
K’).
(II)
as 0,
(12)
267
A. Ronveaux et al. / Orthogonality and linear forms
and the functional comes [l]
equation
for the truncated
weight
,!i = H( x - c)p
(or p = H( c - x)p),
(x-c)+D(.S?)+(x-c)(+‘+rc/).Y=O, or in the standard
be(13)
form
D( $9)
+ &Y= 0,
(14)
where jJ=(x-c)JI-$.
6 = (x - c)G, From the more standard @)‘+$p=o the multiplication
05)
point of view of a weight differential
equation
[15]
7
(16)
by x - c insures
that the new weight
p verifies
the important
lim&5=0. x-c
condition
[15] (17)
Of course the operation of tr_uncation increase in general the class s of the weight [1,13] given as usual by s = max(degree of 1c,- 1, degree of $ - 2). The only exception being the Legendre weight p = 1, a < x < b, which is completely insensitive to a cut at any c inside (a, b). More generally for an arbitrary number of cuts hidden inside a “cutting polynomial” E(x) = lJ,,,( x - ci)( x - dj), relation (15) becomes: \CI=E(x)\l/-E’(x)+
S=E(x)+,
(18)
This shows already that the new family { p,,} remains semiclassical and verifies therefore a second-order linear differential equation [14]. The recurrence coefficients &, and y, can also be obtained for these new polynomials p,. It was shown recently [1,3] how to write in general ,the nonlinear (&, y,) relations using only the relation in the explicit knowledge of 6 and $ given by (18); we write here the recurrence standard form: P n+2
=
(x
-
a+l>P,+,
-
n 2 0,
Yn+lP,,
The distinction between the two different instance are only given by the different relations being identical in both cases. For p = pi,
P1=x-PO,
PO’ 1.
(19)
families generated by ,Zi and p2 in (7) and (8) for initial condition on the &,, the nonlinear (&,, y,,)
d -v(x)
J I%=
dx
Cd / c
; ~(4
(20)
dx
forp=p,, fxp(x) P,=
dx + op(x)
aC J, P(X) dx + i6xp(x)
dx
(21) dx
.
268
A. Ronveaux et al. / Orthogonality and linear forms
These detailed comments emphasize therefore the interest of investigating the family of polynomials orthogonal with respect to the sum of two positive linear forms PI and _Z& generalizing the first case where ZI is singular and the third case where the intersecting support of PI and ZZ is empty. The thud case shows also that in several situations the sum of two (or more than two) semiclassical forms are still semiclassical.
2. Orthogonal polynomials of second category Let Zi, i = 1, 2, be the positive
linear form solution
of the semiclassical
functional
D(G;(x)q) +#;(X)Z=O,
equation
(22)
where I/I~and & are given polynomials in x. The positive form .Y= P’t +PZ, defined by (9, P) = ( Zl, P) + ( Yz, P), P any polynomial, verifies in general a second-order functional equation easily obtained by “differentiation” and elimination of 9, and PZ:
Computing
one more derivative
we obtain
(24) The determinantal
form of the second-order
functional
equation
reads now as
(25) writing for short 9’ This second-order l
is satisfied
+:+Gi=O, This generalized
to a first order in some cases.
(26)
or ~&+6)=d&h+~~).
A=B
This condition
for D(Z). equation reduces
for the following
constraint:
i=l,2.
Legendre
weight corresponds
P=X,H(x-c,)H(d,-x)+A,H(x-c,)H(d,-x),
(27) to a sum of two step functions Ci
[l]: Ai>O.
(28)
But from the fact that 6: + qi = 0 if $: + Gi = 0 in relation (18) (choosing the same cutting polynomial E(x)), we can enlarge the superposition given in (28), cutting each step in the same arbitrary way. b’ =
a” + c,
(29)
A. Ronveaux et al. / Orthogonality and linear forms
when the determinantal adp” + b9’
is the derivative
equation
(25) written
269
as
(30)
+ coLp= 0,
of the first-order
equation
[a_Y’+(b-a’)Z’]‘=O.
(31)
Explicitly, [4Q+*(c - @I’ + 2(4Y42)‘(~
- 4
=[&&(B-A)]“+(AD-BC)+(~~~,)‘(C-D).
(32)
Two important remarks are necessary in order to understand where S? reduces or not to a semiclassical form, the two above conditions being of course only sufficient conditions. The condition AZ1 + BP2 = N(PI +Z’*), cf. (23), with N a polynomial, is necessary and sufficient but not easy to characterize in full generality. Remark 1. If -rP, corresponds to a polynomial modification of ,Epl, then /F= _E” + ZYZ= (1 + 17)=!Z1 is semiclassical, I7 being a polynomial (or a rational function including a linear combination of Dirac masses and derivatives of Dirac masses). The determinantal equation is no longer valid and the first-order differential equation for 9 can easily be obtained from the elimination of Pi in the two relations (23) Z= As an example
(1+ I-I)=Yl,
D(4G,~)
in the two Laguerre
p1 = eCXxa,
= (-4 + nB)%
(33)
weights, 0 G x < cc,
p2 = fxXa+‘,
the sum p = p1 + p2 is semiclassical
(34)
if h is a positive
Remark 2. The second-order differential generalizing the situation given by (29):
equation
integer.
(30) can be decomposed
in the following
(a9’+b~)‘+R(x)(a.=Y’+b~)=O,
in which 2 is of course compute the multiplicative
(35)
semiclassical. factor R.
It is not easy anyway
Definition. When the second-order functional differential equation, we will say that the polynomials are of second category. This is the case for example p=e-xZ+Xe-ax2,
The differential
equation
for the weight, A>O,
(25) reduces
way
to detect
this situation
and to
equation (25) cannot be reduced to a first-order linear form and the corresponding orthogonal
- co < x < cc,
aZ1.
(36)
to
xp” + [ - 1 + 2( a + 1)x2] p’ + 4x3p = 0.
(37)
270
A. Ronveaux et al. / Orthogonality and linear forms
It is interesting to notice, in the spirit of Remark differential equation (37) can be written as
x[p’+2xp]‘+(2x2-l)[p’+2xp] where, of course,
the Hermite
2, that when
a goes to 1 (Hermite
case) the
=o,
(38)
weight p = eCx2 verifies
p’+2xp=o.
(39)
3. Representation of the polynomials Let us call the families { P,(x)}, { Q,(x)} and { R,(x)} polynomials with respect to the functionals A?i, 5P2 and P’, respectively, Vi,
W?7z) =&rWJ2=41,,P,2~
(=%
QnQm> = Ln II Qn II 2 = k,rnd
cc
RJL)
of degree
n orthogonal
(40)
= hn II RrI II * = %,mG. representation of R,. uses in a natural
The Grammian polynomials P,or Q,:
and
symmetrical s
way
the basis
of
or the alternate representation where .Z’i replaces Z2 and Pj is replaced by Qj (A, is a normalizing factor). These two representations of the same polynomials R,(x) give interesting relations between the expansion coefficients. Let us write R,:
5
a,c,nPk = k
k=O
k=O
It is easy to check the following
k
_ ak,n
= j=O
b,,nQ,. relations:
(%9 QjPk)
b,
J4
(_EP~,
p,‘)
9
k bk.n=
-
JFoaj.n
(91, (927
PjQk> Q,‘> ’
OgkgtZ-1.
(42) 4. Stieltjes functions To each functional
S,(x) = -
Pi
corresponds
c y,,:” . ?I>,0
a Stieltjes function
Sj( x) defined
by
(43)
271
A. Ronveaux et al. / Orthogonality and linear forms
The semiclassical character of Pi equation for S;(x):
generates a first-order nonhomogeneous
linear differential
[~iS;(x)]‘+~jS,(x)=I;;(x),
(44)
where E;;(x) is easily computed from c#+and 4; by differentiation and elimination. The second-order differential equation for the Stieltjes function S(x) = S,(x) + S,(x) is again easily obtained by derivation and elimination in the hypothesis of nonreducibility to first order (cf. Remarks 1 and 2) :
(45)
5. Associated Stieltjes functions Let us call R(l) the first associated polynomial of R,(x). Stieltjes functioi corresponding to the family R’,“, and S(x)
s(x) =
YO x - p, -
s(‘)(x) ’
The relation between S(‘)(x), is given by
Yo'l,
the
(46)
if & and y, are the coefficients of the three-term recurrence relation for the (manic) polynomial R,(x) (see (19)). The nonlinear second-order differential equation satisfied by u = y;‘[x fore easily deduced from the linear differential equation satisfied by S(x): aS” + bS’ + CS = F,
- PO- S”‘] is there(47)
and reads as a(~~)“+
b(u2)‘+
cu2 - [6az1’~+ 31x1~- 2Fu3] = 0.
(48)
The associated polynomials (of any order) of orthogonal polynomials of second category are no longer in the Laguerre-Hahn class when a = 0 (Riccati differential equation). This new kind of nonlinear differential equation is now under investigation.
References [l] S. Belmehdi, Formes lineaires et polyn8mes orthogonaux semi-classiques de classe s = 1. Description et classification, These d’Etat, Univ. P. et M. Curie, Paris VI, 1990. [2] S. Belmehdi, On the left multiplication of a regular linear functional by a polynomial, Erice Proc., to appear. [3] S. Belmehdi and A. Ronveaux, About non linear systems satisfied by the recurrence coefficients of semi-classical orthogonal polynomials, Preprint, Univ. Namur. [4] A. Cachafeiro and F. Marcellan, Orthogonal polynomials and modifications, in: Proceedings Segouia 1986, Lecture Notes in Math. 1329 (Springer, Berlin, 1986) 236-240.
272
A. Ronveaux et al. / Orthogonality and linear forms
[5] T.S. Chihara, Orthogonal polynomials and measures with end point masses, Rocky Mountain J. Math 15 (3) (1986) 705-719. [6] W. Gautschi, A survey of Gauss-Christoffel quadrature formulae, in: P.L. Butzer and F. FehCr, Eds., E.B. Christoffel, The Znfluence of his Work in Mathematics and the Physical Sciences (Birkhauser, Basel, 1981) 72-147. [7] E. Hendriksen and H. van Rossum, Semiclassical orthogonal polynomials, in: C. Brezinski et al., Eds., PoZynbmes Orthogonaux et Applications, Lecture Notes in Math. 1171 (Springer, Berlin, 1985) 354-361. [8] R. Koekoek and H.G. Meijer, A generalization of Laguerre polynomials, SIAM .Z. Math. Anal., to appear. [9] T.H. Koornwinder, Orthogonal polynomials with weight function (1 - ~)~(l+ x)O + MS( x + 1) + NS( x - l), Canad. Math. Bull. 27 (2) (1984) 205-214. [lo] H.L. Krall, On orthogonal polynomials satisfying a certain fourth order differential equation, The Pennsylvania State College Studies, No. 6, 1940. [ll] L.L. Littlejohn, The Krall polynomials: A new class of orthogonal polynomials, Quaestiones Math. 5 (1982) 255-265. [12] R. Mach, Orthogonal polynomials with exponential weight in a finite interval and application to optical model, J. Math. Phys. 25 (1984) 2186-2193. [13] P. Maroni, Une caracttrisation des polynames orthogonaux semi-classiques, C.R. Acad. Sci. Paris. Ser. Z Math. 301 (6) (1985) 269-272. [14] P. Maroni, Prolegomenes a l’etude des polyn6mes orthogonaux semi-classiques, Ann. Math. Pura Appl. (4) 149 (1987) 165-184. [15] A. Nikiforov and V.B. Ouvarov, Fonctions Speciales de la Physique Mathematique (MIR, Moscow, 1983). [16] V.B. Ouvarov, Relation between polynomials orthogonal with different weights, Dokl. Akad. Nauk. USSR 126 (1959) 33-36 (in Russian). [17] S. Paszkowski, Sur des transformations dune fonction de poids, in: C. Brezinski et al., Eds., PoZynGmes Orthogonaux et Applications, Lecture Notes in Math 1171 (Springer, Berlin, 1985) 239-247. [18] A. Ronveaux, Sur l’equation differentielle du second ordre satisfaite par une classe de polynames orthogonaux semiclassiques, C.R. Acad. Sci. Paris Ser. Z Math. 305 (1987) 163-166. [19] A. Ronveaux and F. Marcellan, Differential equation for classical type orthogonal polynomials, Canad. Math. Bull. 32 (4) (1989) 404-411. [20] B.A. Shizgal, Eigenvalues of the Lorentz Fokker-Plank equation, J. Chem. Phys. 70 (04) (1979) 1948-1951. [21] B.A. Shizgal, Gaussian quadrature procedure for use in solution of the Boltzmann equation and related problems, J. Comput. Phys. 41 (1981)‘309-328. [22] N.M. Steen, G.D. Byrne and E.M. Gelbrad, Gauss quadratures for the integral /F exp(- x’)f(x) dx and for /texp( - x*)f(x) dx, Math. Comp. 23 (1969) 661-671.
and Applied
Mathematics
265
37 (1991) 265-272
Orthogonality with respect to the sum of two semiclassical regular linear forms A. Ronveaux Physique MathPmatique, Belgium
S. Belmehdi
Facultb
Universitaires
F. Marcellan
Received
61, B-5000 Namur,
*
Laboratoire d’Analyse Numkique, Paris Cedex 05, France
Departamento Spain
Notre Dame de la Paix, rue de Bruxelles
UniversitP Pierre et Marie Curie, Tour 55-65, 5e &age, 4 Place Jussieu, F-75252
**
de Matemciticas,
30 December
ETS de Ingenieros Industriales,
c/JosP
Gutikrrez Abascal 2, E-28006 Madrid,
1990
Abstract Ronveaux, A., S. Belmehdi and F. Marcellkr, Orthogonality with respect to the sum of two semiclassical linear forms, Journal of Computational and Applied Mathematics 37 (1991) 2655272.
regular
The linear form sum of two semiclassical regular linear forms verifies in general a second-order differential equation and we examine some situations where this new form remains semiclassical. The sequences of polynomials orthogonal with respect to this new form, called of second category, have associated polynomials which do not belong to the Laguerre-Hahn class. Keywords: Orthogonal
polynomials,
semiclassical
form, Stieltjes
functions.
1. Introduction
Let { P,(X)} be a family of polynomials P,(x), denoted orthogonal with respect to a positive linear form 2 [1,13]:
P,,, of degree exactly equal to n,
(L-i?, Pn’) =d,Z. (9, P,P,) =O, n#m, In some situations described below it will be convenient to assume that 2 positive Bore1 measure dp of support I:
(2,
.‘=l.
* This author’s research * * This author’s research 0377-0427/91/$03.50
(1)
is represented by a
dl-L.
(2)
has been partially supported by Action Integrado Hispano-Francesa has been supported by CECYT PB89/01981/C02/01.
0 1991 - Elsevier Science Publishers
B.V. AlJ rights reserved
31/90.
A. Ronveaux et al. / Orthogonality and linear forms
266
Modifications polynomials, l
of t_he form 2, and study of the corresponding new family of orthogonal say { P,, }, have already been examined since a long time in the following cases.
S-type modifications
(classical
type, for instance).
~=~+XS(x-cc),
h>O,
(3)
where 6(x - c) is the usual Dirac distribution and h is assumed to be positive that 2 is still positive definite (A > 0 can of course be weakened). l
Polynomial-type P=
modification.
n(x)9,
(4)
where l-I(x) is a rational l
in order to insure
function.
Truncated weights p(x). ,ii=
where H(x)
zH(d,-x)
~H(x-c;)+ ,[ i
j
is the Heaviside
H(x)=O,
p,
dp=p
dx,
(5)
I
step function:
x
H(x)=l,
x20.
(6)
Modifications of the first type appear in [16] (see also [4,5,8-11,191); polynomial-type modifications were already examined by Christoffel [2,6,17,18] when II(x) is a polynomial and more recently by Ouvarov [16] when IJ( x) is a rational function; truncated weights were met by physicists [12,20-223 in several situations and were treated in [l]. These three new forms 2 can also be interpreted as a sum of two linear forms L?=.S’i +LF* where -Ep, is singular in the first case, and equal to (n(x) - l)P in the second case. The third case merits some special attention and some comments. The relative positions of ci and di give several configurations for the new weight p, so let us assume first that there exist only two cuts at x = c and x = d with c < d, c and d inside the support (a, b) of p(x). Two situations appear: &=
[H(x-c)+H(d-x)]p,
(7)
,&=
[H(c-x)+H(x-d)]p.
(8)
p2 for instance can now be interpreted as the sum of two disjoint weights represented by the same function p(x) where the value of p(x) between c and d is reduced to zero [l]. If the starting weight p (or the corresponding linear form 9) is semiclassical [7,13], L? satisfies the functional equation D(+.Y)
+ +Z=
where I$( x) and q(x) ($2, (D(W% Equation
0,
are polynomial
in x and where (10)
PA = (=K Ic,P”)? P,,) = -(G’,
(9) can also be written cpD@‘)
(9)
+ ($’ + #)z=
K’).
(II)
as 0,
(12)
267
A. Ronveaux et al. / Orthogonality and linear forms
and the functional comes [l]
equation
for the truncated
weight
,!i = H( x - c)p
(or p = H( c - x)p),
(x-c)+D(.S?)+(x-c)(+‘+rc/).Y=O, or in the standard
be(13)
form
D( $9)
+ &Y= 0,
(14)
where jJ=(x-c)JI-$.
6 = (x - c)G, From the more standard @)‘+$p=o the multiplication
05)
point of view of a weight differential
equation
[15]
7
(16)
by x - c insures
that the new weight
p verifies
the important
lim&5=0. x-c
condition
[15] (17)
Of course the operation of tr_uncation increase in general the class s of the weight [1,13] given as usual by s = max(degree of 1c,- 1, degree of $ - 2). The only exception being the Legendre weight p = 1, a < x < b, which is completely insensitive to a cut at any c inside (a, b). More generally for an arbitrary number of cuts hidden inside a “cutting polynomial” E(x) = lJ,,,( x - ci)( x - dj), relation (15) becomes: \CI=E(x)\l/-E’(x)+
S=E(x)+,
(18)
This shows already that the new family { p,,} remains semiclassical and verifies therefore a second-order linear differential equation [14]. The recurrence coefficients &, and y, can also be obtained for these new polynomials p,. It was shown recently [1,3] how to write in general ,the nonlinear (&, y,) relations using only the relation in the explicit knowledge of 6 and $ given by (18); we write here the recurrence standard form: P n+2
=
(x
-
a+l>P,+,
-
n 2 0,
Yn+lP,,
The distinction between the two different instance are only given by the different relations being identical in both cases. For p = pi,
P1=x-PO,
PO’ 1.
(19)
families generated by ,Zi and p2 in (7) and (8) for initial condition on the &,, the nonlinear (&,, y,,)
d -v(x)
J I%=
dx
Cd / c
; ~(4
(20)
dx
forp=p,, fxp(x) P,=
dx + op(x)
aC J, P(X) dx + i6xp(x)
dx
(21) dx
.
268
A. Ronveaux et al. / Orthogonality and linear forms
These detailed comments emphasize therefore the interest of investigating the family of polynomials orthogonal with respect to the sum of two positive linear forms PI and _Z& generalizing the first case where ZI is singular and the third case where the intersecting support of PI and ZZ is empty. The thud case shows also that in several situations the sum of two (or more than two) semiclassical forms are still semiclassical.
2. Orthogonal polynomials of second category Let Zi, i = 1, 2, be the positive
linear form solution
of the semiclassical
functional
D(G;(x)q) +#;(X)Z=O,
equation
(22)
where I/I~and & are given polynomials in x. The positive form .Y= P’t +PZ, defined by (9, P) = ( Zl, P) + ( Yz, P), P any polynomial, verifies in general a second-order functional equation easily obtained by “differentiation” and elimination of 9, and PZ:
Computing
one more derivative
we obtain
(24) The determinantal
form of the second-order
functional
equation
reads now as
(25) writing for short 9’ This second-order l
is satisfied
+:+Gi=O, This generalized
to a first order in some cases.
(26)
or ~&+6)=d&h+~~).
A=B
This condition
for D(Z). equation reduces
for the following
constraint:
i=l,2.
Legendre
weight corresponds
P=X,H(x-c,)H(d,-x)+A,H(x-c,)H(d,-x),
(27) to a sum of two step functions Ci
[l]: Ai>O.
(28)
But from the fact that 6: + qi = 0 if $: + Gi = 0 in relation (18) (choosing the same cutting polynomial E(x)), we can enlarge the superposition given in (28), cutting each step in the same arbitrary way. b’ =
a” + c,
(29)
A. Ronveaux et al. / Orthogonality and linear forms
when the determinantal adp” + b9’
is the derivative
equation
(25) written
269
as
(30)
+ coLp= 0,
of the first-order
equation
[a_Y’+(b-a’)Z’]‘=O.
(31)
Explicitly, [4Q+*(c - @I’ + 2(4Y42)‘(~
- 4
=[&&(B-A)]“+(AD-BC)+(~~~,)‘(C-D).
(32)
Two important remarks are necessary in order to understand where S? reduces or not to a semiclassical form, the two above conditions being of course only sufficient conditions. The condition AZ1 + BP2 = N(PI +Z’*), cf. (23), with N a polynomial, is necessary and sufficient but not easy to characterize in full generality. Remark 1. If -rP, corresponds to a polynomial modification of ,Epl, then /F= _E” + ZYZ= (1 + 17)=!Z1 is semiclassical, I7 being a polynomial (or a rational function including a linear combination of Dirac masses and derivatives of Dirac masses). The determinantal equation is no longer valid and the first-order differential equation for 9 can easily be obtained from the elimination of Pi in the two relations (23) Z= As an example
(1+ I-I)=Yl,
D(4G,~)
in the two Laguerre
p1 = eCXxa,
= (-4 + nB)%
(33)
weights, 0 G x < cc,
p2 = fxXa+‘,
the sum p = p1 + p2 is semiclassical
(34)
if h is a positive
Remark 2. The second-order differential generalizing the situation given by (29):
equation
integer.
(30) can be decomposed
in the following
(a9’+b~)‘+R(x)(a.=Y’+b~)=O,
in which 2 is of course compute the multiplicative
(35)
semiclassical. factor R.
It is not easy anyway
Definition. When the second-order functional differential equation, we will say that the polynomials are of second category. This is the case for example p=e-xZ+Xe-ax2,
The differential
equation
for the weight, A>O,
(25) reduces
way
to detect
this situation
and to
equation (25) cannot be reduced to a first-order linear form and the corresponding orthogonal
- co < x < cc,
aZ1.
(36)
to
xp” + [ - 1 + 2( a + 1)x2] p’ + 4x3p = 0.
(37)
270
A. Ronveaux et al. / Orthogonality and linear forms
It is interesting to notice, in the spirit of Remark differential equation (37) can be written as
x[p’+2xp]‘+(2x2-l)[p’+2xp] where, of course,
the Hermite
2, that when
a goes to 1 (Hermite
case) the
=o,
(38)
weight p = eCx2 verifies
p’+2xp=o.
(39)
3. Representation of the polynomials Let us call the families { P,(x)}, { Q,(x)} and { R,(x)} polynomials with respect to the functionals A?i, 5P2 and P’, respectively, Vi,
W?7z) =&rWJ2=41,,P,2~
(=%
QnQm> = Ln II Qn II 2 = k,rnd
cc
RJL)
of degree
n orthogonal
(40)
= hn II RrI II * = %,mG. representation of R,. uses in a natural
The Grammian polynomials P,or Q,:
and
symmetrical s
way
the basis
of
or the alternate representation where .Z’i replaces Z2 and Pj is replaced by Qj (A, is a normalizing factor). These two representations of the same polynomials R,(x) give interesting relations between the expansion coefficients. Let us write R,:
5
a,c,nPk = k
k=O
k=O
It is easy to check the following
k
_ ak,n
= j=O
b,,nQ,. relations:
(%9 QjPk)
b,
J4
(_EP~,
p,‘)
9
k bk.n=
-
JFoaj.n
(91, (927
PjQk> Q,‘> ’
OgkgtZ-1.
(42) 4. Stieltjes functions To each functional
S,(x) = -
Pi
corresponds
c y,,:” . ?I>,0
a Stieltjes function
Sj( x) defined
by
(43)
271
A. Ronveaux et al. / Orthogonality and linear forms
The semiclassical character of Pi equation for S;(x):
generates a first-order nonhomogeneous
linear differential
[~iS;(x)]‘+~jS,(x)=I;;(x),
(44)
where E;;(x) is easily computed from c#+and 4; by differentiation and elimination. The second-order differential equation for the Stieltjes function S(x) = S,(x) + S,(x) is again easily obtained by derivation and elimination in the hypothesis of nonreducibility to first order (cf. Remarks 1 and 2) :
(45)
5. Associated Stieltjes functions Let us call R(l) the first associated polynomial of R,(x). Stieltjes functioi corresponding to the family R’,“, and S(x)
s(x) =
YO x - p, -
s(‘)(x) ’
The relation between S(‘)(x), is given by
Yo'l,
the
(46)
if & and y, are the coefficients of the three-term recurrence relation for the (manic) polynomial R,(x) (see (19)). The nonlinear second-order differential equation satisfied by u = y;‘[x fore easily deduced from the linear differential equation satisfied by S(x): aS” + bS’ + CS = F,
- PO- S”‘] is there(47)
and reads as a(~~)“+
b(u2)‘+
cu2 - [6az1’~+ 31x1~- 2Fu3] = 0.
(48)
The associated polynomials (of any order) of orthogonal polynomials of second category are no longer in the Laguerre-Hahn class when a = 0 (Riccati differential equation). This new kind of nonlinear differential equation is now under investigation.
References [l] S. Belmehdi, Formes lineaires et polyn8mes orthogonaux semi-classiques de classe s = 1. Description et classification, These d’Etat, Univ. P. et M. Curie, Paris VI, 1990. [2] S. Belmehdi, On the left multiplication of a regular linear functional by a polynomial, Erice Proc., to appear. [3] S. Belmehdi and A. Ronveaux, About non linear systems satisfied by the recurrence coefficients of semi-classical orthogonal polynomials, Preprint, Univ. Namur. [4] A. Cachafeiro and F. Marcellan, Orthogonal polynomials and modifications, in: Proceedings Segouia 1986, Lecture Notes in Math. 1329 (Springer, Berlin, 1986) 236-240.
272
A. Ronveaux et al. / Orthogonality and linear forms
[5] T.S. Chihara, Orthogonal polynomials and measures with end point masses, Rocky Mountain J. Math 15 (3) (1986) 705-719. [6] W. Gautschi, A survey of Gauss-Christoffel quadrature formulae, in: P.L. Butzer and F. FehCr, Eds., E.B. Christoffel, The Znfluence of his Work in Mathematics and the Physical Sciences (Birkhauser, Basel, 1981) 72-147. [7] E. Hendriksen and H. van Rossum, Semiclassical orthogonal polynomials, in: C. Brezinski et al., Eds., PoZynbmes Orthogonaux et Applications, Lecture Notes in Math. 1171 (Springer, Berlin, 1985) 354-361. [8] R. Koekoek and H.G. Meijer, A generalization of Laguerre polynomials, SIAM .Z. Math. Anal., to appear. [9] T.H. Koornwinder, Orthogonal polynomials with weight function (1 - ~)~(l+ x)O + MS( x + 1) + NS( x - l), Canad. Math. Bull. 27 (2) (1984) 205-214. [lo] H.L. Krall, On orthogonal polynomials satisfying a certain fourth order differential equation, The Pennsylvania State College Studies, No. 6, 1940. [ll] L.L. Littlejohn, The Krall polynomials: A new class of orthogonal polynomials, Quaestiones Math. 5 (1982) 255-265. [12] R. Mach, Orthogonal polynomials with exponential weight in a finite interval and application to optical model, J. Math. Phys. 25 (1984) 2186-2193. [13] P. Maroni, Une caracttrisation des polynames orthogonaux semi-classiques, C.R. Acad. Sci. Paris. Ser. Z Math. 301 (6) (1985) 269-272. [14] P. Maroni, Prolegomenes a l’etude des polyn6mes orthogonaux semi-classiques, Ann. Math. Pura Appl. (4) 149 (1987) 165-184. [15] A. Nikiforov and V.B. Ouvarov, Fonctions Speciales de la Physique Mathematique (MIR, Moscow, 1983). [16] V.B. Ouvarov, Relation between polynomials orthogonal with different weights, Dokl. Akad. Nauk. USSR 126 (1959) 33-36 (in Russian). [17] S. Paszkowski, Sur des transformations dune fonction de poids, in: C. Brezinski et al., Eds., PoZynGmes Orthogonaux et Applications, Lecture Notes in Math 1171 (Springer, Berlin, 1985) 239-247. [18] A. Ronveaux, Sur l’equation differentielle du second ordre satisfaite par une classe de polynames orthogonaux semiclassiques, C.R. Acad. Sci. Paris Ser. Z Math. 305 (1987) 163-166. [19] A. Ronveaux and F. Marcellan, Differential equation for classical type orthogonal polynomials, Canad. Math. Bull. 32 (4) (1989) 404-411. [20] B.A. Shizgal, Eigenvalues of the Lorentz Fokker-Plank equation, J. Chem. Phys. 70 (04) (1979) 1948-1951. [21] B.A. Shizgal, Gaussian quadrature procedure for use in solution of the Boltzmann equation and related problems, J. Comput. Phys. 41 (1981)‘309-328. [22] N.M. Steen, G.D. Byrne and E.M. Gelbrad, Gauss quadratures for the integral /F exp(- x’)f(x) dx and for /texp( - x*)f(x) dx, Math. Comp. 23 (1969) 661-671.