Partial dierential equations having orthogonal
polynomial solutions
Y.J. Kima, K.H. Kwona;∗, J.K. Leeb
aDepartment of Mathematics, KAIST, Taejon 305-701, South Korea bDepartment of Mathematics, Sunmoon university, Cheonan, South Korea
Received 30 October 1997; received in revised form 19 May 1998
Abstract
We show that if a second order partial dierential equation:
L[u] :=Auxx+ 2Buxy+Cuyy+Dux+Euy=nu
has orthogonal polynomial solutions, then the dierential operator L[·] must be symmetrizable and can not be parabolic in any nonempty open subset of the plane. We also nd Rodrigues type formula for orthogonal polynomial solutions of such dierential equations. c1998 Elsevier Science B.V. All rights reserved.
AMS classication:33C50; 35P99
Keywords:Orthogonal polynomials in two variables; Partial dierential equations; Symmetrizability; Rodrigues type formula
1. Introduction
As a natural generalization of classical orthogonal polynomials of one variable, Krall and Sheer [5] considered orthogonal polynomials in two variables satisfying a second order partial dierential equation with polynomial coecients:
L[u] =Auxx+ 2Buxy+Cuyy+Dux+Euy=nu: (1.1)
Krall and Sheer rst nd necessary and sucient conditions in order for orthogonal polynomials to satisfy the dierential equation (1.1) and then classify such dierential equations, up to a complex linear change of variables, which have at least weak orthogonal polynomials as solutions.
Later, Littlejohn [8] noted that all dierential equations found in [5] are elliptic in the region of orthogonality in case the orthogonalizing weight is known. However, the type of dierential operators cannot be determined properly from the specic forms of the equations given in [5] since the type of a dierential operator is not preserved under a complex change of variables. In fact, Krall and Sheer found at least one dierential equation (see Example 3.3), which is hyperbolic everywhere and has orthogonal polynomials as solutions.
In this work, using the functional calculus on moment functionals developed in [6, 8], we rst prove that if the dierential equation (1.1) has orthogonal polynomials as solutions, thenL[·] cannot be parabolic in any nonempty open subset of the plane and must be symmetrizable. We also establish Rodrigues type formula for orthogonal polynomial solutions to the dierential equation (1.1).
2. Preliminaries
For any integer n¿0, let Pn be the space of real polynomials in two variables of (total)
de-gree 6n and P=S
n¿0Pn. By a polynomial system (PS), we mean a sequence of polynomials
{n−j; j}
is called the normalization of {n}
For any PS {n}
∞
n=0, there is a unique moment functional , called the canonical moment
func-tional of {n}
∞
n=0, dened by the conditions
h;1i= 1 and h; ni= 0; n¿1:
Denition 2.1. A PS {n}
∞
n=0 is called a weak orthogonal polynomial system (WOPS) if there is
a nonzero moment functional such that h; mTni= 0; m6=n. If moreover Hn:=h; nTni; n¿0,
is a nonsingular diagonal matrix, we call {n}
∞
n=0 an orthogonal polynomial system(OPS). In this
case, we say that {n}∞n=0 is a WOPS or an OPS relative to :
Note that if {n}
∞
n=0 is a WOPS relative to , then must be a nonzero constant multiple of the
canonical moment functional of {n}
∞
n=0. If {n}
∞
n=0 is an OPS relative to , then the normalization
{Pn}∞
n=0 of {n}
∞
n=0 is a WOPS, but not necessarily an OPS relative to :
The following is proved in [5] (see also [11]).
Proposition 2.2. For a moment functional ; the following statements are equivalent.
(i) is quasi-denite;
n=0 relative to such that
Hn:=h;PnPTni; n¿0; is nonsingular.
are nonsingular diagonal matrices. Hence {n:=On n}
∞
n=0, is an OPS relative to . In fact, for a
quasi-denite moment functional , let Vn; n¿0, be the vector space of polynomials of degree n,
which are orthogonal to any polynomial of degree ¡ n relative to . Then, {n}
∞
n=0, where n is a
basis (respectively, an orthogonal basis) of Vn, forms a WOPS (respectively, an OPS) relative to
and Hn:=h; nTni; n¿0, are nonsingular (respectively, nonsingular diagonal).
For any moment functional and any polynomial (x; y), we let
h@x; i=−h; xi; h@y; i=−h; yi
h ; i=h; i; ∈P:
Then we have
Lemma 2.3. Let and be moment functionals and R(x; y) a polynomial. Then
Proof. See Lemma 2.3 in [6].
3. Dierential operator L[·]
We now consider the dierential equation (1.1). Krall and Sheer [5] showed that if the dierential equation (1.1) has a PS as solutions, then it must be of the form
L[u] =Auxx+ 2Buxy+Cuyy+Dux+Euy
= (ax2+d
1x+e1y+f1)uxx+ (2axy+d2x+e2y+f2)uxy
+ (ay2+d
3x+e3y+f3)uyy+ (gx+h1)ux+ (gy+h2)uy
=nu; n=an(n−1) +gn: (3.1)
Following Krall and Sheer, we say that the dierential operator L[·] is admissible if m6=n for
m6=n or equivalently n6= 0 for n¿1:
It is shown in [5] that the dierential equation (3.1) has a unique monic PS as solutions if and only if L[·] is admissible.
Theorem 3.1. If the dierential equation (3.1) has an OPS as solutions; then B2 −AC6≡0 in any nonempty open subset of the plane. In other words; the dierential operator L[·] cannot be parabolic in any nonempty open subset of the plane.
In order to prove Theorem 3.1, we rst need the following fact:
Lemma 3.2. If the dierential equation (3.1) has a PS {n}∞n=0 as solutions; then the canonical moment functional of {n}
∞
n=0 must satisfy
L∗[] = 0; (3.2)
h; Di=h; Ei=h; A+xDi=h; C+yEi=h; B+yDi=h; B+xEi= 0; (3.3)
where L∗[·] is the formal Lagrange adjoint of L[·] given by
L∗[u] = (Au)xx+ 2(Bu)xy+ (Cu)yy−(Du)x−(Eu)y: (3.4)
Proof. See Lemma 3.1 in [6].
Proof of Theorem 3.1. Assume that the dierential equation (3.1) has an OPS {n}
∞
n=0 as solutions.
Let be the canonical moment functional of {n}∞n=0. We assume that
B2−AC=−a[d3x3+ (e3−d2)x2y+ (d1−e2)xy2+e1y3]
+(1 4d
2
2−af3−d1d3)x2+ (12d2e2+af2−d1e3−d3e1)xy
+(14e22−e1e3−af1)y2+ (12d2f2−d1f3−f1d3)x
+(1
2e2f2−f1e3−e1f3)y+ 1 4f
2 2 −f1f3
in some nonempty open subset of the plane. Then, all the coecients of B2−AC must be 0. We
now consider the following three cases separately.
Case 1: a+g6= 0 and g= 0: Then, we may assume that6 g= 1 and h1=h2= 0 so that (3.3) gives
10=01= 0 and (a+ 1)20+f1= (a+ 1)02+f3= 2(a+ 1)11+f2= 0:
Therefore
1=2002−211=
1
(a+ 1)2[f1f3− 1 4f
2 2] = 0;
which is a contradiction since 16= 0 when is quasi-denite.
Case 2: a+g6= 0 and g= 0: Then we may assume that a= 1 so that (3.5) gives
d3=e1= 0; d2=e3; d1=e2; f1=14d21; f2=12d1d2; f3=14d22:
Hence the dierential equation (3.1) becomes, after replacing x+1
2d1 by x and y+ 1
2d2 by y,
x2uxx+ 2xyuxy+y2uyy=n(n−1)u: (3.6)
Then we have by (3.3), 20=02=11= 0 so that 1= 0, which is a contradiction.
Case3:a+g= 0 andg6= 0: Then we may assume thata=−1; g= 1, andh1=h2= 0. In this case,
it is easy to see that if the dierential equation (3.1) has a PS as solutions, thenf1=f2=f3= 0. We
then have from (3.5) d1=e1=d2=e2=d3=e3= 0 so that the dierential equation (3.1) becomes
x2uxx+ 2xyuxy+y2uyy−xux−yuy=n(n−2)u:
Then we have from (3.2)
Amn:=hL∗[]; xmyni=h; L[xmyn]i= (m+n)(m+n−2)mn= 0; m; n¿0:
Hence mn= 0 for m+n¿3 so that 2= 0, which is a contradiction.
While Krall and Sheer considered only admissible dierential equations, we do not assume the admissibility of L[·] in Theorem 3.1 since we do not know whether L[·] must be admissible or not when the dierential equation (3.1) has an OPS as solutions.
When L[·] is not admissible, the dierential equation (3.1) may have either no PS as solutions or innitely many distinct monic PS’s as solutions. For example, the dierential equation (3.6) has innitely many monic PS’s {Pn}∞
n=0 as solutions;
Pmn(x; y) =xmyn for m+n= 16 and P10=x+; P01=y+;
where and are arbitrary constants. Note that for any choice of and ; {Pn}∞
n=0 can not be an
OPS but for == 0; {xmyn}∞
m; n=0 is a WOPS relative to (x; y):
Example 3.3. Krall and Sheer [5] showed that the dierential equation
L[u] = 3yuxx+ 2uxy−xux−yuy=−nu (3.7)
has an OPS as solutions. Since B2−AC= 1¿0, the dierential operator L[·] in (3.7) is hyperbolic
Littlejohn [8] observed that all dierential equations found by Krall and Sheer [5], that have orthogonal polynomial solutions whose weight functions are known, are elliptic in the region of orthogonality.
However, since Krall and Sheer used a complex linear change of variables in their classication, the type of the dierential equation cannot be properly determined from the specic forms of the equations given in [5].
For example, Krall and Sheer obtained the following dierential equation (see the equation (5.13) in [5]) by a complex linear change of variables
uxx+uyy−xux−yuy=−nu (3.8)
from the equation (see Eq. (5.12) in [5])
f2uxy+gxux+gyuy=gnu: (3.9)
Note that since f2(6= 0) is real, the dierential equation (3.9) is hyperbolic everywhere while
Eq. (3.8) is elliptic everywhere.
Proposition 3.4. Let {n}∞n=0 be a PS satisfying the dierential equation (3:1). If {n}∞n=0 is at least a WOPS; then the canonical moment functional of {n}
∞
n=0 satises L∗[] = 0 and
M1[] := (A)x+ (B)y−D= 0; (3.10)
M2[] := (B)x+ (C)y−E= 0: (3.11)
Proof. See Corollary 3.3 in [6].
Lemma 3.5. For the dierential operator L[·] in(3.1) and any moment functional ;the following statements are equivalent.
(i) M1[] =M2[] = 0;
(ii) L[·] is formally symmetric on polynomials; that is;
hL[P]; Qi=hL[Q]; Pi; P and Q in P:
Furthermore; if L[P] =P and L[Q] =Q for 6=; then h; PQi= 0 for any moment functional
satisfying M1[] =M2[] = 0.
Proof. See Lemma 3.6 in [6].
We call the functional equations M1[] =M2[] = 0 in (3.10) and (3.11) the moment equations
for the dierential equation (3.1).
Lemma 3.6. If the dierential equation (3.1) is admissible and has a WOPS as solutions; then the moment equations M1[] =M2[] = 0 can have only one solution up to a nonzero constant multiple.
As a consequence of Theorem 3.1 and Lemma 3.5, we can now prove:
Corollary 3.7. If the dierential equation (3.1) has an OPS {n}
∞
n=0 as solutions; then
(i) the moment equations M1[] =M2[] = 0 can have only one solution up to a nonzero constant multiple; which must be quasi-denite;
(ii) the dierential equation (3.1) can have only one monic WOPS as solutions (but may have in-nitely many other monic PS’s as solutions;which are not WOPS’s;unlessL[·] is admissible).
Proof. Let be the canonical moment functional of {n}
∞
n=0 and {Pn}
∞
n=0 the normalization of
{n}
∞
n=0. Then is quasi-denite and M1[] =M2[] = 0 by Proposition 3.4 and {Pn}
∞
n=0 is a monic
WOPS satisfying the dierential equation (3.1).
(i) If L[·] is admissible, then it follows from Lemma 3.6. We now assume that L[·] is not admissible, that is, N= 0 for some N¿1. Let be any moment functional satisfying the moment
equationsM1[] =M2[] = 0:Then Lemma 3.5 impliesh; ni= 0; n6= 0; N sincen6= 0 forn6= 0; N.
Therefore we have, by Lemma 2.3(iii), =(x; y) for some polynomial (x; y) of degree 6N: Then
M1[] =M1[] = (xA+yB)+M1[] = (xA+yB)= 0;
M2[] =M2[] = (xB+yC)+M2[] = (xB+yC)= 0:
Hence we have by Lemma 2.3(ii)
xA+yB= 0 and xB+yC= 0;
which implies thatx=y= 0, that is,(x; y) must be a constant sinceB2−AC6≡0 by Theorem 3.1.
(ii) Let {Qn}∞
n=0 be any monic WOPS relative to satisfying the dierential equation (3.1). Then
satises the moment equationsM1[] =M2[] = 0 by Proposition 3.4. Hence by (i),=cfor some
nonzero constantc. Therefore{Qn}∞
n=0 is also a monic WOPS relative to so that{Qn}
∞
n=0={Pn}
∞
n=0
by Proposition 2.2 since is quasi-denite.
Corollary 3.7 provides a unique existence of orthogonal polynomial solutions to the dierential equation (3.1) in the monic level whetherL[·] is admissible or not. However, the dierential equation (3.1) may have innitely many distinct OPS’s, which are not monic, as solutions if it has at least one.
Denition 3.8 (Littlejohn [8]). The dierential operator L[·] in (3.1) is symmetric if L[·] =L∗[·]. We say that L[·] is symmetrizable if there is a non-trivial function s(x; y) such that s(x; y) is C2 in
some open set and sL[·] is symmetric.
In this case, we call s(x; y) a symmetry factor of L[·]:
It is easy to see (cf. [8]) that a function s(x; y) is a symmetry factor of L[·] if and only if s(x; y) satises
M1[s] = (As)x+ (Bs)y−Ds= 0; (3.12)
We call the simultaneous equations (3.12) and (3.13) the symmetry equations for L[·]. We rst see by an example that not every dierential operator L[·] is symmetrizable.
Example 3.9 (Littlejohn [8]). Consider the dierential equation
L[u] =x2u
xx+ 2xyuxy+y2uyy+ (gx+h1)ux+ (gy+h2)uy=n(n−1 +g)u: (3.14)
Note rst that the dierential (3.14) is parabolic everywhere so that it cannot have an OPS as solutions by Theorem 3.1. The corresponding symmetry equations are
x2sx+xysy+ ((3−g)x−h1)s= 0;
xysx+y2sy+ ((3−g)y−h2)s= 0
so that (h2x−h1y)s= 0: Hence, if h21+h226= 0, then s(x; y) = 0 on {(x; y)∈R2|h2x−h1y6= 0} and
soL[·] cannot be symmetrizable. However, if g= 3 and h1=h2= 0, then L[·] is symmetrizable with
s(x; y) = arctan(y=x) as a symmetry factor.
Proposition 3.10. Assume that B2−AC6≡0. Then the dierential operator L[·] is symmetrizable if and only if
@ @y
C(A
x+By−D)−B(Bx+Cy−E)
B2−AC
= @ @x
A(B
x+Cy−E)−B(Ax+By−D)
B2−AC
: (3.15)
We call (3.15) the compatibility condition for the symmetrizability of L[·]:
Proof. (⇒): We may solve the symmetry equations (3.12) and (3.13) for sx and sy as
(B2−AC)sx= [C(Ax+By−D)−B(Bx+Cy−E)]s; (3.16)
(B2−AC)s
y= [A(Bx+Cy−E)−B(Ax+By−D)]s; (3.17)
from which (3.15) follows.
(⇐): Assume that the compatibility condition (3.15) is satised. Choose any open rectangle R= (x0−r; x0+r)×(y0−r; y0+r) (r ¿0) in the plane such that (B2−AC)(x; y)6= 0 for all (x; y)
in R. If we set
s(x; y) = exp
Z x
x0
f(t; y) dt+
Z y
y0
g(x0; t) dt
; (x; y)∈R;
where f(x; y) = [C(Ax+By−D)−B(Bx+Cy−E)](B2−AC)
−1 and g(x; y) = [A(B
x+Cy−E)−
B(Ax+By−D)](B2−AC)
−1
, then s(x; y) satises (3.12) and (3.13). Hence L[·] is symmetrizable with s(x; y) as a symmetry factor.
Example 3.11. Consider the following dierential equation
L[u] =x2u
Then B2−AC= 2xy+ 16≡0 but L[·] is not symmetrizable since the compatibility condition (3.15)
is not satised.
Theorem 3.12. If the dierential equation (3.1) has an OPS {n}
∞
n=0 as solutions; then L[·] must be symmetrizable.
Proof. Let be the canonical moment functional of {n}n=0∞ . Then satises M1[] =M2[] = 0
by Proposition 3.4. Solving the moment equations M1[] =M2[] = 0 for x and y, we obtain
x= and y=; (3.19)
where =B2−AC; =C(A
x+By−D)−B(Bx+Cy−E); =A(Bx+Cy−E)−B(Ax+By−D): Then
(x; y)6≡0 by Theorem 3.1 and we have from (3.19) (x)y−(y)x=()y−()x and so
(y−x)= (y−x): Hence, we have by Lemma 2.3(ii), y−y=x−x since is
quasi-denite. Therefore the compatibility condition (3.15) is satised so that L[·] is symmetrizable by Proposition 3.10.
The symmetrizability of ordinary dierential equations of arbitrary order having OPS’s in one variable as solutions is recently proved in [7].
4. Rodrigues type formula
From now on, we may and shall assume (see Theorem 3.12) that the dierential operator L[·] in (3.1) is symmetrizable and let w(x; y)(6≡0) be a symmetry factor of L[·]. That is, w(x; y) is any nonzero solution of the symmetry equations
M1[w] = (Aw)x+ (Bw)y−Dw= 0; M2[w] = (Bw)x+ (Cw)y−Ew= 0: (4.1)
Solving M1[w] =M2[w] = 0 for wx and wy yields (cf. (3.16) and (3.17))
wx=w and wy=w: (4.2)
Note that deg()63; deg()62 and deg()62: Decompose A; B, and C into
A=A1A2; B=A1B1C1; C=C1C2; (4.3)
where A16≡0; C16≡0: Then
=A1C10; 0=A1B21C1−A2C2;
=C10; 0=A1B1(E−Bx−Cy)−C2(D−Ax−By);
=A10; 0=B1C1(D−Ax−By)−A2(E−Bx−Cy)
so that Eq. (4.2) become
pwx=0w and qwy=0w; (4.4)
Note that p6≡0; q6≡0 and deg(p)63;deg(q)63;deg(0)62;deg(0)62:
Lemma 4.1 (cf. Suetin [9], p. 183). If
(A1)y= (C1)x= 0; (4.5)
then for any integers m and n¿0
1 w@
m x@
n y(p
mqn
w) := mn(x; y) (4.6)
is a polynomial of degree 62(m+n): If moreover;
deg(p);deg(q)62 and deg(0);deg(0)61; (4.7)
then deg( mn)6m+n:
Proof. Assume that the condition (4.5) holds. Then for any polynomial (x; y) and any integers m and n, we have
@x(pmqnw) = (pm
−1qn
1)w and @y(pmqnw) = (pmqn
−1 2)w;
where
1=mpx+nA1(0)x+px+0;
2=mC1(0)y+nqy+qy+0:
Since deg(1)6deg() + max{deg(p) − 1;deg(0)} and deg(2)6deg() + max{deg(q) − 1;
deg(0)}, the conclusions follow easily by induction on m and n.
However, in general, mn need not satisfy the dierential equation (3.1) and we cannot say
anything on orthogonality of { mn}, unless w(x; y) is an orthogonalizing weight function for an
OPS satisfying the dierential equation (3.1).
Hence we now formalize Lemma 4.1 by using, instead of a symmetry factor w(x), a moment functional solution of the moment equations M1[] =M2[] = 0:
Theorem 4.2. Assume that the dierential equation (3.1) has a WOPS {n}∞n=0 as solutions and let be the canonical moment functional of {n}
∞
n=0. Assume that satises
px=0 and qy=0: (4.8)
(i) If the condition (4.5) holds; then for any integers m and n¿0
@xm@yn(pmqn) = mn; (4.9)
where mn(x; y) is a polynomial of degree 62(m+n); and
h; xkyl mn(x; y)i= 0; 06k+l6m+n and (k; l)6= (m; n): (4.10)
(iii) If is quasi-denite and the conditions (4.5) and (4.7) hold; then { n}
∞
n=0;where n={ n0; n−1;1; : : : ; 0n}; is a WOPS relative to and satises the dierential equation (3.1).
Before proving Theorem 4.2, we rst note that under the assumptions as in Theorem 4.2, satises the moment equationsM1[] =M2[] = 0 by Proposition 3.4, which, however, do not imply
(4.8) in general.
Proof of Theorem 4.2. The proof of (i) and (ii) except (4.10) is essentially the same as that of Lemma 4.1. Now,
h; xky‘ mn(x; y)i=h mn; xky‘i= (−1)m+nhpmqn; @xm@ n y(x
ky‘)i
from which (4.10) follows. Finally, we further assume that is quasi-denite. We rst claim that deg( mn) =m+n and h; xmyn mni 6= 0 form andn¿0. Assume that deg( mn)6m+n−1 for some
n=0 satisfy the dierential equation (3.1), {Pn}
∞
n=0 and{Qn}
∞
n=0
also satisfy the dierential equation (3.1).
Combining Lemma 4.1 and Theorem 4.2, we now have:
Theorem 4.3. Assume that the dierential equation (3.1) has an OPS {n}∞n=0 relative to as solutions and let w(x; y) be a symmetry factor of L[·]: If the conditions (4.5), (4.7), and (4.8)
hold; then the PS { mn} dened by (4.6) is a WOPS relative to and satises the dierential equation (3.1).
However, h; n nTi, with { n}
∞
n=0 as in Theorem 4.3, is nonsingular but need not be diagonal,
that is, { n}
∞
n=0 is a WOPS but need not be an OPS in general (see Example 4.5 below).
We may call (4.6) and (4.9) Rodrigues type formulas for orthogonal polynomial solutions of the dierential equations (3.1).
Example 4.4. Assume that Ay=B=Cx= 0 so that the dierential equation (3.1) is of the form
where A(x) =d1x+f1; C(y) =e3y+f3; D(x) =gx+h1; E(y) =gy+h2 andg6= 0:Then it is shown
so that the conditions (4.5) and (4.7) hold. On the other hand, the canonical moment func-tional of {Pn}∞
so that the Rodrigues type formula (4.14) is nothing but the tensor product of one dimensional Rodrigues formulas for{Pn0(x)}
∞
n=0 and{P0n(y)}
∞
n=0(see [1, 2, 10] for Rodrigues formula for classical
orthogonal polynomials of one variable).
We may, of course, replace by a symmetry factor w(x; y) =w1(x)w2(y) of the dierential
equation (4.11), where w1(x) and w2(y) are symmetry factors of the dierential equations (4.12)
and (4.13), respectively.
Example 4.5. Consider the dierential equation for circle polynomials:
L[u] = (x2−1)u
2 is a symmetry factor of the
For example, we have 00(x; y) = 1 and
10(x; y) = (g−1)x; 01(x; y) = (g−1)y;
20(x; y) = (g+ 1)(gx2+y2−1); 11(x; y) = (g+ 1)(g−1)xy;
02(x; y) = (g−1)(x2+gy2−1):
However, if g6= 1;0;−1; : : :, then the dierential equation (4.15) has an OPS {n}∞n=0 (called the
circle polynomials) as solutions. Then the canonical moment functional of {n}
∞
n=0 satises the
condition (4.8) so that by Theorem 4.3, { mn} is a WOPS. But, even in this case, { mn} is not an
OPS. For if we let be the canonical moment functional of { mn}, then satises L∗[] = 0 (cf.
Lemma 3.2), that is,
(m+n)(m+n−1 +g)mn−m(m−1)m−2; n−n(n−1)m; n−2= 0; m; n¿0
so that
00= 1; 10=01=11=30=21=12=03=31=13= 0;
20=02=
1
g+ 1; 40=04=
3
(g+ 1)(g+ 3); 22=
1
(g+ 1)(g+ 3):
Hence, we have
H2=h; 2 2Ti=
(g+ 1)(g−1) g+ 3
2g 0 2
0 g−1 0 2 0 2g
;
which is nonsingular but not diagonal.
Example 4.6. Consider the dierential equation for triangle polynomials:
L[u] = (x2−x)uxx+ 2xyuxy+ (y2−y)uyy+ [(a+b+c+ 3)x−(1 +a)]ux
+ [(a+b+c+ 3)y−(1 +b)]uy=nu: (4.16)
In decomposition (4.3), we take A1=x; C1=y so that
0=x+y−1; 0= (a+c)x+ay−a; 0=bx+ (b+c)y−b;
p=x(x+y−1); q=y(x+y−1):
Then by Lemma 4.1,
1 w@
m x@
n y[x
myn(x+y−1)m+nw] =
mn(x; y); m; n¿0 (4.17)
is a polynomial of degree 6m+n, where w(x; y) =xayb(x+y −1)c is a symmetry factor of
(4.16) has an OPS {n}
∞
n=0 (called the triangle polynomials) as solutions and let be the
canon-ical moment functional of {n}
∞
n=0. Then satises the moment equations M1[] =M2[] = 0 or
equivalently
xy(x+y−1)x=y((a+c)x+ay−a);
xy(x+y−1)y=x(bx+ (b+c)y−b): (4.18)
Let
u:=x(x+y−1)x−[(a+c)x+ay−a]=px−0;
v:=y(x+y−1)y−[bx+ (b+c)y−b]=qy−0:
Then, (4.18) becomes
yu=xv= 0: (4.19)
On the other hand, we can see by using M1[] =M2[] = 0 that
u=−v=xyx−xyy+ (bx−ay): (4.20)
Hence, we have from (4.19) and (4.20)
hu; xmyni=hv; xmyni= 0; m+n¿1: (4.21)
We also have from (4.20)
hu;1i=−hv;1i=hxyx−xyy+ (bx−ay);1i= (b+ 1)10−(a+ 1)01
and from (3.3)
h; Di= (a+b+c+ 3)10−(1 +a) = 0;
h; Ei= (a+b+c+ 3)01−(1 +b) = 0
so that hu;1i=hv;1i= 0. Therefore, together with (4.21), we have u=v= 0, that is, satises the condition (4.8). Hence, by Theorem 4.3,{ mn} given by (4.17) is a WOPS relative toand satises
the dierential equation (4.16).
Rodrigues type formulas for circle and triangle polynomials have been known before (see [3, chapter 12] and [4, Section 4]).
We nally give a negative example for which Rodrigues type formula (4.6) does not hold.
Example 4.7. Krall and Sheer [5] showed that the dierential equation
L[u] = 3yuxx+ 2uxy−xux−yuy=−nu (4.22)
has an OPS as solutions. In decomposition (4.3), we must choose A1= 1 and C1= 1 in order for
the condition (4.5) to be satised. Then
so that by Lemma 4.1,
1 w@
m x@
nw
y = mn; m; n¿0
is a polynomial of degree 62(m+n), where w(x; y) = exp(y3−xy) is a symmetry factor of the
dierential equation (4.22). For example,
10(x; y) =−y and 01(x; y) = 3y2−x
and 10 satises the dierential equation (4.22) but 01 does not satisfy the dierential equation
(4.22).
Acknowledgements
This work is partially supported by KOSEF(95-0701-02-01-3), GARC, and the Korean Ministry of Education (BSRI 97-1420). Authors thank referees for giving many valuable suggestions and directing attention to Refs. [3, 4].
References
[1] W.C. Brenke, On polynomial solutions of a class of linear dierential equations of the second order, Bull. Amer. Math. Soc. 36 (1930) 77– 84.
[2] C.W. Cryer, Rodrigues formulas and the classical orthogonal polynomials, Boll. Unione. Mat. Ital. 25 (1970) 1–11. [3] A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, vol. II, McGraw-Hill,
New York, 1953.
[4] T.H. Koornwinder, Two-variable analogues of the classical orthogonal polynomials, in: R.A. Askey (Ed.), Theory and Applications of Special functions, Academic Press, New York, 1975, pp. 435– 495.
[5] H.L. Krall, I.M. Sheer, Orthogonal polynomials in two variables, Ann. Mat. Pura Appl. Serie 4 76 (1967) 325–376. [6] Y.J. Kim, K.H. Kwon, J.K. Lee, Orthogonal polynomials in two variables and second order partial dierential
equations, J. Comput. Appl. Math. 82 (1997) 239 – 260.
[7] K.H. Kwon, G.J. Yoon, Symmetrizability of dierential equations having orthogonal polynomial solutions, J. Comput. Appl. Math. 83 (1997) 257– 268.
[8] L.L. Littlejohn, Orthogonal polynomial solutions to ordinary and partial dierential equations, in: M. Alfaro et al. (Eds.), Proc. 2nd Internat. Symp. Orthogonal Polynomials and their Applications, Segovia (Spain), 1986, Lect. Notes Math., vol. 1329, Springer, Berlin, 1988, pp. 98 –124.
[9] P.K. Suetin, Orthogonal Polynomials in Two Variables, Nauka, Moscow, 1988 (in Russian).
[10] F. Marcellan, A. Branquinho, J. Petronilho, Classical orthogonal polynomials: a functional approach, Acta Appl. Math. 34 (1994) 283–303.
