some systems of multivariate orthogonal polynomialsorthogonal polynomials of two variables from...

Some systems of multivariate orthogonalpolynomials

Iván AreaUniversidade de Vigo

AIMS-Volkswagen Stiftung Workshop onIntroduction to Orthogonal Polynomials and Applications

Douala/Limbe, Cameroon, October 5–12, 2018

Iván Area Universidade de Vigo Multivariate orthogonal polynomials Douala, 2018 1 / 95

Example: Hermite polynomials

1 Hermite polynomials were defined by Pierre-Simon Laplace in1810 —in a different form—.

2 They were studied in detail by Pafnuty Chebyshev in 1859.Chebyshev’s work was overlooked and they were named laterafter Charles Hermite.

3 Hermite wrote on the polynomials in 1864 describing them asnew. They were not new!

4 In later 1865 papers Hermite was the first to define themultidimensional polynomials.


∫ ∞−∞

Hn(x)Hm(x)e−x2/2dx = d2n δn,m =

{d2

n , n = m,0, n 6= m.

Rodrigues’ representation Hn(x) =(−1)n

e−x2

dn

dxn e−x2.

Recurrence relation Hn+1(x) = 2xHn(x)− 2nHn−1(x).

Explicit representation Hn(x) = n!

n/2∑m=0

(−1)m

m!(n − 2m)!(2x)n−2m.

Generating function exp(2xt − t2) =∞∑

n=0

Hn(x)tn

n!.

Differential equation y ′′(x)− 2xy ′(x) + ny(x) = 0.

Similarly, for the other families, they were introduced and later obtainedsome relations or generalized.


Characterizations

1 Three-term recurrence relation2 Rodrigues formula3 Hahn’s characterization4 Tricomi’s characterization5 Bochner’s characterization...

So that, we have a general framework. To me it is important to noticethat all of the known families are solution of a second order lineardifferential equation of hypergeometric type

σ(x)y ′′(x) + τ(x)y ′(x) + λy(x) = 0

where σ and τ are polynomials degree at most two and one,respectively, and λ is a constant.


Rodrigues formula

O. Rodrigues (1816) stated that for Legendre polynomials

pn(x) =1

2nn!1dn

dxn

[(x2 − 1)n1

]In general (Tricomi, 1955),

pn(x) =An

%(x)

dn

dxn [%(x)σn(x)]

where %(x) is the symmetrization factor of the differential operator i.e.

(σ(x)%(x))′ = %(x)τ(x),

and An is a normalizing constant.


Let us have a look to a very classical paper of W.A. Al-Salam.


1 Of course, not all the weights are here. We are imposing someconditions —which are equivalent!— Let me recall the lecturegiven by Prof. Marcellán on tuesday about semiclassicalorthogonal polynomials, or even more, who are the orthogonalpolynomials with respect to the skew normal distribution? (out ofany known scheme in our domain but very useful in statistics)

2 In particular, I am interested in the second-order linear differentialoperator of hypergeometric type (cfr. also, the lecture of Prof.David Gómez-Ullate on wednesday).

3 Let me insist that in the univariate case we are using the word"classical" for many different equivalent concepts.


Several variables: Historical development

Already mentioned, in 1865 papers Hermite was the first to introducemultivariable orthogonal polynomials with respect to themultidimensional gaussian.Basic reference:

P. Appell, J. Kampé de Fériet, Fonctions Hypergéométriques etHypersphériques. Polynômes d’Hermite, Gauthier-Villars, Paris, 1926,

which contains a number of properties about generalizations ofHermite polynomials to several variables. It is a very classical book(almost 100 years ago) and it is not easy to get a copy. Thus, e.g. forn = 2 they analized orthogonal polynomials with domain exactly R2,with orthogonality weight function defined as the bivariate gaussian.


Orthogonal polynomials on a triangular region were introduced byProriol:

J. Proriol. Sur une famille de polynômes à deux variables orthogonauxdans un triangle C.R. Acad. Sci. Paris 245(5), 2459–2461, (1957).

They were applied to the problem of solving the Schrödinger equationfor the Helium atom.

G. Munschy. Résolution de l’équation de Schrödinger des atomes àdeux électrons III. J. Phys. Radium 8(18), 552–558 (1957).

G. Munschy and P. Pluvininage. Résolution de l’équation deSchrödinger des atomes à deux électrons II. J. Phys. Radium 8(18),157–160 (1957).


The same class was independently obtained by Karlin and McGregor

S. Karlin and J. McGregor. On some stochastic models in genetics. In“Stochastic models in medicine and biology", J. Gurland editor,University of Wisconsin Press, Madison, 1964 245–271.

in view of applications to genetics, as indicated by Koornwinder, firstsystematic study on classical orthogonal polynomials of two variables.

T. Koornwinder. Two–variable analogues of the classical orthogonalpolynomials. In “Theory and Application of Special Functions", R.Askey ed., Proc. Adv. Semin., The University of Wisconsin–Madison,Academic Press 1975, 435–495.

More concretely, Koornwinder gave a general method of generatingorthogonal polynomials of two variables from orthogonal polynomialsof one variable. This method was also discussed by Dunkl and Xu intheir excellent book (available in the room).Ch. F. Dunkl and Y. Xu Orthogonal Polynomials of Several variables .Encyclopedia of Mathematics and Its Applications 81. (CUP, 2001).


Krall and Sheffer and independently Engelis considered sometwo–dimensional analogues of classical orthogonal polynomials which aresolutions of linear partial differential equations of the second order.

H.L. Krall and I.M. Sheffer. Orthogonal polynomials in two variables. Ann.Mat. Pura Appl. 4(76), 325–376 (1967).

G.K. Engelis. On some two–dimensional analogues of the classicalorthogonal polynomials (in Russian). Latviıskiı Matematiceskiı Ezegodnik 15,169–202 (1974).

Multidimensional problems of approximation theory, numerical analysis, andprobability theory require orthogonal polynomials of several discretevariables:

S. Karlin and J. McGregor. Linear growth models with many types andmultidimensional Hahn polynomials. Theory Appl. spec. Funct., Proc. adv.Semin., Madison 1975, 261–288 (1975).

They formulate some genetic models in which the state space ismulti–dimensional and discrete and they introduce the multivariable Hahnpolynomials.


M.V. Tratnik. Multivariable Meixner, Krawtchouk, andMeixner–Pollaczek polynomials. J. Math. Phys. 30 (12), 2740–2749(1989).

As already mentioned by Prof. Luc Vinet, Tratnik presented amultivariable biorthogonal generalization for Meixner, Kravchuk andMeixner–Pollaczek. Tratnik showed that these families of polynomialsare orthogonal with respect to subspaces of lower degree andbiorthogonal within a given subspace.

Moreover, in M.V. Tratnik. Some multivariable orthogonal polynomialsof the Askey tableau–discrete families. J. Math. Phys. 32(9),2337–2342 (1991). an extension of the previously known multivariableHahn polynomials to all of the remaining discrete families of the Askeysqueme was given.Do not forget that previously R.C. Griffiths, Orthogonal polynomials onthe multinomial distribution, Austral. J. Statist. 13 (1971) 27-35.Corrigenda (1972) Austral. J. Statist. 14, 270.

Let us have a look to some of the papers of Tratnik.Iván Area Universidade de Vigo Multivariate orthogonal polynomials Douala, 2018 12 / 95

Univariate (continuous) classical orthogonal polynomials can becharacterized in terms of the second order linear differential equationof hypergeometric type

σ(x)y ′′ + τ(x)y ′ + λy = 0,

where σ(x) and τ(x) are polynomials of at most the second and firstdegree respectively and λ is a constant. The solutions of the aboveequation have the property that derivatives of these solutions of anyorder also satisfy an equation of the same type.

A.F. Nikiforov, S.K. Suslov, and V.B. Uvarov. Classical OrthogonalPolynomials of a Discrete Variable. Springer Series in ComputationalPhysics. (Springer, Berlin, 1991).


As a generalization, in Lyskova considered a special class of linearpartial differential equations, called basic class,

n∑i,j=1

aij(x)∂2u∂xi∂xj

+n∑

i=1

bi(x)∂u∂xi

+ λu = 0,

where aij(x) = aji(x) and the coefficients aij(x) and bi(x) are chosenso that the derivatives of any order of the solutions of the equation arealso solutions of an equation of the same type.

A.S. Lyskova. Orthogonal polynomials in several variables. Sov. Math.,Dokl. 43(1), 264–268 (1991).

A.S. Lyskova. On some properties of orthogonal polynomials inseveral variables. Russ. Math. Surv. 52(4), 840–841 (1997).

Let us have a look to the works of Lyskova


H.L. Krall and L.M. Sheffer. Orthogonal polynomials in two variables.Ann. Math. Pure Appl. 76 (1967), 325–376.

Krall and Sheffer studied the problem of finding all polynomialeigenfunctions of second order linear differential operators in twovariables having polynomial coefficients of degree equal to the order ofderivative under certain further restrictions relating to itssymmetrizability and the orthogonality of their eigenfunctions. Theyclassified all possible normal forms of the operators satisfying therequired properties. It was shown by Vinet and Zhedanov that, for allthese types, there correspond quantum mechanical systems on aEuclidean (pseudo–Euclidean) plane, two–dimensional sphere, orhyperboloid.

L. Vinet and A. Zhedanov. Two–Dimensional Krall–Sheffer polynomialsand quantum systems on spaces with constant curvature. Letters inMathematical Physics, 65(2), 83–94 (2003).


Key concepts

Admissible: if there exists a sequence {λn} (n = 0,1, . . . ) such that forλ = λn, there are precisely n + 1 linearly independent solutions in theform of polynomials of total degree n and has no non-trivial solutions inthe set of polynomials whose total degree is less than n. This conceptwas introduced by Krall and Sheffer in the case of second order partialdifferential equations and also by Xu in the case of second orderpartial difference equations.

H.L. Krall, L.M. Sheffer, Orthogonal polynomials in two variables, Ann.Math. Pure Appl. 76 (1967) 325–376.

Y. Xu, Second order difference equations and discrete orthogonalpolynomials of two variables, Internat. Math. Res. Notices 8 (2005)449–475.


Framework

1 Admissible.2 Hypergeometric.3 Differential/difference/q-difference/divided-difference operators


From 1 to 2

One essential difference between polynomials in one variable and inseveral variables is the lack of an obvious basis in the latter.

One possibility to avoid this problem is to consider gradedlexicographical order and use the matrix vector representation, firstintroduced by M.A. Kowalski (1982) and afterwards considered by Y.Xu (1993):

xn = (xn−kyk )T, 0 ≤ k ≤ n, n ∈ N0.

x1 = (x , y)T, x2 = (x2, xy , y2)T, . . . .

Let {Pnn−k ,k (x , y)} be a sequence of polynomials in the space Π2

n of allpolynomials of total degree at most n in two variables, x = (x , y), withreal coefficients. Such polynomials are finite sums of terms of the formaxn−kyk , where a ∈ R.


From 1 to 2

From now on the (column) vector representation (Kowalski) will beadopted, so that Pn will denote the (column) polynomial vector

Pn = (Pnn,0(x , y),Pn

n−1,1(x , y), . . . ,Pn1,n−1(x , y),Pn

0,n(x , y))T.

Then, each polynomial vector Pn can be written in terms of the xn as:

Pn = Gn,nxn + Gn,n−1xn−1 + · · ·+ Gn,0 x0,

where Gn,j are matrices of size (n + 1)× (j + 1) and the leading matrixcoefficient Gn,n is a nonsingular square matrix of size (n + 1)× (n + 1).


From 1 to 2

Definition (Monic polynomial vector)

A polynomial vector Pn is said to be monic if its leading matrixcoefficient Gn,n is the identity matrix (of size (n + 1)× (n + 1)); i.e.:

Pn = xn + Gn,n−1xn−1 + · · ·+ Gn,0 x0 .

Then, each of its polynomial entries Pnn−k ,k (x , y) are of the form:

Pnn−k ,k (x , y) = xn−kyk + terms of lower total degree .

“Hat" notation Pn will be used for monic polynomials.

For instance

P1 = {x + a, y + b}T,

P2 = {x2 + cx + dy + e, xy + fx + gy + h, y2 + mx + ny + u}T


Bivariate orthogonality

Definition (Orthogonality)

Let L be a moment linear functional acting on Π2n. A sequence of

polynomials {Pnn−k ,k (x , y)} ⊂ Π2

n, is said to be orthogonal with respectto L if ∀n ∈ N0 there exist an invertible matrix Hn of size n + 1 s.t.

L[(xmPT

n)]

= 0 ∈M(m+1,n+1), n > m, L[(xnPT

n)]

= Hn ∈M(n+1,n+1)

If there exists an integral representation of this orthogonality functionalL, then its action can be written in terms of a weight function% := %(x , y) defined in a certain domain D ⊂ R2:

L(P) =

∫∫D

P(x , y)%(x , y) dx dy , P ∈ Π2n ,

which is defined in the set Π2n provided that all the above integrals

exist. Then, the family {Pn}n≥0 is said to be orthogonal with respect to% in the domain D.


Some partial differential operators

Following Krall & Sheffer (1967) and Kwon, Lee & Littlejohn (2001)these are essentially the classical OP’s in two variables:

1 Hn−k (x)Hk (y), w.r.t. exp(−x2 − y2) in R2, andL = ∂xx + ∂yy − 2(x∂x + y∂y ).

2 Lαn−k (x)Lβk (y), w.r.t. xαyβe−x−y in (0,+∞)2, andL = x∂xx + y∂yy + (1 + α− x)∂x + (1 + β − y)∂y .

3 Hn−k (x)Lβk (y), w.r.t. yβe−x2−y in R× (0,+∞), andL = 1

2∂xx + y∂yy − x∂x + (1 + β − y)∂y .


1

2

3

4 P(α,β+γ+2k+1)n−k (1− 2x)(1− x)kP(β,γ)

k (1− 2y/(1− x)), w.r.t.xαyβ(1− x − y)γ in the triangular region x , y > 0, x + y < 1,L = x(1− x)∂xx + y(1− y)∂yy − 2xy∂xy+(α + 1−$x)∂x + (β + 1−$y)∂y , $ = α + β + γ + 3.

5 P(α+k+1/2,α+k+1/2)n−k (x)(1− x2)k/2P(α,α)

k (y(1− x2)−1/2), w.r.t.(1− x2 − y2)α in the disk x2 + y2 < 1, andL = (1− x2)∂xx + (1− y2)∂yy − 2xy∂xy − (2α + 3)(x∂x + y∂y ).


Let us have a look to a very classical book and highly recommended:P.K. Suetin, first edited in Russian in 1988 and translated into Englishin 1999Starting from moments, some examples are given and later theso-called classical Appell’s orthogonal polynomials are studied. Later,admissible differential equations for polynomials orthogonal over adomain is studied in detail.


Partial differential, difference and q-differenceequations

Many families of orthogonal polynomials of a discrete variable(q-analogues and in nonuniform lattices were existing in the literature)so ...

a11(x , y)∂2u(x , y)

∂2x+ a12(x , y)

∂2u(x , y)

∂x∂y+ a22(x , y)

∂2u(x , y)

∂2y

+ b1(x , y)∂u(x , y)

∂x+ b2(x , y)

∂u(x , y)

∂y+ λu(x , y) = 0.

Using finite difference schemes we may approximate the first– andsecond–order partial derivatives, using the linear combination of thebackward and forward difference quotients, with error O(h2) + O(k2)for h→ 0 and k → 0. The quantities h > 0 and k > 0 are thediscretization parameters or mesh widths.


∂u∂x

(x , y) ≈ 12

[u(x + h, y)− u(x , y)

h+

u(x , y)− u(x − h, y)

h

],

∂u∂y

(x , y) ≈ 12

[u(x , y + k)− u(x , y)

k+

u(x , y)− u(x , y − k)

k

],

∂2u∂x2 (x , y) ≈ 1

h

[u(x + h, y)− u(x , y)

h− u(x , y)− u(x − h, y)

h

],

∂2u∂y2 (x , y) ≈ 1

k

[u(x , y + k)− u(x , y)

k− u(x , y)− u(x , y − k)

k

],

∂2u∂x∂y

(x , y) ≈ 12k

[u(x + h, y)− u(x , y)

h− u(x + h, y − k)− u(x , y − k)

h

]+

12h

[u(x , y + k)− u(x , y)

k− u(x − h, y + k)− u(x − h, y)

k

],


For h = k = 1 we obtain

σ11(x , y)∆1∇1u(x , y) + σ12(x , y)∆1∇2u(x , y)

+ σ21(x , y)∆2∇1u(x , y) + σ22(x , y)∆2∇2u(x , y)

+ τ1(x , y)∆1u(x , y) + τ2(x , y)∆2u(x , y) + λu(x , y) = 0 .


Partial differential, difference and q-differenceequations

a11(x , y)∂2u(x , y)

∂2x+ a12(x , y)

∂2u(x , y)

∂x∂y+ a22(x , y)

∂2u(x , y)

∂2y

+ b1(x , y)∂u(x , y)

∂x+ b2(x , y)

∂u(x , y)

∂y+ λu(x , y) = 0.

This equation uses the simplest q-difference schemes of thesecond-order precision

a11(x , y)D1qD1

q−1u(x , y) + a22(x , y)D2qD2

q−1u(x , y)

+ a12a(x , y)D1qD2

qu(x , y) + a12d (x , y)D1q−1D2

q−1u(x , y)

+ b1(x , y)D1qu(x , y) + b2(x , y)D2

qu(x , y) + λu(x , y) = 0.


Hypergeometric equations

DefinitionWe say that the partial differential equation is hypergeometric if all thederivatives uα(x) = Dr

1Ds2u(x) of the solutions u = u(x) of the equation

are also solutions of an equation of the same type.

This concept was introduced by Lyskova (1991) as basic class in themultivariable continuous case. In the discrete, uα(x) = ∆r

1∆s2u(x). In

the q-case, uα(x) = [D1q ]r [D2

q ]su(x).

This concept applied to the equation implies some restrictions on thecoefficients aij and bi . For instance, in the continuous case

a11(x , y) = a11(x) = a1x2 + b1x + c1 + 0y2 + 0xy + 0y ,b1(x , y) = b1(x) = f1x + g1 + 0y

a12(x , y) = 0x2 + 0y2 + s12xy + t12x + u12y + v12


Hypergeometric and admissible equation

DefinitionThe hypergeometric difference equation will be called admissible ifthere exists a sequence {λn} (n = 0,1, . . . ) such that for λ = λn, thereare precisely n + 1 linearly independent solutions in the form ofpolynomials of total degree n and has no non-trivial solutions in the setof polynomials whose total degree is less than n.

This condition imposed on the equation implies some restrictions onthe coefficients of the polynomials aij and bi . For instance, in thecontinuous case we have

a11(x) = ax2 + b1x + c1, a22(y) = ay2 + b2y + c2

a12(x , y) = axy + · · · , b1(x) = f x + g1, b2(y) = f y + g2


Potentially self-adjoint equation

Du(x) = σ11(x)∆1∇1u(x)

+ σ12(x)∆1∇2u(x) + σ21(x)∆2∇1u(x)

+ σ22(x)∆2∇2u(x) + τ1(x)∆1u(x) + τ2(x)∆2u(x)

The adjoint operator D† of D is defined by

D†u = ∆1∇1(σ11u) + ∆1∇2(σ21u) + ∆2∇1(σ12u) + ∆2∇2(σ22u)

−∇1(τ1u)−∇2(τ2u).

The operator D is self-adjoint if D† = D.


Potentially self-adjoint operators

DefinitionThe operator D is potentially self-adjoint in a domain G if there exists inthis domain a positive real function %(x) = %(x , y) such that theoperator %(x)D is self-adjoint in the domain G.

In order that D be potentially self-adjoint, we multiply the equationthrough by a positive function %(x) in some domain G. Then, theoperator is self-adjoint provided [RGA, 2007] the following conditions


Potentially self-adjoint operators

%(x , y)

%(x + 1, y)=

σ11(x + 1, y) + σ21(x + 1, y)

σ11(x , y) + σ12(x , y) + τ1(x , y),

%(x , y)

%(x + 1, y − 1)=σ21(x + 1, y − 1)

σ12(x , y),

%(x , y)

%(x , y + 1)=

σ12(x , y + 1) + σ22(x , y + 1)

σ21(x , y) + σ22(x , y) + τ2(x , y).

∆1 [σ11%] + ∆2 [σ12%] = τ1%,

∆1 [σ21%] + ∆2 [σ22%] = τ2%,

∆1∇2 [σ21%] = ∆2∇1 [σ12%] .


Weight function for orthogonality

$1(x , y) = σ11(x , y) + σ21(x , y),

$2(x , y) = σ22(x , y) + σ12(x , y),

$3(x , y) = σ11(x , y) + σ12(x , y) + τ1(x , y),

$4(x , y) = σ21(x , y) + σ22(x , y) + τ2(x , y).

Then,

G1(x , y) =$3(x , y)

$1(x + 1, y), G2(x , y) =

$4(x , y)

$2(x , y + 1),

%(x , y) = κ

y−1∏i=y0

G2(x , i)x−1∏j=x0

G1(j , y0) .


Example

The linear partial difference equation

(p1 − 1)x∆1∇1u(x , y) + (p2 − 1)y∆2∇2u(x , y)

+ p1y∆1∇2u(x , y) + p2x∆2∇1u(x , y)

+ (x − Np1)∆1u(x , y) + (y − Np2)∆2u(x , y)− (n1 + n2)u(x , y) = 0 ,

has as a solution the bivariate Kravchuk polynomials of total degreen1 + n2, defined by Tratnik as the generalized Kampé de Fériethypergeometric series

K p1,p2n1,n2

(x , y ; N) = (x + y − N)n1+n2

×F 0:2;21:0;0

(− : −n1,−x ;−n2,−y

−n1 − n2 − x − y + N + 1 : −;−p1 + p2 − 1

p1,p1 + p2 − 1

p2

),

where N is a non–negative integer and p1, p2 are real parameterssatisfying p1 > 0, p2 > 0, 0 < p1 + p2 < 1.


%(x , y) = κ

y−1∏i=y0

G2(x , i)x−1∏j=x0

G1(j , y0) .

For this specific equation, the approach gives us as orthogonalityweight function

%(x , y) =N!

x!y !(N − x − y)!px

1py2(1− p1 − p2)N−x−y

which is the trinomial distribution (bivariate extension of binomialdistribution), in the triangular domain G given by

x ≥ 0, y ≥ 0, 0 ≤ x + y ≤ N,

with p1 > 0, p2 > 0, 0 < p1 + p2 < 1.


The monic bivariate Kravchuk polynomials (Tratnik 1991)

K p1,p2n1,n2

(x , y ; N) = (−1)n1+n2pn11 pn2

2 (N − n1 − n2 + 1)n1+n2

× F 0:2;21:0;0

(− : −n1,−x ;−n2,−y

−N : −;−1p1,

1p2

),

the non-monic bivariate Kravchuk polynomials (Tratnik 1991), definedas a generalized Kampé de Fériet series

K p1,p2n1,n2

(x , y ; N) = (x + y − N)n1+n2

×F 0:2;21:0;0

(− : −n1,−x ;−n2,−y

−n1 − n2 − x − y + N + 1 : −;−p1 + p2 − 1

p1,p1 + p2 − 1

p2

),


the non-monic bivariate Kravchuk polynomials defined as a product ofKravchuk polynomials

K p1,p2n1,n2

(x , y ; N)

=(N − n1)!

N! (n1 − N)n2

Kn1(x ; p1/(p1+p2), x+y) Kn2(x+y−n1; p1+p2,N−n1) ,

where for 0 < p < 1 and n = 0,1, . . . ,N, the univariate Kravchukpolynomials are normalized as

Kn(x ; p,N) = (−N)n 2F1

(−n,−x−N

∣∣∣ 1p

),

and the non-monic bivariate Kravchuk polynomials (Geronimo andIliev, 2010) defined also as a product of univariate Kravchukpolynomials

K2(n1,n2; x , y ; p1,p2; N)

=1

(−N)n1+n2

Kn1(x ; p1,N − n2) Kn2(y ; p2/(1− p1),N − x).

Also ... as Rodrigues formula!Iván Area Universidade de Vigo Multivariate orthogonal polynomials Douala, 2018 38 / 95

Partial differential equations

So, in the continuous, discrete and even for the q-analogues we areconsidering equations of this type

a11(x , y)∂2u(x , y)

∂2x+ a12(x , y)

∂2u(x , y)

∂x∂y+ a22(x , y)

∂2u(x , y)

∂2y

+ b1(x , y)∂u(x , y)

∂x+ b2(x , y)

∂u(x , y)

∂y+ λnu(x , y) = 0,

which we assume that are hypergeometric, admissible and potentiallyself-adjoint.Why to restrict to this class? There are families out of here! (e.g. in thebook of Suetin). First answer from a quite recent paper. Secondanswer at the end of the talk.


Recurrence relations

The idea is to use the (column) vector representation (Kowalski).

Pn = (Pnn,0(x , y),Pn

n−1,1(x , y), . . . ,Pn1,n−1(x , y),Pn

0,n(x , y))T.

Pn = Gn,nxn + Gn,n−1xn−1 + · · ·+ Gn,0 x0,

where Gn,j are matrices of size (n + 1)× (j + 1) and Gn,n is anonsingular square matrix of size (n + 1)× (n + 1).As mentioned some slides before, monic if its leading matrix coefficientGn,n is the identity matrix (of size (n + 1)× (n + 1)); i.e.:

Pn = xn + Gn,n−1xn−1 + · · ·+ Gn,0 x0 .


Known results

Then, each of its polynomial entries Pnn−k ,k (x , y) are of the form:

Pnn−k ,k (x , y) = xn−kyk + terms of lower total degree .


Known results

Theorem (Dunkl and Xu)Let L be a positive definite moment linear functional acting on thespace Π2

n of all polynomials of total degree at most n in two variables,and {Pn}n≥0 be an orthogonal family with respect to L. Then, forn ≥ 0, there exist unique matrices An,j of size (n + 1)× (n + 2), Bn,j ofsize (n + 1)× (n + 1),and Cn,j of size (n + 1)× n, such that

xjPn = An,jPn+1 + Bn,jPn + Cn,jPn−1, j = 1,2

with the initial conditions P−1 = 0 and P0 = 1. Here the notationx1 = x, x2 = y is used.


TheoremThe explicit expressions of the matrices An,j , Bn,j and Cn,j (j = 1,2) interms of the values of the leading coefficients Gn,n, Gn,n−1 and Gn,n−2in the expansions are given by

An,j = Gn,nLn,jG−1n+1,n+1, n ≥ 0,

B0,j = −A0,jG1,0,

Bn,j = (Gn,n−1Ln−1,j − An,jGn+1,n)G−1n,n, n ≥ 1,

C1,j = −(A1,jG2,0 + B1,jG1,0),

Cn,j = (Gn,n−2Ln−2,j − An,jGn+1,n−1 − Bn,jGn,n−1)G−1n−1,n−1, n ≥ 2 .


The matrices Ln,j of the size (n + 1)× (n + 2)

Ln,1 =

1 i 0. . .

...i 1 0

and Ln,2 =

0 1 i...

. . .0 i 1

,

so thatx xn = Ln,1xn+1, y xn = Ln,2xn+1,

and

x2 xn = Ln,1Ln+1,1xn+2, y2 xn = Ln,2Ln+1,2xn+2 ,

Ln,2Ln+1,1 = Ln,1Ln+1,2,

and for j = 1,2,Ln,j LT

n,j = In+1,

where In+1 denotes the identity matrix of the size n + 1.


Moreover,∂jxn = En,j xn−1, j = 1,2,

where the matrices En,j of the size (n + 1)× n are given by

En,1 =

n i

n − 1. . .i 1

0 . . . 0 0

, En,2 =

0 . . . 01 i

2. . .i n

.


Discrete situation

Explicit expression for the matrices Gn,n−1 and Gn,n−2:

Gn,n−1 = SnF−1n−1(λn),

Gn,n−2 =(Tn + Gn,n−1Sn−1

)F−1

n−2(λn),

where the nonsingular matrix Fn(λ`) is given by

Fn(λ`) = (λn − λ`)In+1,

In+1 denotes the identity matrix of size (n + 1)× (n + 1)


Moreover, using the left inverse D†n of the joint matrix Ln

D†n =

1 0

1/2 © 1/2 ©. . . . . .

© 1/2 © 1/20 1

,

we can write a recursive formula for the monic orthogonal polynomials

Pn+1 = D†n

[(xy

)⊗ In+1 − Bn

]Pn − D†nCnPn−1, n ≥ 0,

with the initial conditions P−1 = 0, P0 = 1, where ⊗ denotes theKronecker product and

Bn =(

BTn,1 ,B

Tn,2

)T, Cn =

(CT

n,1 ,CTn,2

)T,

are matrices of size (2n + 2)× (n + 1) and (2n + 2)× n, respectively.Iván Area Universidade de Vigo Multivariate orthogonal polynomials Douala, 2018 47 / 95

Bivariate big q-Jacobi polynomials

Non-monic

Pn,k (x , y ; a,b, c,d ; q) := Pn−k (y ; a,bcq2k+1,dqk ; q)

× yk (dq/y ; q)kPk (x/y ; c,b,d/y ; q),

n ∈ N , k = 0,1, . . . ,n, q ∈ (0,1), 0 < aq,bq, cq < 1, d < 0,

where the univariate big q-Jacobi are (under some restrictions on theparameters)

Pm(t ; A,B,C; q) := 3φ2

(q−m, ABqm+1, t

Aq, Cq

∣∣∣q; q).

Potentially self-adjoint admissible second order partial q-differenceequation of hypergeometric type

a11(x) =√

q (d q − x)(a c q2 − x

),a22(y) =

√q (a q − y) (d q − y) ,

a12a(x , y) = a c q4 (d − b x) (1− y) ,a12d (x , y) = (d q − x) (a q − y) ,

[...].


Bivariate big q-Jacobi polynomials

Lewanowicz and Wozny proved that these polynomials satisfy thefollowing orthogonality relation∫ aq

dq

∫ cqy

dqW (x , y ; a,b, c,d ; q)Pn,k (x , y ; q)Pm,l(x , y ; q)dqxdqy

= Hn,k (a,b, c,d ; q)δn,mδk ,l ,

where 0 < aq,bq, cq < 1, d < 0, and the weight function is defined by

W (x , y ; a,b, c,d ; q) :=(dq/y , c−1x/y , x/d , y/a, y/d ; q)∞

y(c−1d/y , cqy/d , x/y ,bx/d , y ; q)∞,

with the notations

(a1, . . . ,ar ; q)∞ = (a1; q)∞ · · · (ar ; q)∞,

and

(a; q)∞ =∞∏

j=0

(1− aqj).


Monic solutions

Bn,1

dq−i+n+2(

acq2i−1(

q−i+n+1(

b(

acqi+n+1 + q−i+n+2 − q − 1)− q − 1

)+ 1)

+ 1)

(abcq2n+1 − 1

) (abcq2n+3 − 1

)+

acqn+2(−b(q + 1)qn−i

(acq2i + q

)+ ab(b + 1)cq2n+2 + b + 1

)(abcq2n+1 − 1

) (abcq2n+3 − 1

) , (i = j),

−

(qi−1 − 1

) (aqi−1 − 1

)q−i+n+2

(abcdq2n+2 − abc(q + 1)qn+1 + d

)(abcq2n+1 − 1

) (abcq2n+3 − 1

) , (i = j + 1),

0, otherwise,

Bn,2

q−i

q[i]q(

aqi − 1) (

dqn+1 − 1)

abcq2n+3 − 1+

q[i − 1]q(

q − aqi) (

dqn − 1)

abcq2n+1 − 1+ q

, (i = j),

−acqi+1

(q−i+n+1 − 1

) (bq−i+n+1 − 1

) (abcq2(n+1) − d(q + 1)qn + 1

)(abcq2n+1 − 1

) (abcq2n+3 − 1

) , (i = j − 1),

0, otherwise,


Monic solutions

Pn,m(x, y ; a, b, c, d ; q) =

(db

)n(aq; q)m(bq; q)n(dqn+1; q)m(abcqm+2/d ; q)n

(abcqm+n+2; q)n+m

×n∑

i=0

m∑j=0

(−1)−i−j[

ni

]q

[mj

]q

q12 (i(i−2n+1)+j(j−2m+1))

(abcqm+n+2; q)i+j

(aq; q)j (bq; q)i (dqn+1; q)j (abcqm+2/d ; q)i(bx/d ; q)i (y ; q)j

=

(db

)n(aq; q)m(bq; q)n(dqn+1; q)m(abcqm+2/d ; q)n

(abcqm+n+2; q)n+m

× Φ1:2;20:2;2

[abcqn+m+2 : q−n

, bx/d ; q−m, y

− : bq, abcqm+2/d ; aq, dqn+1

q : q, q0, 0, 0

],

where the generalized bivariate basic hypergeometric series is defined by

Φλ:r ;sµ:u;v

[α1, . . . , αλ : a1, . . . , ar ; c1, . . . , csβ1, . . . , βµ : b1, . . . , bu ; d1, . . . , dv

q : x, yi, j, k

]

=∞∑

m,n=0

(α1, . . . , αλ; q)m+n(a1, . . . , ar ; q)m(c1, . . . , cs ; q)n

(β1, . . . , βµ; q)m+n(b1, . . . , bu ; q)m(d1, . . . , dv ; q)n

xmynqi(m

2

)+j(n

2

)+kmn

(q; q)m (q; q)n.


Third solution

Pn,m(x , y ; a,b, c,d ; q) =Λn,m

%(x , y)[D1

q ](n)[D2q ](m)[

%(x , y)x2ny2m(dq/x ; q)n(aq/y ; q)m(x/y ; q)m(cqy/x ; q)n

],


Limit as q ↑ 1

(x2 − 1

) ∂2

∂x2 f +(

y2 − 1) ∂2

∂y2 f + 2 ((x + 1)(y − 1))∂2

∂x∂yf

+ (x(α + β + γ + 3) + α− β + γ + 1)∂

∂xf

+ (y(α + β + γ + 3) + α− β − γ − 1)∂

∂yf

− n(α + β + γ + n + 2)f = 0.

The orthogonality weight function for the polynomial solutions of theabove equation can be computed following AGRZ giving rise to

%(α,β,γ)(x , y) = (1− y)α(x + 1)β(y − x)γ ,

in the triangular domain

R = {(x , y) ∈ R2 | x ≤ y ≤ 1, −1 ≤ x ≤ 1}.


At least three orthogonal polynomial solutions of the latter PDE withrespect to the same weight function %(α,β,γ)(x , y) on the same domainR

1 The monic polynomial solutions of the latter PDE satisfythree-term recurrence relations where the matrix coefficients canbe easily computed by considering the limit as q ↑ 1 of ourmatrices for a = qα, b = qβ, c = qγ and d = −qδ, or eventuallyfrom AGRZ.Moreover, we have the following representation in terms ofgeneralized Kampé de Fériet hypergeometric series as

A(α,β,γ)n,m (x , y) = (−1)n2n+m (α + 1)m (β + 1)n

(α + β + γ + m + n + 2)n+m

× F 1:1;10:1;1

(α + β + γ + m + n + 2 : −n;−m

− : β + 1;α + 1x + 1

2,1− y

2

).


Partial differential equation

1 Monic2 The non-monic polynomials which can be computed from the

Rodrigues formula (exactly coincide with those in the book ofSuetin, Chapter III)

%(α,β,γ)(x , y) A(α,β,γ)n,m (x , y)

=∂n+m

∂xn ∂ym

[(x + 1)β+n (1− y)α+m (y − x)γ+n+m

].


Partial differential equation

1 Monic (recurrence relation in matrix form and explicit rep.)2 Non-monic Rodrigues’s3 And a third non-monic polynomial solution

Jn,m(x , y ;α, β, γ) =m!(y + 1)m(n −m)!

(γ + 1)m(α + 1)n−m

× P(γ,β)m

(2x − y + 1

y + 1

)P(α,β+γ+2m+1)

n−m (y),

where for a,b > −1,

P(a,b)n (x) =

(a + 1)n

n!2F1

(−n,n + a + b + 1

a + 1

∣∣∣ 1− x2

),

are the (classical and univariate) Jacobi polynomials.


Limits!!!

1 limq↑1

W (x , y ; qα,qβ,qγ ,−qδ; q) = %(α,β,γ)(x , y).

2 Matrices ↑ 1 Matrices!!3 Monic

limq→1

Pn,m(x , y ; qα,qβ,qγ ,−qδ; q) = A(α,β,γ)n,m (x , y).

4 Non-monic

limq→1

Pn,m(x , y ; qα,qβ,qγ ,−qδ; q) = Jn,m(x , y ;α, β, γ).

5 Non monic (Rodrigues’)

limq→1

Pn,m(x , y ; qα,qβ,qγ ,−qδ; q) = A(α,β,γ)n,m (x , y).


In this framework we know that from the second-order linear partialdifferential equation of hypergeometric type we obtain

1 Three-term recurrence relation2 Rodrigues formula3 Pearson’s system4 Structure relation(s)5 ...


Next step: nonuniform lattices.

Extreme difficulties appear.


Table: S. Suslov. The theory of difference analogues of special functions ofhypergeometric type. 1989 Russ. Math. Surv. 44 227


Nonuniform lattices

Univariate Racah polynomials can be defined in terms ofhypergeometric series as

rn(α, β, γ, δ; s) = rn(s) = (α + 1)n (β + δ + 1)n (γ + 1)n

× 4F3

(−n,n + α + β + 1,−s, s + γ + δ + 1

α + 1, β + δ + 1, γ + 1

∣∣∣1) , n = 0,1, . . . ,N,

where rn(α, β, γ, δ; s) is a polynomial of degree 2n in s and of degree nin the quadratic lattice

η(s) = s(s + γ + δ + 1),

and (A)n = A(A + 1) · · · (A + n − 1) with (A)0 = 1 denotes thePochhammer symbol.


Univariate Racah polynomials satisfy the following second-order lineardivided-difference equation

φ(η(s))D2ηrn(s) + τ(η(s))SηDηrn(s) + λnrn(s) = 0,

where φ is a polynomial of degree two in the lattice η(s) given by

φ(η(s)) = −(η(s))2

+12

(−α(2β + δ + γ + 3) + β(δ − γ − 3)− 2(δγ + δ + γ + 2))η(s)

− 12

(α + 1)(γ + 1)(β + δ + 1)(δ + γ + 1),

τ is a polynomial of degree one in the lattice η(s) given by

τ(η(s)) = −(α + β + 2)η(s)− (α + 1)(γ + 1)(β + δ + 1),

the eigenvalues λn are given by

λn = n(α + β + n + 1),


and the difference operators Dη and Sη are defined by

Dηf (s) =f (s + 1/2)− f (s − 1/2)

η(s + 1/2)− η(s − 1/2), Sηf (s) =

f (s + 1/2) + f (s − 1/2)

2.

Notice that the above operators transform polynomials of degree n inthe lattice η(s) into polynomials of respectively degree n − 1 and n inthe same variable η(s).We would like to notice here that

Dηrn(α, β, γ, δ; s)

= n(n + α + β + 1)rn−1(α + 1, β + 1, γ + 1, δ; s − 1/2)


Multivariable Racah polynomials have been introduced by Tratnik anddeeply analyzed by Geronimo and Iliev, where they construct acommutative algebra Ax of difference operators in Rp, depending onp + 3 parameters, which is diagonalized by the multivariable Racahpolynomials considered by Tratnik. In the particular case p = 2, thebivariate Racah polynomials are defined in terms of univariate Racahpolynomials as

Rn,m(s, t ;β0, β1, β2, β3,N) = rn(β1−β0−1, β2−β1−1,−t−1, β1 +t ; s)

× rm(2n + β2 − β0 − 1, β3 − β2 − 1,n − N − 1,n + β2 + N; t − n),

which are polynomials in the lattices x(s) = s(s + β1) andy(t) = t(t + β2). These polynomials coincide with the bivariate Racahpolynomials of parameters a1, a2, a3, γ, and η introduced by Tratnikafter the substitutions

β0 = a1−η−1, β1 = a1, β2 = a1 +a2, β3 = a1 +a2 +a3, N = −γ−1.


Let us a look to some highlights in the nice paper of Geronimo andIliev.


Theorem

The bivariate Racah polynomials are solution of the followingfourth-order linear partial divided-difference equation

f1(x(s), y(t))D2xD2

yRn,m(s, t) + f2(x(s), y(t))SxDxD2yRn,m(s, t)

+ f3(x(s), y(t))SyDyD2xRn,m(s, t) + f4(x(s), y(t))SxDxSyDyRn,m(s, t)

+ f5(x(s))D2xRn,m(s, t) + f6(y(t))D2

yRn,m(s, t) + f7(x(s))SxDxRn,m(s, t)

+ f8(y(t))SyDyRn,m(s, t) + (m + n)(β3−β0 + m + n−1)Rn,m(s, t) = 0,

where Rn,m(s, t) := Rn,m(s, t ;β0, β1, β2, β3,N), and the coefficients fi ,i = 1, . . . ,8 are polynomials in the lattices x(s) and y(t) given by


f8(y(t)) = (β0 − β3)y(t)− N(β0 − β2)(β3 + N),

f7(x(s)) = (β0 − β3)x(s)− N(β0 − β1)(β3 + N),

f6(y(t)) = −(y(t))2 +12

(2N2 + 2β3(β0 + N)

− β2(β3 + β0))y(t)− 12

Nβ2(β0 − β2)(β3 + N),

f5(x(s)) = −(x(s))2 +12

(2β3(N + β0)

+ 2N2 − β1(β3 + β0))x(s)

− 12

Nβ1(β0 − β1)(β3 + N),

f4(x(s), y(t)) = −2x(s)y(t)

+ (2N2 + β2(1− β0) + β3(β0 − 1 + 2N))x(s)

+ (β0 − β1)(β3 + 1)y(t)− N(β0 − β1)(β2 + 1)(β3 + N),

[...]


Theorem

The polynomial R(1,0)n,m (s, t) := DxRn,m(s, t) is solution of the following

fourth-order linear partial divided-difference equation

f11(x(s), y(t))D2xD2

yR(1,0)n,m (s, t) + f21(x(s), y(t))SxDxD2

yR(1,0)n,m (s, t)

+ f31(x(s), y(t))SyDyD2xR(1,0)

n,m (s, t) + f41(x(s), y(t))SxDxSyDyR(1,0)n,m (s, t)

+ f51(x(s))D2xR(1,0)

n,m (s, t) + f61(y(t))D2yR(1,0)

n,m (s, t)

+ f71(x(s))SxDxR(1,0)n,m (s, t) + f81(y(t))SyDyR(1,0)

n,m (s, t)

+ (m + n − 1)(β3 − β0 + m + n)R(1,0)n,m (s, t) = 0,

where the coefficients fi1, i = 1, . . . ,8 are polynomials in the latticesx(s) and y(t)


Theorem

The polynomial R(0,1)n,m (s, t) := DyRn,m(x(s), y(t)) is solution of the

following fourth-order linear partial divided-difference equation


yR(0,1)n,m (s, t) + f22(x(s), y(t))SxDxD2

yR(0,1)n,m (s, t)

+ f32(x(s), y(t))SyDyD2xR(0,1)

n,m (s, t) + f42(x(s), y(t))SxDxSyDyR(0,1)n,m (s, t)

+ f52(x(s))D2xR(0,1)

n,m (s, t) + f62(y(t))D2yR(0,1)

n,m (s, t)

+ f72(x(s))SxDxR(0,1)n,m (s, t) + f82(y(t))SyDyR(0,1)

n,m (s, t)

+ (m + n − 1)(β3 − β0 + m + n)R(0,1)n,m (s, t) = 0,

where the coefficients fi2, i = 1, . . . ,8 are polynomials in the latticesx(s) and y(t).


Notice that for i = 1, . . . ,8 we have

fi1(x(s), y(t))

= fi(x(s − 1/2), y(t − 1);β0,1 + β1,2 + β2,2 + β3,N − 1),

and

fi2(x(s), y(t)) = fi(x(s), y(t − 1/2);β0, β1, β2 + 1, β3 + 2,N − 1),

where fi = fi(x(s), y(t);β0, β1, β2, β3,N) can be explicitly given.As a consequence of the latter equalities, we obtain the followingrelation

DxRn,m(s, t ;β0, β1, β2, β3,N) = n(n − β0 + β2 − 1)

× Rn−1,m(s − 1/2, t − 1;β0, β1 + 1, β2 + 2, β3 + 2,N − 1).


We have recently proposed a conjecture about the shape of the formof the hypergeometric-type divided-difference equation satisfied by thep-variate Racah polynomials.


q-quadratic lattices!

Bivariate Askey-Wilson polynomials defined by

Pn,m(s, t ; a,b, c,d ,e2|q) = pn(x(s); a,b,e2qt ,e2q−t |q)

× pm(y(t); ae2qn,be2qn, c,d |q),

Pn,m(s, t ; a,b, c,d ,e2|q) = pn(x(s); ce2qm,de2qm,a,b|q)

× pm(y(t); c,d ,e2qs,e2q−s|q),

have been introduced by Gasper and Rahman in 2005. They arepolynomials of total degree n + m in the variables x(s) and y(t) := x(t).The univariate Askey-Wilson polynomials are defined by

pn(x(s); a,b, c,d |q)

=(ab,ac,ad ; q)n

an 4φ3

(q−n,abcdqn−1,aqs,aq−s

ab,ac,ad

∣∣∣∣∣q; q

);

they are polynomials of degree n in the q-quadratic lattice

x(s) = cos θ =qs + q−s

2, qs = eiθ.


Theorem

Let

x(s) =qs + q−s

2= cos θ1, y(t) =

qt + q−t

2= cos θ2.

The bivariate Askey-Wilson polynomials are solution of the followingfourth-order linear partial divided-difference equation

f1(x(s), y(t))D2xD2

yPn,m(s, t) + f2(x(s), y(t))SxDxD2yPn,m(s, t)

+ f3(x(s), y(t))SyDyD2xPn,m(s, t) + f4(x(s), y(t))SxDxSyDyPn,m(s, t)

+ f5(x(s))D2xPn,m(s, t) + f6(y(t))D2

yPn,m(s, t) + f7(x(s))SxDxPn,m(s, t)

+ f8(y(t))SyDyPn,m(s, t) + λn,mPn,m(s, t) = 0,

where λn,m = 16q−(m+n)+3(1− qm+n)(1− abcde22qm+n−1), Pn,m(s, t)

stands for Pn,m(s, t ; a,b, c,d ,e2|q) or Pn,m(s, t ; a,b, c,d ,e2|q),


f8(y(t)) = 8q2(q − 1)(

2(1− abcde2

2)y(t) +

(d + c

)(abe2

2 − 1)

+(cd − 1

)(a + b

)e2),

f7(x(s)) = 8q2(q − 1)(

2(1− abcde2

2)x(s) +

(cde2

2 − 1)(

a + b)

+(d + c

)(ab − 1

)e2),

f6(y(t)) = 4q3/2(q − 1)2(− 2(abcde2

2 + 1)(

y(t))2

+((

d + c)(

abe22 + 1

)+(cd + 1

)(a + b

)e2

)y(t)

+(cd − 1

)(abe2

2 − 1)−(d + c

)(a + b

)e2

),

f5(x(s)) = 4q3/2(q − 1)2(− 2(abcde2

2 + 1)(

x(s))2

+((

cde22 + 1

)(a + b

)+(d + c

)(ab + 1

)e2

)x(s)

+(cde2

2 − 1)(

ab − 1)−(d + c

)(a + b

)e2

),

[...]


Proposition

The difference derivative of the bivariate Askey-Wilson polynomials

DxPn,m(s, t ; a,b, c,d ,e2|q) := P(1,0)n,m (s, t)

are solution of a fourth-order linear partial divided-difference equation


yP(1,0)n,m (s, t) + f12(x(s), y(t))SxDxD2

yP(1,0)n,m (s, t)

+f13(x(s), y(t))SyDyD2xP(1,0)

n,m (s, t) + f14(x(s), y(t))SxDxSyDyP(1,0)n,m (s, t)

+f15(x(s))D2xP(1,0)

n,m (s, t) + f16(y(t))D2yP(1,0)

n,m (s, t)

+f17(x(s))SxDxP(1,0)n,m (s, t) + f18(y(t))SyDyP(1,0)

n,m (s, t)

+16 q−m−n+4 (qm+n−1 − 1) (

abcde22qm+n − 1

)P(1,0)

n,m (s, t) = 0,

with

f1i(x(s), y(t)) = fi(s, t ; aq12 ,bq

12 , c,d ,e2q

12 ), i = 1, . . . ,8,


PropositionThe difference derivative of the bivariate Askey-Wilson polynomials

DyPn,m(s, t ; a,b, c,d ,e2|q) := P(0,1)n,m (s, t)

are solution of a fourth-order linear partial divided-difference equation


yP(0,1)n,m (s, t) + f22(x(s), y(t))SxDxD2

yP(0,1)n,m (s, t)

+f23(x(s), y(t))SyDyD2xP(0,1)

n,m (s, t) + f24(x(s), y(t))SxDxSyDyP(0,1)n,m (s, t)

+f25(x(s))D2xP(0,1)

n,m (s, t) + f26(y(t))D2yP(0,1)

n,m (s, t)

+f27(x(s))SxDxP(0,1)n,m (s, t) + f28(y(t))SyDyP(0,1)

n,m (s, t)

+16 q4−n−m (qm+n−1 − 1) (

abcde22qm+n − 1

)P(0,1)

n,m (s, t) = 0,

with

f2i(x(s), y(t)) = fi(s, t ; a,b, cq12 ,dq

12 ,e2q

12 ), i = 1, . . . ,8.


By using the same ideas as for the continuous case, discrete case andtheir q-analogues we have succeeded to obtain the explicit forms ofthe matrices appearing in the three-term recurrence relation satisfiedby all the known families of bivariate orthogonal polynomials onnonuniform lattices.


We denote by

xn =(

x(s)n, x(s)n−1y(t), · · · , x(s)y(t)n−1, y(t)n)T,

the column vector of the monomials x(s)n−ky(t)k . Let Pn denote the(column) polynomial vector of the polynomials Pn−k ,k (x(s), y(t)) oftotal degree n,

Pn = (Pn,0(x(s), y(t)),Pn−1,1(x(s), y(t)), . . . ,P0,n(x(s), y(t)))T.

Then,

Pn = Gn,nxn + Gn,n−1xn−1 + Gn,n−2xn−2 + · · ·+ Gn,0 x0, (1)

where Gn,j are matrices of size (n + 1)× (j + 1) and Gn,n is anonsingular square matrix of size (n + 1)× (n + 1).


We have the relations

x(s)xn = Ln,1xn+1, y(t)xn = Ln,2xn+1, (2)

where Ln,1 and Ln,2 are (n + 1)× (n + 2) size matrices given by

Ln,1 =

1 0 . . . . . . 0

0 1. . .

......

. . . . . . . . ....

0 . . . 0 1 0

, Ln,2 =

0 1 0 . . . 0...

. . . 1. . .

......

. . . . . . 00 . . . . . . 0 1

.


TheoremThe polynomials Pn are solution of the three-term recurrence relations

xjPn = An,jPn+1 + Bn,jPn + Cn,jPn−1, j = 1,2, (3)

with the initial conditions P−1 = 0 and P0 = 1, where x1 = x(s) andx2 = y(t) and the matrices An,j of size (n + 1)× (n + 2), Bn,j of size(n + 1)× (n + 1), and Cn,j of size (n + 1)× n are given by

An,j = Gn,nLn,jG−1n+1,n+1, n ≥ 0,

B0,j =(−A0,jG1,0

)G−1

0,0,

Bn,j =(Gn,n−1Ln−1,j − An,jGn+1,n

)G−1

n,n, n ≥ 1,

C1,j =(−A1,jG2,0 − B1,jG1,0

)G−1

0,0,

Cn,j =(Gn,n−2Ln−2,j − An,jGn+1,n−1 − Bn,jGn,n−1

)G−1

n−1,n−1, n ≥ 2.

It follows clearly that to compute the coefficients An,j , Bn,j and Cn,j , weneed to know explicitly the coefficients Gn,n, Gn,n−1, Gn,n−2 of Pn.


From the definition of the bivariate Askey-Wilson polynomials, we get

Pn,m(s, t ; a,b, c,d ,e2|q) =

C(n,m)(x(s)ny(t)m, x(s)n−1y(t)m+1, . . . , x(s)y(t)m+n−1, y(t)m+n)T

+ · · · ,

where

C(n,m) =(Ai(n,m))0≤i≤n,

is the leading coefficient vector of Pn,m(s, t ; a,b, c,d ,e2|q) in the basisx(s)iy(t)j with

Ai(n,m) = (−2 a)n+m q12 (n+m)(m+n−1)e2

i+mC2(n,m,m)C1(n,n − i),

for i = 0,1, . . . ,n, where

C1(n, i) =(ab; q)n

(q−n,abe2

2qn−1; q)

i q i

(ab,q; q)i an ,

C2(n,m, j) =

(abe2

2q2 n,ace2qn,ade2qn; q)

m

(q−m,abcde2

2q2 n+m−1; q)

j q j(abe2

2q2 n,ace2qn,ade2qn,q; q)

j (ae2qn)m .


For the second bivariate Askey-Wilson polynomial family, we have

Pn,m(s, t ; a,b, c,d ,e2|q) =

C(n,m)(x(s)ny(t)m, x(s)n+1y(t)m−1, . . . , x(s)n+m−1y(t), x(s)n+m)T

+ · · · ,

whereC(n,m) = (Aj(n,m))0≤j≤m

and

Aj(n,m) = (−2 c)m+n q1/2 (m+n)(m+n−1)e2n+jC1(n,m,n)C2(m,m − j),

where

C1(n,m, i) =

(cde2

2q2 m,ace2qm,bce2qm; q)

n

(q−n,abcde2

2q2 m+n−1; q)

i q i

(cde22q2 m,ace2qm,bce2qm,q; q)i (ce2qm)n ,

C2(m, j) =(cd ; q)m

(q−m, cde2

2qm−1; q)

j q j

cm (cd ,q; q)j.


Since

Pn−j,j(x(s), y(t)) =

n−j∑i=0

Ai(n − j , j)x(s)n−j−iy(t)i+j + · · · ,

and

Pn−j,j(x(s), y(t)) =

j∑k=0

Ak (n − j , j)x(s)n−j+ky(t)j−k + · · · ,

it follows that Gn,n = (gk ,l(n))0≤k ,l≤n is an upper triangular matrixwhere

gk,l (n) = Al−k (n − k , k) = (−2)n q12 n2+ 1

2 (−2 k+1)n+k2+k−le2l−k

×(ab; q)n−k

(qk−n; q

)n−l

(q; q)n−l (q; q)k

×(abe2

2qn−k−1; q)

n−l

(q−k ; q

)k

(abcde2

2q2 n−k−1; q)

k

(ab; q)n−l,


and Gn,n = (gk ,l(n))0≤k ,l≤n is a lower triangular matrix where

gk,l (n) = Ak−l (n − k , k) = (−2)n q12 n2+ 1

2 (−2 k+1)n+k2−k+le2k−l

×(q−n+k ,abcde2

2qk+n−1; q)

n−k (dc; q)k

(q−k , cde2

2qk−1; q)

l

(q; q)n−k (dc; q)l (q; q)l,

where we consider

Pn = (Pn−k ,k (x(s), y(t)))Tk=0,...,n = Gn,nxn + Gn,n−1xn−1 + · · · .


Connection!

Let Pn be the column vector of monic bivariate Askey-Wilsonpolynomials. Then, we have

Pn = Gn,n Pn, Pn = Gn,n Pn.

As a consequence, we obtain the following connection formulaebetween the two families of bivariate Askey-Wilson polynomials

Pn = Gn,n(Gn,n)−1 Pn, n ≥ 0.

It is therefore sufficient to get the recurrence relation for Pn since theone of Pn follows from this connection formula. This connectionformula can also be written as

Pn−j,j(x(s), y(t)) =n∑

l=0

bj,l Pn−l,l(x(s), y(t)), j = 0,1, . . . ,n,

where (bj,0,bj,1, . . . ,bj,n) is the j th row of the matrix Gn,n(Gn,n)−1.Iván Area Universidade de Vigo Multivariate orthogonal polynomials Douala, 2018 85 / 95

To compute the coefficients Gn,n−1 and Gn,n−2, we use the fourth-orderlinear partial divided-difference equations satisfied by the bivariateAskey-Wilson polynomials and the following results.The action of the operators Dx and Sx on x(s)n is given for i = 1,2 by

Dxx(s)n = Hn,n−1,ix(s)n−1 + Hn,n−2,ix(s)n−2 + Hn,n−3,ix(s)n−3 + · · · ,Sxx(s)n = Qn,n,ix(s)n + Qn,n−1,ix(s)n−1 + Qn,n−2,ix(s)n−2 + · · · ,

where

Hn,n−1,i =

√q

q − 1(q

n2 − q−

n2 ), Hn,n−2,i = 0,

Hn,n−3,i =(−2 + n)

(q1/2−n/2 − q1/2+n/2)+ n

(q−1/2+n/2 − q3/2−n/2)

4 q − 4,

Qn,n,i =12

(qn/2 + q−n/2

),

Qn,n−1,i = 0, Qn,n−2,i = −18

nq−n2 (q − 1)(qn−1 − 1).


It follows that

Dxxn = En,n−1,1xn−1 + En,n−2,1xn−2 + En,n−3,1xn−3 + · · · ,Sxxn = Mn,n,1xn + Mn,n−1,1xn−1 + Mn,n−2,1xn−2 + · · · ,Dyxn = En,n−1,2xn−1 + En,n−2,2xn−2 + En,n−3,2xn−3 + · · · ,Syxn = Mn,n,2xn + Mn,n−1,2xn−1 + Mn,n−2,2xn−2 + · · · ,

where En,j,i = (pn,j,k ,l,i)1≤k≤n+1,1≤l≤j+1, i = 1,2,(j = n − 1,n − 2,n − 3) are matrices of size (n + 1)× (j + 1) andMn,j,i = (rn,j,k ,l,i)1≤k≤n+1,1≤l≤j+1, i = 1,2, (j = n,n − 1,n − 2) arematrices of size (n + 1)× (j + 1) given by

pn,j,k,l,1 =

{Hn−k+1,j−k+1,1, k = l0, k 6= l, , pn,j,k,l,2 =

{Hn−j+l−1,l−1,2, k = l + n − j0, k 6= l + n − j,

rn,j,k,l,1 =

{Qn−k+1,j−k+1,1, k = l0, k 6= l, , rn,j,k,l,2 =

{Qn−j+l−1,l−1,2, k = l + n − j0, k 6= l + n − j .


We substitute the expansion in the divided-difference equation, we firstuse Dxxn, Sxxn, Dyxn, Syxn and then x(s)xn, y(t)xn. By equating thecoefficients of xn−1 and xn−2 we obtain the explicit expressions of thematrices Gn,n−1 and Gn,n−2 in terms of the nonsingular matrix Gn,n,

Gn,n−1 =1

λn−1 − λnGn,nSn,

Gn,n−2 =1

λn−2 − λn(Gn,nTn + Gn,n−1Sn−1),

(4)

where λn = 16q3−n (1− qn)(1− abcde2

2qn−1).


The matrix Sn of size (n + 1)× n is given by

Sn =

s1,1 0 . . . 0

s2,1 s2,2. . .

...

0 s3,2. . . 0

.... . . . . . sn,n

0 . . . 0 sn+1,n

(n ≥ 1),

where for k = 1, . . . ,n,

sk,k = 8 q1−n (qn−k+1 − 1)(

(a + b) (cde22qn+k − q2) + q (c + d)

(qnab − qk)e2

),

sk+1,k = 8(qk − 1

)q−n−k+2(

(c + d) (abe22q2 n − qk+1) + e2 qn (a + b)

(cdqk − q

) ).


The matrix Tn of size (n + 1)× (n − 1) is given by

Tn =

t1,1 0 · · · · · · 0

t2,1 t2,2. . .

...

t3,1 t3,2. . . . . .

...

0 t4,2. . . . . . 0

.... . . . . . . . . tn−1,n−1

.... . . . . . tn,n−1

0 · · · · · · 0 tn+1,n−1

(n ≥ 2),

where, for 1 ≤ k ≤ n − 1,

tk,k = q−2k−n (4(q + 1)qk+n ((ab + q)(cde22q2k + q3) + e2(a + b)(c + d)qk+2)

+4q2n (−cde22q2k (ab(q + 1)(k(q − 1) + n(−q) + n + 1) + q2)

−e2(a + b)(c + d)qk+3 − abq4)− 4e2(a + b)(c + d)q3k+2 − 4q2k+3(ab + q2(−k + n + 1) + k − n + q

)− 4cde2

2q4k+1) ,and we can also give the expression for tk+1,k .


More connection

P. Iliev and Y. Xu. Connection coefficients for classical orthogonalpolynomials of several variables. Advances in Mathematics 310(13)290–326 (2017).


Some references:1 P. Appell and J. Kampé de Fériet. Fonctions hypergéométriques et

hypersphériques. Polynomes d’Hermite. VII + 434 p. Paris,Gauthier-Villars, 1926.

2 IA, N. Atakishiyev, E. Godoy, and J. Rodal. Linear partialq-difference equations on q-linear lattices and their bivariateq-orthogonal polynomial solutions. Applied Mathematics andComputation 223: 520–536, 2013.

3 IA, E. Godoy, and J. Rodal. On a class of bivariate second-orderlinear partial difference equations and their monic orthogonalpolynomial solutions. J. Math. Anal. Appl., 389:165–178, 2012.

4 IA, E. Godoy, A. Ronveaux, and A. Zarzo. Bivariate second-orderlinear partial differential equations and orthogonal polynomialsolutions. J. Math. Anal. Appl., 387(2):1188–1208, 2012.

5 IA and E. Godoy. On limit relations between some families ofbivariate hypergeometric orthogonal polynomials. J. Phys. A:Math. Theor., 46(035202):11 pp, 2013.


6 C. F. Dunkl and Y. Xu. Orthogonal polynomials of severalvariables, volume 81 of Encyclopedia of Mathematics and itsApplications. Cambridge University Press, Cambridge, 2001.

7 J. S. Geronimo and P. Iliev. Multivariable Askey-Wilson functionand bispectrality. Ramanujan J. 24:3, 273–287, 2011.

8 H. L. Krall and I. M. Sheffer. Orthogonal polynomials in twovariables. Ann. Mat. Pura Appl. (4), 76:325–376, 1967.

9 S. Lewanowicz and P. Wozny. Two-variable orthogonalpolynomials of big q-Jacobi type. J. Comput. Appl. Math.,233(6):1554–1561, 2010.

10 A.S. Lyskova. Orthogonal polynomials in several variables. Sov.Math., Dokl., 43(1):264–268, 1991.

11 J. Rodal, IA, and E. Godoy. Orthogonal polynomials of twodiscrete variables on the simplex. Integral Transforms Spec.Funct., 16(3):263–280, 2005.

12 J. Rodal, IA, and E. Godoy. Linear partial difference equations ofhypergeometric type: orthogonal polynomial solutions in twodiscrete variables. J. Comput. Appl. Math., 200(2):722–748, 2007.


13 J. Rodal, IA, and E. Godoy. Structure relations for monicorthogonal polynomials in two discrete variables. J. Math. Anal.Appl., 340(2):825–844, 2008.

14 P. K. Suetin. Orthogonal polynomials in two variables, volume 3 ofAnalytical Methods and Special Functions. Gordon and BreachScience Publishers, Amsterdam, 1999.

15 D.D. Tcheutia, Y. Guemo Tefo, M. Foupouagnigni, E. Godoy, IA.Linear partial divided-difference equation satisfied by multivariateorthogonal polynomials on quadratic lattices. MathematicalModelling of Natural Phenomena 12(3) (2017) 14–43.

16 D.D. Tcheutia, M. Foupouagnigni, Y. Guemo Tefo, IA.Divided-difference equation and three-term recurrence relations ofsome systems of bivariate q-orthogonal polynomials. Journal ofDifference Equations and Applications 23(12) 2004–2036 (2017).


Thank you


some systems of multivariate orthogonal polynomialsorthogonal polynomials of two variables from...

Documents