当前位置:文档之家› Properties of some families of hypergeometric orthogonal polynomials in several variables

Properties of some families of hypergeometric orthogonal polynomials in several variables

Properties of some families of hypergeometric orthogonal polynomials in several variables
Properties of some families of hypergeometric orthogonal polynomials in several variables

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PROPERTIES OF SOME F AMILIES OF HYPERGEOMETRIC ORTHOGONAL POLYNOMIALS IN SEVERAL VARIABLES J.F.VAN DIEJEN Abstract.Limiting cases are studied of the Koornwinder-Macdonald multivari-able generalization of the Askey-Wilson polynomials.We recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables consisting of multivariable Wilson,continuous Hahn and Jacobi type poly-nomials,respectively.For each class of polynomials we provide systems of di?erence (or di?erential)equations,recurrence relations,and expressions for the norms of the polynomials in terms of the norm of the constant polynomial.1.Introduction It is to date over a decade ago that Askey and Wilson released their famous memoir [AW],in which they introduced a four-parameter family of basic hypergeometric polynomials nowadays commonly referred to as the Askey-Wilson polynomials [GR].These polynomials,which are de?ned explicitly in terms of a terminating 4φ3series,have been shown to exhibit a number of interesting properties.Among other things,it was demonstrated that they satisfy a second order di?erence equation,a three-term recurrence relation,and that—in a suitable parameter regime—they constitute an orthogonal system with respect to an explicitly given positive weight function with support on a ?nite interval (or on the unit circle,depending on how the coordinates are chosen).Many (basic)hypergeometric orthogonal polynomials studied in the literature arise

as special (limiting)cases of the Askey-Wilson polynomials and have been collected in the so-called (q -)Askey scheme [AW,KS].For instance,if the step size parameter of the di?erence equation is scaled to zero,then the Askey-Wilson polynomials go over in Jacobi polynomials:well-known classical hypergeometric orthogonal polynomials satisfying a second order di?erential equation instead of a di?erence equation.One may also consider the transition from orthogonal polynomials on a ?nite interval to

2J. F.VAN DIEJEN

orthogonal polynomials on a(semi-)in?nite interval.This way one arrives at Wilson polynomials(semi-in?nite interval)and at continuous Hahn polynomials(in?nite interval).

The purpose of the present paper is to generalize this state of a?airs from one to several variables.Starting point is a recently introduced multivariable generalization of the Askey-Wilson polynomials,found for special parameters by Macdonald[M2] and in full generality(involving?ve parameters)by Koornwinder[K].By means of limiting transitions similar to those in the one-variable case,we arrive at multivariable Jacobi polynomials[V,De](see also[BO]and reference therein)and at multivariable Wilson and continuous Hahn polynomials[D3].

The(q-)Askey scheme involves many more limits and special cases of the Askey-Wilson polynomials than those described above.For instance,one also considers transitions from certain polynomials in the scheme to similar polynomials with less parameters and transitions from polynomials with a continuous orthogonality mea-sure to polynomials with a discrete orthogonality measure.Such transitions(or rather their multivariable analogues)will not be considered here.We refer instead to[D1, Sec. 5.2]for the transition from multivariable Askey-Wilson polynomials to Mac-donald’s q-Jack polynomials(i.e.,multivariable q-ultraspherical polynomials)[M4] (as an example of a limit leading to similar polynomials but with less parameters), and to[SK]for the transition from multivariable Askey-Wilson polynomials to mul-tivariable big and little q-Jacobi polynomials[S](as an example of a limit leading to multivariable polynomials with a discrete orthogonality measure).

Whenever one is dealing with orthogonal polynomials an important question arises as to the explicit computation of the normalization constants converting the poly-nomials into an orthonormal system.For Jacobi polynomials calculating the or-thonormalization constants boils down to the evaluation of(standard)beta integrals, whereas Askey-Wilson polynomials give rise to q-beta integrals.In the case of sev-eral variables one has to deal with Selberg type integrals(Jacobi case)and q-Selberg type integrals(Askey-Wilson case),respectively.For these multiple integrals explicit evaluations have been conjectured by Macdonald that were recently checked using techniques involving so-called shift operators[Op,HS,C1,M5].(Roughly speaking these shift operators allow one to relate the values of the(q-)Selberg integral for di?erent values of the parameters separated by unit shifts;the integral can then be solved,?rst for nonnegative integer-valued parameters by shifting the parameters to zero in which case the integrand becomes trivial,and then for arbitrary nonnegative parameters using an analyticity argument(viz.Carlson’s theorem).)

Very recently,the author observed that Koornwinder’s second order di?erence equation for the multivariable Askey-Wilson polynomials may be extended to a sys-tem of n(=number of variables)independent di?erence equations[D1]and that the polynomials also satisfy a system of n independent recurrence relations[D5].(To date

HYPERGEOMETRIC POLYNOMIALS IN SEVERAL VARIABLES3 a complete proof for these recurrence relations is only available for a self-dual four-parameter subfamily of the?ve-parameter multivariable Askey-Wilson polynomials.) It turns out that the recurrence relations,combined with the known evaluation for the norm of the unit polynomial(i.e.,the constant term integral)[Gu,Ka],may also be used to verify Macdonald’s formulas for the orthonormalization constants of the multivariable Askey-Wilson polynomials[D5].Below,we will use these results to arrive at systems of di?erence(or di?erential)equations,recurrence relations and expressions for the orthonormalization constants,for all three limiting cases of the multivariable Askey-Wilson polynomials considered in this paper(Wilson,continuous Hahn and Jacobi type).

We would like to emphasize that much of the presented material admits a phys-ical interpretation in terms of Calogero-Sutherland type exactly solvable quantum n-particle models related to classical root systems[OP]or their Ruijsenaars type dif-ference versions[R1,R2,D2].The point is that the second order di?erential equation for the multivariable Jacobi polynomials may be seen as the eigenvalue equation for a trigonometric quantum Calogero-Sutherland system related to the root system BC n [OP].From this viewpoint the second order di?erence equation for the multivariable Askey-Wilson polynomials corresponds to the eigenvalue equation for a Ruijsenaars type di?erence version of the BC n-type quantum Calogero-Sutherland system[D4]. The transitions to the multivariable continuous Hahn and Wilson polynomials amount to rational limits leading to(the eigenfunctions of)similar di?erence versions of the A n?1-type rational Calogero model with harmonic term(continuous Hahn case)and its B(C)n-type counterpart(Wilson case)[D3].For further details regarding these connections with the Calogero-Sutherland and Ruijsenaars type quantum integrable n-particle systems the reader is referred to[D2,D3,D4].

The material is organized as follows.First we de?ne our families of multivari-able(basic)hypergeometric polynomials in Section2and recall their second order di?erence equation(Askey-Wilson,Wilson,continuous Hahn type)or second order di?erential equation(Jacobi type)in Section3.Next,in Section4,limit transitions from the Askey-Wilson type family to the Wilson,continuous Hahn and Jacobi type families are discussed.We study the behavior of our recently introduced systems of di?erence equations and recurrence relations for the multivariable Askey-Wilson type polynomials with respect to these limits in Sections5and6,respectively.The recur-rence relations for the Wilson,continuous Hahn and Jacobi type polynomials thus obtained in Section6are then employed in Section7to derive explicit expressions for the(squared)norms of the corresponding polynomials in terms of the(squared) norm of the unit polynomial.

4J. F.VAN DIEJEN

2.Multivariable(basic)hypergeometric polynomials

In this section multivariable versions of some orthogonal families of(basic)hyper-geometric polynomials are characterized.The general idea of the construction(which is standard,see e.g.[V,M2,K,SK])is to start with an algebra of(symmetric)poly-nomials H spanned by a basis of(symmetric)monomials{mλ}λ∈Λ,with the setΛlabeling the basis elements being partially ordered in such a way that for allλ∈Λthe subspaces Hλ≡Span{mμ}μ∈Λ,μ≤λare?nite-dimensional.It is furthermore assumed that the space H is endowed with an L2inner product ·,· ?characterized by a cer-tain weight function?.To such a con?guration we associate a basis{pλ}λ∈Λof H consisting of the polynomials pλ,λ∈Λ,determined(uniquely)by the two conditions i.pλ=mλ+ μ∈Λ,μ<λcλ,μmμ,cλ,μ∈C;

ii. pλ,mμ ?=0ifμ<λ.

In other words,the polynomial pλconsists of the monomial mλminus its orthogonal projection with respect to the inner product ·,· ?onto the?nite-dimensional sub-space Span{mμ}μ∈Λ,μ<λ.By varying the concrete choices for the space H,the basis {mλ}λ∈Λand the inner product ·,· ?,we recover certain(previously introduced) multivariable generalizations of the Askey-Wilson,Wilson,continuous Hahn and Ja-cobi polynomials,respectively.Below we will specify the relevant data determining these families.The fact that in the case of one variable the corresponding polynomi-als pλindeed reduce to the well-known one-variable polynomials studied extensively in the literature is immediate from the weight function.The normalization for the polynomials is determined by the fact that(by de?nition)pλis monic in the sense that the coe?cient of the leading monomial mλin pλis equal to one.

It turns out that in all of our cases the basis{mλ}λ∈Λcan be conveniently expressed in terms of the monomial symmetric functions

m sym,λ(z1,...,z n)= μ∈S n(λ)zμ11···zμn n,λ∈Λ,(2.1) where

Λ={λ∈Z n|λ1≥λ2≥···≥λn≥0}.(2.2) In(2.1)the summation is meant over the orbit ofλunder the action of the permuta-tion group S n(acting on the vector componentsλ1,...,λn).As partial order of the integral coneΛ(2.2)we will always take the dominance order de?ned by

μ≤λi?

m

j=1μj≤m j=1λj for m=1,...,n(2.3)

(andμ<λi?μ≤λandμ=λ).

HYPERGEOMETRIC POLYNOMIALS IN SEVERAL VARIABLES 5

Note:In order to avoid confusion between the various families we will often equip the polynomials and other objects of interest with the superscripts ‘AW’,‘W’,‘cH’or ‘J’to indicate Askey-Wilson,Wilson,continuous Hahn or Jacobi type polynomials,respectively.Sometimes,however,these superscripts will be suppressed when dis-cussing more general properties of the polynomials that hold simultaneously for all families.

2.1.Askey-Wilson type.To arrive at multivariable Askey-Wilson type polynomi-als one considers a space H AW consisting of even and permutation invariant trigono-metric polynomials.Speci?cally,the space H AW is spanned by the monomials

m AW λ(x )=m sym,λ(e iαx 1+e ?iαx 1,···,e iαx n +e ?iαx n ),λ∈Λ(2.4)

(with Λgiven by

(2.2)).

The

relevant inner

product on H AW is determined by

m AW λ,m AW μ ?AW = π/α

?π/α···

π/α?π/αm AW λ(x )

(t e iα(ε1x j +ε2x k );q )∞

(2.6)

× 1≤j ≤n ε=±1

(e 2iαεx j ;q )∞t l 0(t 0t 1t 2t 3q l ?1;q )l 4φ3 q ?l ,t 0t 1t 2t 3q l ?1,t 0e iαx ,t 0e ?iαx t 0t 1,t 0t 2,t 0t 3;q,q .(2.8)

6J. F.VAN DIEJEN

2.2.Wilson type.In the Wilson case the appropriate space H W consists of even and permutation invariant polynomials and is spanned by the monomials

m Wλ(x)=m sym,λ(x21,···,x2n),λ∈Λ.(2.9) The inner product on H W is now determined by

m Wλ,m Wμ ?W= ∞?∞··· ∞?∞m Wλ(x)

Γ(i(ε1x j+ε2x k))

(2.11)

× 1≤j≤nε=±1Γ(ν0+iεx j)Γ(ν1+iεx j)Γ(ν2+iεx j)Γ(ν3+iεx j)

(?1)l(ν0+ν1+ν2+ν3+l?1)l

×(2.13)

4F3 ?l,ν0+ν1+ν2+ν3+l?1,ν0+ix,ν0?ix

ν0+ν1,ν0+ν2,ν0+ν3

;1 .

2.3.Continuous Hahn type.The space H cH is very similar to that of the Wilson case but instead of only the even sector it now consists of all permutation invariant polynomials.The monomial basis for the space H cH then becomes

m cHλ(x)=m sym,λ(x1,···,x n),λ∈Λ.(2.14) The inner product is of the same form as for the Wilson case

m cHλ,m cHμ ?cH= ∞?∞··· ∞?∞m cHλ(x)

Γ(i(x j?x k))

Γ(ν+i(x k?x j))

HYPERGEOMETRIC POLYNOMIALS IN SEVERAL VARIABLES7

where

ν≥0,Re(ν±0),Re(ν±1)>0,ν?0=ν+1.(2.17) Just as in the case of Wilson type polynomials,the polynomials p cHλcorresponding to the weight function(2.16)were introduced in[D3].For n=1they reduce to monic continuous Hahn polynomials[AW,KS]

p cH l(x)=

i l(ν+0+ν?0,ν+0+ν?1)l

2(x j+x k)sin

α

2

x j) 2ν0 cos(α

(ν0+ν1+l)l2F1 ?l,ν0+ν1+l

ν0+1/2

;sin2 αx

(b1,...,b s)k

z k

(b1,...,b s;q)k

z k

8J. F.VAN DIEJEN

for k=1,2,3,....

3.Second order difference or differential equations

As it turns out,all families of polynomials{pλ}λ∈Λintroduced in the previous section satisfy an eigenvalue equation of the form

D pλ=Eλpλ,λ∈Λ,(3.1) where D:H→H denotes a certain second order di?erence operator(Askey-Wilson, Wilson and continuous Hahn case)or a second order di?erential operator(Jacobi case).Below we will list for each family the relevant operator D together with its eigenvalues Eλ,λ∈Λ.In each case the proof that the polynomials pλindeed satisfy the corresponding eigenvalue equations boils down to demonstrating that the operator D:H→H maps the?nite-dimensional subspaces Hλ=Span{mμ}μ∈Λ,μ≤λinto themselves(triangularity)and that it is symmetric with respect to the inner product ·,· ?.In other words,one has to show that

Triangularity

D mλ= μ∈Λ,μ≤λ[D]λ,μmμ,with[D]λ,μ∈C(3.2)

and that

Symmetry

Dmλ,mμ ?= mλ,Dmμ ?.(3.3) It is immediate from these two properties and the de?nition of the polynomial pλthat Dpλlies in Hλand is orthogonal with respect to ·,· ?to all monomials mμ,μ∈Λwithμ<λ.But then comparison with the de?nition of pλshows that Dpλmust be proportional to pλ,i.e.,pλis an eigenfunction of D.The corresponding eigenvalue Eλis obtained via an explicit computation of the diagonal matrix element[D]λ,λin Expansion(3.2).

For the Jacobi case a proof of the second order di?erential equation along the above lines was given by Vretare[V].In the Askey-Wilson case the proof was given by Macdonald[M2]and(in general)Koornwinder[K].The proof for the Wilson and continuous Hahn case is very similar to that of the Askey-Wilson case and has been outlined in[D3].

3.1.Askey-Wilson type.The second order(q-)di?erence operator diagonalized by the polynomials p AW

,λ∈Λ,is given by

λ

D AW= 1≤j≤n V AW j(x)(T j,q?1)+V AW?j(x)(T?1j,q?1) (3.4)

HYPERGEOMETRIC POLYNOMIALS IN SEVERAL VARIABLES

9with

V AW ±j (x )= 0≤

r ≤3(1?t r e ±iαx j )

1?e iα(±x j +x k )

1?te iα(±x j ?x k )

(1?e 2iαx )(1?qe 2iαx )

p AW l (qx )?p AW l (x )

+

(3.9) 0≤r ≤3(1?t r e ?iαx )(±2ix j )(±2ix

j ?1)(3.11)

×

1≤k ≤n,k =j iν±x j +x k ±x j ?x k

10J. F.VAN DIEJEN

and the action of T j is given by a unit shift of the j th variable along the imaginary axis

(T j f )(x 1,...,x n )=f (x 1,...,x j ?1,x j +i,x j +1,...,x n ).

(3.12)The corresponding eigenvalues now read E W λ= 1≤j ≤n λj λj +ν0+ν1+ν2+ν3?1+2(n ?j )ν .

(3.13)

Proposition 3.2([D3]).The multivariable Wilson polynomials p W λ,λ∈Λ(2.2),satisfy the second order di?erence equation

D W p W λ=

E W λp W λ.(3.14)

For n =1,Equation (3.14)reduces to the second order di?erence equation for the one-variable Wilson polynomials [KS] 0≤r ≤3(iνr +x )

2ix (2ix +1)

p W l (x ?i )?p W l (x ) =l (l +ν0+ν1+ν2+ν3?1)p W l (x ).

3.3.Continuous Hahn type.For the continuous Hahn type one has D cH = 1≤j ≤n V cH j,?(x )(T j ?1)+V cH j,+(x )(T ?1j ?1)

(3.16)

with

V cH j,+(x )=(ν+0+ix j )(ν+1+ix j )

1≤k ≤n,k =j 1+νi (x j ?x k ) .(3.18)

The action of T j is the same as in the Wilson case (cf.(3.12))and the eigenvalues are given by E cH λ= 1≤j ≤n

λj λj +ν+0+ν+1+ν?0+ν?1?1+2(n ?j )ν .(3.19)

Proposition 3.3([D3]).The multivariable continuous Hahn polynomials p cH λ,λ∈Λ(2.2),satisfy the second order di?erence equation

D cH p cH λ=

E cH λp cH λ.(3.20)

HYPERGEOMETRIC POLYNOMIALS IN SEVERAL VARIABLES 11

For n =1,Equation (3.20)reduces to the second order di?erence equation for the one-variable continuous Hahn polynomials [KS](ν?0?ix )(ν?1?ix ) p cH l (x +i )?p W l (x ) +(3.21)(ν+0+ix )(ν+1+ix ) p cH l (x ?i )?p cH l (x ) =l l +ν+0+ν+1+ν?0+ν?1?1 p cH l (x ).

3.4.Jacobi type.In the case of multivariable Jacobi type polynomials the operator D diagonalized by p λis given by a second order di?erential operator of the form D J =? 1≤j ≤n ?2j ?α 1≤j ≤n ν0cot(αx j 2) ?j (3.22)?αν 1≤j

cot(α2(x j ?x k ))(?j ??k ) where ?j ≡?/?x j .The eigenvalue of D J on p J λtakes the value E J λ= 1≤j ≤n λj λj +ν0+ν1+2(n ?j )ν .

(3.23)

Proposition 3.4([V,De]).The multivariable Jacobi polynomials p J λ,λ∈Λ(2.2),satisfy the second order di?erential equation

D J p J λ=

E J λp J λ.(3.24)

For n =1,Equation (3.24)reduces to the second order di?erential equation for the one-variable Jacobi polynomials [AS,KS]

?d 2p J λ

2

)?ν1tan(αx dx (x )(3.25)

=l (l +ν0+ν1)p J l (x ).4.Limit transitions

The operator D of the previous section can be used to arrive at the following useful representation for the polynomials p λ(cf.[M2,D5,SK])p λ=

μ∈Λ,μ<λ

D ?

E μ

12J. F.VAN DIEJEN

since for parameter values indicated in Section2one has that(see[D1,Sec.5.2]and

[SK])

μ<λ=?Eμ

all mμwithμ<λfollows from the symmetry of D and the fact that the operator in

the numerator—viz. μ∈Λ,μ<λ(D?Eμ)—annihilates the subspace Span{mμ}μ∈Λ,μ<λin view of the Cayley-Hamilton theorem.)

Below we will use Formula(4.1)to derive limit transitions from the Askey-Wilson

type to the Wilson,continuous Hahn and Jacobi type families,respectively.The transition‘Askey-Wilson→Jacobi’has already been considered before in[M2,D1, SK]and is included here mainly for the sake of completeness.It will be put to use in Section6when deriving a system of recurrence relations for the multivariable Jacobi type polynomials.

4.1.Askey-Wilson→Wilson.When studying the limit p AW

λ→p Wλit is convenient

to?rst express the multivariable Askey-Wilson polynomials in terms of a slightly modi?ed monomial basis consisting of the functions

?m AW λ(x)=(2/α)2|λ|m sym,λ(sin2(αx1/2),...,sin2(αx n/2)),λ∈Λ,

(4.3)

where|λ|≡λ1+···+λn.Notice that

lim α→0?m AW

λ

(x)=m Wλ(x)(4.4)

whereas the original monomials m AW

λ(x)(2.4)all reduce to constant functions in this

https://www.doczj.com/doc/e93092480.html,ing the relation sin2(αx j/2)=1/2?(e iαx j+e?iαx j)/4one easily infers that

the bases{?m AW

λ}λ∈Λand{m AW

λ

}λ∈Λare related by a triangular transformation of

the form

?mλ=(?1/α2)|λ|m AW

λ

+ μ∈Λ,μ<λaλ,μmμwith aλ,μ∈R.(4.5)

It is clear that in Formula(4.1)we may always replace the monomial basis{mλ}λ∈Λby a di?erent basis that is related by a unitriangular transformation,since(cf.above)the operator μ∈Λ,μ<λ(D?Eμ)in the numerator of the r.h.s.annihilates the subspace Span{mμ}μ∈Λ,μ<λbecause of the Cayley-Hamilton theorem.Hence,by taking in account the diagonal matrix elements in the basis transformation(4.5),one sees that

Formula(4.1)can rewritten in terms of?m AW

λ

as

p AW

λ

=(?α2)|λ| μ∈Λ,μ<λD AW?E AWμ

HYPERGEOMETRIC POLYNOMIALS IN SEVERAL VARIABLES13 If we now substitute

q=e?α,t=e?αν,t r=e?ανr(r=0,1,2,3)(4.7) in D AW(3.4)and E Wλ(3.7),then we have that

lim α→0α?2D AW=D W,lim

α→0

α?2E AW

λ

=E Wλ.(4.8)

(Notice to this end that for q=e?αthe action of T j,q(3.6)on trigonometric poly-nomials is the same as that of T j(3.12),i.e.,the action amounts to a shift of the variable x j over an imaginary unit:x j→x j+i.To infer then that in the limitα→0 the di?erence operatorα?2D AW formally goes to D W boils down to checking that the coe?cients of the operator converge as advertised.)

By applying the limits(4.4)and(4.8)to Formula(4.6)we end up with the fol-lowing limiting relation between the multivariable Askwey-Wilson and Wilson type polynomials.

Proposition4.1.For Askey-Wilson parameters given by(4.7)one has

p Wλ(x)=lim

α→0(?1/α2)|λ|p AW

λ

(x),λ∈Λ(4.9)

(with|λ|≡λ1+···+λn).

4.2.Askey-Wilson→continuous Hahn.Just like in the previous subsection,the

derivation of the transition p AW

λ→p cHλhinges again on Formula(4.1).If we shift the

variables x1,...,x n over a half period by setting

x j→x j?π/(2α),j=1,...,n(4.10) and substitute parameters in the following way

q=e?α,t=e?αν,

t0=?ie?αν+0,t1=?ie?αν+1,t2=ie?αν?0,t3=ie?αν?1,

(4.11)

then the version of Formula(4.1)for the multivariable Askey-Wilson polynomials takes the form(e j denotes the j th unit vector in the standard basis of R n)

p AW

λ x?πE AWλ?E AWμ ?m AWλ(x)

(4.12)

where

?D AW= 1≤j≤n ?V AW j(x)(T j?1)+?V AW?j(x)(T?1j?1) ,

14J. F.VAN DIEJEN with

?V AW j (x)=

(1+e?αν+0e iαx j)(1+e?αν+1e iαx j)(1?e?αν?0e iαx j)(1?e?αν?1e iαx j)

1+e iα(x j+x k)

1?e?ανe iα(x j?x k)

(1+e?2iαx j)(1+e?αe?2iαx j)

× 1≤k≤n,k=j 1+e?ανe?iα(x j+x k)1?e?iα(x j?x k)

and

?m AW

λ

(x)≡m sym,λ(2sin(αx1),...,2sin(αx n)).

(Just as in the case of the transition Askey-Wilson→Wilson we have rewritten the operators T j,q(3.6)for q=e?αas T j(3.12).)After dividing by(2α)|λ|the r.h.s. of(4.12)goes forα→0to the corresponding formula for the continuous Hahn

polynomials(i.e.with?D AW→D cH,E AW

λ→E cHλand(2α)?|λ|?m AW

λ

→m cHλ).

Hence,we now arrive at the following limiting relation between the multivariable Askey-Wilson and continuous Hahn type polynomials.

Proposition4.2.For Askey-Wilson parameters given by(4.11)one has

p cHλ(x)=lim

α→01

ω ,λ∈Λ(4.13)

whereω≡e1+···+e n(with e j denoting the j th unit vector in the standard basis of R n).

4.3.Askey-Wilson→Jacobi.To recover the Jacobi type polynomials we substitute the Askey-Wilson parameters

t=q g,t0=q g0,t1=?q g1,t2=q g′0+1/2,t3=?q g′1+1/2.

(4.14) With these parameters the formula of the Form(4.1)for the Askey-Wilson type poly-nomials reduces in the limit q→1to the corresponding formula for the Jacobi type

polynomials(i.e.,the di?erence operator D AW with eigenvalues E AW

λgets replaced

by the di?erential operator D J with eigenvalues E Jλ).The limit q→1amounts to sending the di?erence step size to zero.In order to analyze the behavior of the operator D AW for q→1in detail it is convenient to substitute q=e?αβ(so the action of T j,q(3.6)on trigonometric polynomials amounts to the shift x j→x j+iβ) and then write formally T j,q=exp(iβ?j).A formal expansion inβthen shows that D AW~β2D J and that E AW

λ

~β2E Jλforβ→0.Here the Jacobi parametersν,νr

HYPERGEOMETRIC POLYNOMIALS IN SEVERAL VARIABLES15 are related to the parameters g,g(′)r in(4.14)viaν=g,ν0=g0+g′0andν1=g1+g′1. As a consequence we obtain the following limiting relation between the multivariable Askey-Wilson and Jacobi type polynomials.

Proposition4.3.For Askey-Wilson parameters given by(4.14)one has

p Jλ(x)=lim

q→1p AW

λ

(x)(4.15)

with the Jacobi parametersν,ν0andν1taking the value g,g0+g′0and g1+g′1, respectively.

Remarks:i.In the above derivations of Propositions4.1,4.2and4.3we have used that the Askey-Wilson type di?erence operator converges formally(i.e.,without specifying the domains of the operators of interest)to the corresponding operators

connected with the Wilson,continuous Hahn and Jacobi type polynomials,respec-tively.In our case such formal limits get their precise meaning when being applied to Formula(4.1).

ii.For all our four families AW,W,cH and J the dependence of(the coe?cients of) the operator D and of the eigenvalues Eλis polynomial in the parameters t,t0,t1,t2,t3

(AW),ν,ν0,ν1,ν2,ν3(W),ν,ν±0,ν±0(cH)andν,ν0,ν1(J),respectively.Hence it is clear from Formula(4.1)that(the coe?cients of)the polynomials pλare rational in these parameters.We may thus extend the parameter domains for the polynomials given in Section2to generic(complex)values by alternatively characterizing pλas the

polynomial of the form pλ=mλ+ μ∈λ,μ<λcλ,μmμsatisfying the eigenvalue equation Dpλ=Eλpλ.It is clear that the limit transitions discussed in this section then extend to these larger parameter domains of generic(complex)parameter values.

iii.In the case of one variable the limit transitions from Askey-Wilson polynomials to Wilson,continuous Hahn and Jacobi polynomials were collected in[KS](together with many other limits between the various(basic)hypergeometric orthogonal fami-lies appearing in the(q-)Askey scheme).

5.Higher order difference or differential equations

In[D1]it was shown that the second order di?erence equation for the multivariable Askey-Wilson type polynomials can be extended to a system of di?erence equations having the structure of eigenvalue equations of the form

D r pλ=

E r,λpλ,r=1,...,n,(5.1) for n independent commuting di?erence operators D1,D2,...,D n of order2,4,...,2n, respectively.For r=1one recovers the second order di?erence equation discussed in Section3.1.After recalling the explicit expressions for the Askey-Wilson type di?er-

ence operators D AW

r and their eigenvalues E AW

r,λ

,we will apply the limit transitions

of Section4to arrive at similar systems of di?erence equations for the multivariable

16J. F.VAN DIEJEN

Wilson and continuous Hahn type polynomials.In case of the transition‘Askey-Wilson→Jacobi’the step size is sent to zero and the system of di?erence equations degenerates to a system of hypergeometric di?erential equations,thus generalizing the state of a?airs for the second order operator in the previous section.This limit from Askey-Wilson type di?erence equations to Jacobi type di?erential equations has already been discussed in detail in[D1,Sec.4],so here we will merely state the results and refrain from presenting a complete treatment of this case.

5.1.Askey-Wilson type.The di?erence operators diagonalized by the multivari-able Askey-Wilson polynomials via Eq.(5.1)are given by

D AW

r

= J?{1,...,n},0≤|J|≤r

εj=±1,j∈J U AW

J c,r?|J|

(x)V AW

εJ,J c

(x)TεJ,q r=1,...,n,(5.2)

with

TεJ,q= j∈J Tεj j,q

V AW

εJ,K

(x)= j∈J w AW(εj x j) j,j′∈J j

× j∈J k∈K v AW(εj x j+x k)v AW(εj x j?x k),

U AW

K,p

(x)=

(?1)p L?K,|L|=pεl=±1,l∈L l∈L w AW(εl x l) l,l′∈L l

and

v AW(z)=t?1/2 1?t e iαz

(1?e2iαz)(1?q e2iαz)

.(5.4)

Here the action of the operators T±1j,q is de?ned in accordance with(3.6).The sum-mation in(5.2)is over all index sets J?{1,...,n}with cardinality|J|≤r and over all con?gurations of signsεj∈{+1,?1}with j∈J.Furthermore,by convention empty products are taken to be equal to one and U K,p≡1for p=0.

HYPERGEOMETRIC POLYNOMIALS IN SEVERAL VARIABLES17

The corresponding eigenvalue of D AW

r on p AW

λ

has the value

E AW r,λ=E r(τ1qλ1+τ?11q?λ1,...,τn qλn+τ?1n q?λn;τr+τ?1r,...,τn+τ?1n)

(5.5)

where

E r(x1,...,x n;y r,...,y n)≡(5.6)

J?{1,...,n}

0≤|J|≤r

(?1)r?|J| j∈J x j r≤l1≤···≤l r?|J|≤n y l1···y l r?|J| and

τj=t n?j(t0t1t2t3q?1)1/2,j=1,...,n.(5.7) (The second sum in(5.6)is understood to be equal to1when|J|=r.) Summarizing,we have the following theorem generalizing Proposition3.1.

Theorem5.1([D1,D4]).The multivariable Askey-Wilson polynomials p AW

λ,λ∈Λ

(2.2)satisfy a system of di?erence equations of the form

D AW

r p AW

λ

=E AW

r,λ

p AW

λ

,r=1,...,n.(5.8)

For r=1the Di?erence equation(5.8)goes over in the second order Di?erence equation(3.8)after multiplication by a constant with value t n?1(t0t1t2t3q?1)1/2.More generally,one may multiply the Di?erence equation(5.8)for arbitrary r by the con-stant factor t r(n?1)?r(r?1)/2(t0t1t2t3q?1)r/2to obtain a di?erence equation that is poly-nomial in the parameters t,t0,...,t3and rational in q.Such multiplication amounts

to omitting the factors t?1/2and(t0t1t2t3q?1)?1/2in the de?nition of v AW(z)(5.3) and w AW(x)(5.4)and to replacing the eigenvalues by

E AW

r,λ

→t?r(r?1)/2

×E r(τ+1qλ1+τ?1q?λ1,...,τ+n qλn+τ?n q?λn;τ+r+τ?r,...,τ+n+τ?n) with

τ+j=t n?1(t0t1t2t3q?1)1/2τj=t0t1t2t3q?1t2n?1?j,

τ?j=t n?1(t0t1t2t3q?1)1/2τ?1j=t j?1.

5.2.Wilson type.If we substitute Askey-Wilson parameters of the form(4.7)and

divide byα2r,then forα→0the operator D AW

r

goes over in

D W r= J?{1,...,n},0≤|J|≤r

εj=±1,j∈J

U W J c,r?|J|(x)V WεJ,J c(x)TεJ r=1,...,n,(5.9)

18J. F.VAN DIEJEN

with

TεJ= j∈J Tεj j

V WεJ,K(x)= j∈J w W(εj x j) j,j′∈J j

× j∈J k∈K v W(εj x j+x k)v W(εj x j?x k),

U W K,p(x)=

(?1)p L?K,|L|=pεl=±1,l∈L l∈L w W(εl x l) l,l′∈L l

× l∈L k∈K\L v W(εl x l+x k)v W(εl x l?x k) ,

and

v W(z)=iν+z

2iz(2iz?1)

.(5.10)

Here the action of T±j is taken to be in accordance with(3.12).To verify this limit it su?ces to recall that for q=e?αthe action of T j,q(3.6)amounts to that of T j (3.12)and to observe that for parameters(4.7)one has limα→v AW(z)=v W(z)and limα→α?2w AW(z)=w W(z).

The eigenvalues become in this limit

E W r,λ=E r (ρW1+λ1)2,...,(ρW n+λn)2;(ρW r)2,...,(ρW n)2 (5.11) with E r(···;···)taken from(5.6)and

ρW j=(n?j)ν+(ν0+ν1+ν2+ν3?1)/2.(5.12) For the eigenvalues the computation verifying the limit is a bit more complicated than for the di?erence operators;it hinges on the following lemma.

Lemma5.2([D1,Sec.4.2]).One has

lim

α→0

α?2r E r(eαx1+e?αx1,...,eαx n+e?αx n;eαy r+e?αy r,...,eαy n+e?αy n)

=E r(x21,...,x2n;y2r,...,y2n).

We may thus conclude that the transition AW→W gives rise to the following generalization of Proposition3.2.

HYPERGEOMETRIC POLYNOMIALS IN SEVERAL VARIABLES 19

Theorem 5.3.The multivariable Wilson polynomials p W

λ,λ∈Λ(2.2)satisfy a sys-

tem of di?erence equations of the

form D W r p W λ=E W r,λp W λ,r =1,...,n.(5.13)

For r =1the Di?erence equation (5.13)coincides with the second order Di?erence equation (3.14).

5.3.Continuous Hahn type.After shifting over a half period as in (4.10)and choosing Askey-Wilson parameters of the form (4.11),the operators α?2r D AW

r go for α→0

over in D cH r = J +,J ??{1,...,n }J +∩J ?=?,|J +|+|J ?|≤r

U cH

J c +∩J c ?,r ?|J +|?|J ?|(x )V cH

J +,J ?;J c +∩J c ?(x )T J +,J ?

(5.14)r =1,...,n ,where

T J +,J ?=

j ∈J +T ?1j j ∈J ?

T j V cH

J +,J ?;K (x )= j ∈J +w cH +(x j )

j ∈J ?w cH ?(x j ) j ∈J +,j ′∈J ?

v cH (x j ?x j ′)v cH (x j ?x j ′?i )× j ∈J +k ∈K v cH (x j ?x k ) j ∈J

?

k ∈K v cH (x k ?x j ),

U cH K,p (x )=

(?1)p

L +,L ??K,L +∩L

?=?

|L +|+|L ?|=p l ∈L +w cH +(x l ) l ∈L ?w cH ?(x l ) l ∈L +l ′∈L ?

v cH (x l ?x l ′)v cH (x l ′?x l +i )× l ∈L +k ∈K \L +∪L ?v cH (x l ?x k ) l

∈L

?

k ∈K \L +∪L ?v cH (x k ?x l )

,

and

v cH (z )=(1+ν

20J. F.VAN DIEJEN

with E r(···;···)taken from(5.6)and

ρcH j=(n?j)ν+(ν+0+ν+1+ν?0+ν?1?1)/2.(5.18) Hence,we arrive the following generalization of Proposition3.3.

Theorem5.4.The multivariable continuous Hahn polynomials p cH

λ,λ∈Λ(2.2)

satisfy a system of di?erence equations of the form

D cH r p cHλ=

E cH r,λp cHλ,r=1,...,n.(5.19) For r=1the Di?erence equation(5.19)coincides with the second order Di?erence equation(3.20).

5.4.Jacobi type.Ater substituting Askey-Wilson parameters given by(4.14)and dividing by a constant with value(1?q)2r the r th di?erence equation in Theorem5.1 for the Askey-Wilson type polynomials goes for q→1over in a di?erential equation D J r p Jλ=E J r,λp Jλof order2r for the multivariable Jacobi type polynomials[D1,Sec.

4].The computation of the eigenvalue E J r,λ=lim q→1(1?q)?2r E AW

r,λhinges again on

Lemma5.2and the result is

E J r,λ=E r (ρJ1+λ1)2,...,(ρJ n+λn)2;(ρJ r)2,...,(ρJ n)2 (5.20) with E r(···;···)taken from(5.6)and

ρJ j=(n?j)ν+(ν0+ν1)/2,(5.21) whereν=g,ν0=g0+g′0andν1=g1+g′1.For r=1the di?erential operator

D J r=lim q→1(1?q)?2r D AW

r is given by D J(3.22).More generally one has that D J r

is of the form

D J r= J?{1,...,n}|J|=r j∈J?2j+l.o.(5.22)

(where l.o.stands for the parts of lower order in the partials),but it seems di?cult to obtain the relevant di?erential operators for arbitrary r in explicit form starting

from D AW

r

(5.2).

Theorem5.5([D1,Sec.4]).The multivariable Jacobi polynomials p J

λ,λ∈Λ(2.2)

satisfy a system of di?erential equations of the form

D J r p Jλ=

E J r,λp Jλ,r=1,...,n,(5.23)

where D J r=lim q→1(1?q)?2r D AW

r is of the form(5.22)and the corresponding eigen-

values are given by(5.20).

For r=1the Di?erential equation(5.23)coincides with the second order Di?er-ential equation(3.24).

Remarks:i.The proof in[D1,D4]demonstrating that the multivariable Askey-Wilson type polynomials satisfy the system of di?erence equations in Theorem5.1

初中英语介词用法归纳总结

初中英语介词用法归纳总结 常用介词基本用法辨析 表示方位的介词:in, to, on 1. in 表示在某地范围之内。 Shanghai is/lies in the east of China. 上海在中国的东部。 2. to 表示在某地范围之外。 Japan is/lies to the east of China. 日本位于中国的东面。 3. on 表示与某地相邻或接壤。 Mongolia is/lies on the north of China. 蒙古国位于中国北边。 表示计量的介词:at, for, by 1. at 表示“以……速度”“以……价格”。 It flies at about 900 kilometers an hour. 它以每小时900公里的速度飞行。 I sold my car at a high price. 我以高价出售了我的汽车。 2. for 表示“用……交换,以……为代价”。 He sold his car for 500 dollars. 他以五百元把车卖了。

注意:at表示单价(price) ,for表示总钱数。 3. by 表示“以……计”,后跟度量单位。 They paid him by the month. 他们按月给他计酬。 Here eggs are sold by weight. 在这里鸡蛋是按重量卖的。 表示材料的介词:of, from, in 1. of 成品仍可看出原料。 This box is made of paper. 这个盒子是纸做的。 2. from 成品已看不出原料。 Wine is made from grapes. 葡萄酒是葡萄酿成的。 3. in 表示用某种材料或语言。 Please fill in the form in pencil first. 请先用铅笔填写这个表格。They talk in English. 他们用英语交谈。 表示工具或手段的介词:by, with, on 1. by 用某种方式,多用于交通。 I went there by bus. 我坐公共汽车去那儿。 2. with表示“用某种工具”。

some和any的用法与练习题

some和 any 的用法及练习题( 一) 一、用法: some意思为:一些。可用来修饰可数名词和不可数名词,常常用于肯定句 . any 意思为:任何一些。它可以修饰可数名词和不可数名词,当修饰可数名词 时要用复数形式。常用于否定句和疑问句。 注意: 1、在表示请求和邀请时,some也可以用在疑问句中。 2、表示“任何”或“任何一个”时,也可以用在肯定句中。 3、和后没有名词时,用作代词,也可用作副词。 二、练习题: 1.There are ()newspapers on the table. 2.Is there ( )bread on the plate. 3.Are there () boats on the river? 4.---Do you have () brothers ?---Yes ,I have two brothers. 5.---Is there () tea in the cup? --- Yes,there is () tea in it ,but there isn’t milk. 6.I want to ask you() questions. 7.My little boy wants ()water to drink. 8.There are () tables in the room ,but there aren’t ( )chairs. 9.Would you like () milk? 10.Will you give me () paper? 复合不定代词的用法及练习 一.定义: 由 some,any,no,every 加上 -body,-one,-thing,-where构成的不定代词,叫做复合不定代词 . 二. 分类: 1.指人:含 -body 或 -one 的复合不定代词指人 . 2.含-thing 的复合不定代词指物。 3.含-where 的复合不定代词指地点。 三:复合不定代词: somebody =someone某人 something 某物,某事,某东西 somewhere在某处,到某处 anybody= anyone 任何人,无论谁 anything任何事物,无论何事,任何东西 anywhere 在任何地方 nobody=no one 无一人 nothing 无一物,没有任何东西 everybody =everyone每人,大家,人人 everything每一个事物,一切 everywhere 到处 , 处处 , 每一处

初中英语名词练习题与详解

名词 判断对错 1、[误] Please give me a paper. [正] Please give me a piece of paper. [析]不要认为可以数的名词就是可数名词,这种原因是对英语中可数与不可数名词的概念 与中文中的能数与不能数相混淆了,所以造成了这样的错误,因paper 在英语中是属于物质名词一类,是不可数名词。而不可数名词要表达数量时,要用与之相关的量词来表达,如: two pieces of paper. 2、[误] Please give me two letter papers. [正] Please give me two pieces of letter paper. [析] paper 作为纸讲是不可数名词,而作为报纸、考卷、文章讲时则是可数名词,如:Each student should write a paper on what he has learnt. 3、[误] My glasses is broken. [正] My glasses are broken. 4、[误] I want to buy two shoes. [正] I want to buy two pairs of shoes. [析]英语中glasses—眼镜, shoes—鞋, trousers—裤子等由两部分组成的名词一般要用复 数形式。如果要表示一副眼镜应用 a pair of glasses 而这时的谓语动词应与量词相一致。如:5、This pair of glasses is very good. [误] May I borrow two radioes? [正] May I borrow two radios? [析]以o 结尾的名词大都是用加es 来表示其复数形式,但如果 o 前面是一个元音字母或外来语时则只加s 就可以了。这样的词有zoo— zoos,piano—pianos. 6、[误] This is a Mary's dictionary. [正] This is Mary's dictionary. [析]如名词前有指示代词this, that, these those, 及其他修饰词our,some, every, which,或所有格时,则不要再加冠词。 7、[误] There are much people in the garden. [正] There are many people in the garden. [析]可数名词前应用 many, few, a few, a lot of 来修饰,而 people 是可数名词,而且是复数名词,如: The people are planting trees here. 8、[误] I want a few water. [正] I want a little water. [析]不可数名词前可以用 a little, little, a lot of, some来修饰,但不可用many,few 来修饰。 9、[误] Thank you very much. Y our family is very kind to me. [正] Thank you very much. Y our family are very kind to me. 10、[误] Tom's and Mary's family are waiting for us. [正] Tom's and Mary's families are waiting for us. 11、[误] I'm sorry . I have to go. Tom's families are waiting for me. [正] I'm sorry. I have to go. Tom's family are waiting for me. [析]集合名词如果指某个集合的整体,则应视为单数,如指某个集合体中的个体则应视为 复数。如 :My family is a big family. When I came in, Tom's family were watching TV. 即汤姆一家人正在看电视。这样的集合名词有:family class, team 等。

初中英语介词用法总结

初中英语介词用法总结 介词(preposition):也叫前置词。在英语里,它的搭配能力最强。但不能单独做句子成分需要和名词或代词(或相当于名词的其他词类、短语及从句)构成介词短语,才能在句中充当成分。 介词是一种虚词,不能独立充当句子成分,需与动词、形容词和名词搭配,才能在句子中充当成分。介词是用于名词或代词之前,表示词与词之间关系的词类,介词常与动词、形容词和名词搭配表示不同意义。介词短语中介词后接名词、代词或可以替代名词的词(如:动名词v-ing).介词后的代词永远为宾格形式。介词的种类: (1)简单介词:about, across, after, against, among, around, at, before, behind, below, beside, but, by, down, during, for, from, in, of, on, over, near, round, since, to, under, up, with等等。 (2)合成介词:inside, into, outside, throughout, upon, without, within (3)短语介词:according to, along with, apart from, because of, in front of, in spite of, instead of, owing to, up to, with reguard to (4)分词介词:considering, reguarding, including, concerning 介词短语:构成 介词+名词We go to school from Monday to Saturday. 介词+代词Could you look for it instead of me? 介词+动名词He insisted on staying home. 介词+连接代/副词I was thinking of how we could get there. 介词+不定式/从句He gives us some advice on how to finish it. 介词的用法: 一、介词to的常见用法 1.动词+to a)动词+ to adjust to适应, attend to处理;照料, agree to赞同,

some和any的用法

some和any的用法: (1)两者修饰可数单数名词,表某一个;任何一个;修饰可数复数名词和不可数名词,表一些;有些。〔2)一般的用法:some用于肯定句;any用于疑问句,否定句或条件句。 I am looking for some matches. Do you have any matches? I do not have any matches. (3)特殊的用法: (A) 在期望对方肯定的回答时,问句也用some。 Will you lend me some money? (=Please lend me some money.) (B) any表任何或任何一个时,也可用于肯定句。 Come any day you like. (4)some和any后没有名词时,当做代名词,此外两者也可做副词。 Some of them are my students.〔代名词) Is your mother any better?(副词) 3. many和much的用法: (1)many修饰复数可数名词,表许多; much修饰不可数名词,表量或程度。 He has many friends, but few true ones. There hasn't been much good weather recently. (2)many a: many a和many同义,但语气比较强,并且要与单数名词及单数形动词连用。 Many a prisoner has been set free. (=Many prisoners have been set free.) (3)as many和so many均等于the same number of。前有as, like时, 只用so many。 These are not all the books I have. These are as many more upstairs.

some与any的用法区别教案资料

s o m e与a n y的用法 区别

some与any的用法区别 一、一般说来,some用于肯定句,any用于否定句和疑问句。例如: She wants some chalk. She doesn’t want any chalk. Here are some beautiful flowers for you. Here aren’t any beautiful flowers. 二、any可与not以外其他有否定含义的词连用,表达否定概念。例如: He never had any regular schooling. In no case should any such idea be allowed to spread unchecked. The young accountant seldom (rarely, hardly, scarcely) makes any error in his books. I can answer your questions without any hesitation. 三、any可以用于表达疑问概念的条件句中。例如: If you are looking for any stamps, you can find them in my drawer. If there are any good apples in the shop, bring me two pounds of them. If you have any trouble, please let me know. 四、在下列场合,some也可用于疑问句。 1、说话人认为对方的答复将是肯定的。例如: Are you expecting some visitors this afternoon?(说话人认为下午有人要求,所以用some)

50套初中英语数词

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一、名词的分类 名词可分为普通名词和专有名词两大类。 1. 普通名词又可分为: (1)个体名词。如:cup,desk,student等。一般可数,有单复数形式。 (2)集体名词。如:class,team,family等。一般可数,有单复数形式。 (3)物质名词。如:rice,water,cotton等。一般不可数,没有单复数之分。 (4)抽象名词。如:love,work,life等。一般不可数,没有单复数之分。 2. 专有名词:如:China,Newton,London等。 二、名词的数 (一)可数名词的复数形式的构成规则 1. 一般情况下在名词的词尾加s,如:book books,pencil pencils. 2. 以-s,-x,-ch,-sh结尾的名词加-es,其读音为[iz]。如:bus buses,box boxes,watch watches,dish dishes等。 3. 以-y结尾的名词: (1)以“辅音字母+y”结尾的名词,把y改为i再加es,读音为[iz],如:factory factories,company companies等。 (2)以“元音字母+y”结尾的名词,或专有名词以y结尾,直接在词尾加-s,读音为[z]。如:key keys,Henry Henrys等。 4. 以-f和-fe结尾的名词: (1)变-f或-fe为v再加-es,读音为[vz]。如:thief thieves,wife wives,half halves等。 (2)直接在词尾加-s,如:roof roofs,gulf gulfs,chief chiefs,proof proofs等。 (3)两者均可。如:handkerchief handkerchiefs或handkerchieves. 5. 以-o结尾的名词:

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