Constructing bispectral orthogonal polynomials from the classical discrete families of Charlier, Meixner and Krawtchouk
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Constructing bispectral orthogonal polynomials from the classical discrete families of Charlier, Meixner and Krawtchouk1 Manuel Dom´ınguez de la Iglesia Instituto de Matem´aticas, C.U. Universidad Nacional Aut´onoma de M´exico Constructive Functions 2014 Vanderbilt University, Nashville, May 26, 2014 1joint work with Antonio J. Dur´an
Introduction Methodology Examples Outline 1Introduction Classical discrete orthogonal polynomials Krall orthogonal polynomials 2Methodology D-operators Choice of polynomials Identifying the measure 3Examples Charlier, Meixner and Krawtchouk polynomials
Introduction Methodology Examples Outline 1Introduction Classical discrete orthogonal polynomials Krall orthogonal polynomials 2Methodology D-operators Choice of polynomials Identifying the measure 3Examples Charlier, Meixner and Krawtchouk polynomials
Introduction Methodology Examples Discrete orthogonal polynomials A system of polynomials (pn)nis orthogonal with respect to a discrete measure ω(x) = Px∈S axδtx,S ⊂ Nif hpn,pmiω=X x∈S axpn(tx)pm(tx) = kpnk2 ωδnm,n,m≥0 Every family of OP’s (pn)nsatisfy a three-term recurrence relation xpn(x) = an+1pn+1(x) + bnpn(x) + cnpn−1(x),n≥1 where an,cn6= 0, bn∈Rand p0(x) = 1,p−1(x) = 0. Jacobi operator (tridiagonal): Jp = b0a1 c1b1a2 c2b2a3 ......... p0(x) p1(x) p2(x) . . . =x p0(x) p1(x) p2(x) . . . =xp,x∈ S The converse result is also true (Favard’s or spectral theorem)
Introduction Methodology Examples Discrete orthogonal polynomials A system of polynomials (pn)nis orthogonal with respect to a discrete measure ω(x) = Px∈S axδtx,S ⊂ Nif hpn,pmiω=X x∈S axpn(tx)pm(tx) = kpnk2 ωδnm,n,m≥0 Every family of OP’s (pn)nsatisfy a three-term recurrence relation xpn(x) = an+1pn+1(x) + bnpn(x) + cnpn−1(x),n≥1 where an,cn6= 0, bn∈Rand p0(x) = 1,p−1(x) = 0. Jacobi operator (tridiagonal): Jp = b0a1 c1b1a2 c2b2a3 ......... p0(x) p1(x) p2(x) . . . =x p0(x) p1(x) p2(x) . . . =xp,x∈ S The converse result is also true (Favard’s or spectral theorem)
Introduction Methodology Examples Discrete orthogonal polynomials A system of polynomials (pn)nis orthogonal with respect to a discrete measure ω(x) = Px∈S axδtx,S ⊂ Nif hpn,pmiω=X x∈S axpn(tx)pm(tx) = kpnk2 ωδnm,n,m≥0 Every family of OP’s (pn)nsatisfy a three-term recurrence relation xpn(x) = an+1pn+1(x) + bnpn(x) + cnpn−1(x),n≥1 where an,cn6= 0, bn∈Rand p0(x) = 1,p−1(x) = 0. Jacobi operator (tridiagonal): Jp = b0a1 c1b1a2 c2b2a3 ......... p0(x) p1(x) p2(x) . . . =x p0(x) p1(x) p2(x) . . . =xp,x∈ S The converse result is also true (Favard’s or spectral theorem)
Introduction Methodology Examples Discrete orthogonal polynomials A system of polynomials (pn)nis orthogonal with respect to a discrete measure ω(x) = Px∈S axδtx,S ⊂ Nif hpn,pmiω=X x∈S axpn(tx)pm(tx) = kpnk2 ωδnm,n,m≥0 Every family of OP’s (pn)nsatisfy a three-term recurrence relation xpn(x) = an+1pn+1(x) + bnpn(x) + cnpn−1(x),n≥1 where an,cn6= 0, bn∈Rand p0(x) = 1,p−1(x) = 0. Jacobi operator (tridiagonal): Jp = b0a1 c1b1a2 c2b2a3 ......... p0(x) p1(x) p2(x) . . . =x p0(x) p1(x) p2(x) . . . =xp,x∈ S The converse result is also true (Favard’s or spectral theorem)
Introduction Methodology Examples Classical families If we set ∆f(x) = f(x+ 1) −f(x),∇f(x) = f(x)−f(x−1) the classification problem is to find discrete OP’s (pn)n Lancaster, 1941 σ(x)∆∇pn(x) + τ(x)∆pn(x) + λnpn(x)=0,x∈ S ⊂ N degσ≤2,degτ= 1 In other words, if we call the shift operator sjf(x) = f(x+j) the difference equation reads [σ(x) + τ(x)]s1pn(x)−[2σ(x) + τ(x)]s0pn(x) +σ(x)s−1pn(x) + λnpn(x)=0,x∈ S ⊂ N
Introduction Methodology Examples Classical families Charlier (Poisson): S={0,1,2, . . .}. ωa(x) = ∞ X x=0 ax x!δx,a>0 aca n(x+ 1) −(x+a)ca n(x) + xca n(x−1) = −nca n(x) Meixner (Pascal, Geometric): S={0,1,2, . . .}. ωa,c(x) = Γ(c)(1 −a)c ∞ X x=0 (c)xax x!δx,0<a<1,c>0 (i)j=i(i+ 1) · · · (i+j−1) is the Pochhammer symbol a(x+c)ma,c n(x+ 1) −(x+a(x+c))ma,c n(x) +xma,c n(x−1) = n(a−1)ma,c n(x)
Introduction Methodology Examples Krall polynomials (discrete case) The same question arise in the discrete setting, i.e. find families of OP’s (qn)nwhich are also eigenfunctions of a higher order difference operator Dd= s X j=r hj(x)sj,hs,hr6= 0,⇒Dd(qn) = λnqn Bavinck-van Haeringen-Koekoek, 1994: adding deltas at the endpoints of the support does not work (infinite order difference operator). Surprisingly, it has not been until very recently (Dur´an, 2012) when the first examples appeared. Also s−r= 2m. (qn)nare typically orthogonal with respect to the measure ωF(x) = Y f∈F (x−f)ω(x) where ωis a discrete classical weight and Fis a finite set of numbers. This is also called a Christoffel transform of ω.
Introduction Methodology Examples Krall polynomials (discrete case) The same question arise in the discrete setting, i.e. find families of OP’s (qn)nwhich are also eigenfunctions of a higher order difference operator Dd= s X j=r hj(x)sj,hs,hr6= 0,⇒Dd(qn) = λnqn Bavinck-van Haeringen-Koekoek, 1994: adding deltas at the endpoints of the support does not work (infinite order difference operator). Surprisingly, it has not been until very recently (Dur´an, 2012) when the first examples appeared. Also s−r= 2m. (qn)nare typically orthogonal with respect to the measure ωF(x) = Y f∈F (x−f)ω(x) where ωis a discrete classical weight and Fis a finite set of numbers. This is also called a Christoffel transform of ω.
Introduction Methodology Examples Krall polynomials (discrete case) The same question arise in the discrete setting, i.e. find families of OP’s (qn)nwhich are also eigenfunctions of a higher order difference operator Dd= s X j=r hj(x)sj,hs,hr6= 0,⇒Dd(qn) = λnqn Bavinck-van Haeringen-Koekoek, 1994: adding deltas at the endpoints of the support does not work (infinite order difference operator). Surprisingly, it has not been until very recently (Dur´an, 2012) when the first examples appeared. Also s−r= 2m. (qn)nare typically orthogonal with respect to the measure ωF(x) = Y f∈F (x−f)ω(x) where ωis a discrete classical weight and Fis a finite set of numbers. This is also called a Christoffel transform of ω.
Introduction Methodology Examples Krall polynomials (discrete case) The same question arise in the discrete setting, i.e. find families of OP’s (qn)nwhich are also eigenfunctions of a higher order difference operator Dd= s X j=r hj(x)sj,hs,hr6= 0,⇒Dd(qn) = λnqn Bavinck-van Haeringen-Koekoek, 1994: adding deltas at the endpoints of the support does not work (infinite order difference operator). Surprisingly, it has not been until very recently (Dur´an, 2012) when the first examples appeared. Also s−r= 2m. (qn)nare typically orthogonal with respect to the measure ωF(x) = Y f∈F (x−f)ω(x) where ωis a discrete classical weight and Fis a finite set of numbers. This is also called a Christoffel transform of ω.
Introduction Methodology Examples Conjectures (Dur´ an, 2012) For a finite set Fconsider rF=Pf∈Ff− nF(nF−1) 2+ 1, where nF= #(F). Conjecture A: Let ωabe the Charlier weight and consider (Ffinite) ωF a=Y f∈F (x−f)ωa The OP’s (qn)nwith respect to ωF aare eigenfunctions of a higher-order difference operator with −s=r=rF. Conjecture B: Let ωa,cbe the Meixner weight and consider (F1,F2finite) ωF1,F2 a,c=Y f∈F1 (x+c+f)Y f∈F2 (x−f)ωa,c The OP’s (qn)nwith respect to ωF1,F2 aare eigenfunctions of a higher-order difference operator with −s=r=rF1+rF2 −1. Conjecture C: Let ωa,Nbe the Krawtchouk weight and consider (F1,F2finite) ωF1,F2 a,N=Y f∈F1 (x−f)Y f∈F2 (N−1−f−x)ωa,N The OP’s (qn)nwith respect to ωF1,F2 aare eigenfunctions of a higher-order difference operator with −s=r=rF1+rF2 −1.
Introduction Methodology Examples Conjectures (Dur´ an, 2012) For a finite set Fconsider rF=Pf∈Ff− nF(nF−1) 2+ 1, where nF= #(F). Conjecture A: Let ωabe the Charlier weight and consider (Ffinite) ωF a=Y f∈F (x−f)ωa The OP’s (qn)nwith respect to ωF aare eigenfunctions of a higher-order difference operator with −s=r=rF. Conjecture B: Let ωa,cbe the Meixner weight and consider (F1,F2finite) ωF1,F2 a,c=Y f∈F1 (x+c+f)Y f∈F2 (x−f)ωa,c The OP’s (qn)nwith respect to ωF1,F2 aare eigenfunctions of a higher-order difference operator with −s=r=rF1+rF2 −1. Conjecture C: Let ωa,Nbe the Krawtchouk weight and consider (F1,F2finite) ωF1,F2 a,N=Y f∈F1 (x−f)Y f∈F2 (N−1−f−x)ωa,N The OP’s (qn)nwith respect to ωF1,F2 aare eigenfunctions of a higher-order difference operator with −s=r=rF1+rF2 −1.
Introduction Methodology Examples Conjectures (Dur´ an, 2012) For a finite set Fconsider rF=Pf∈Ff− nF(nF−1) 2+ 1, where nF= #(F). Conjecture A: Let ωabe the Charlier weight and consider (Ffinite) ωF a=Y f∈F (x−f)ωa The OP’s (qn)nwith respect to ωF aare eigenfunctions of a higher-order difference operator with −s=r=rF. Conjecture B: Let ωa,cbe the Meixner weight and consider (F1,F2finite) ωF1,F2 a,c=Y f∈F1 (x+c+f)Y f∈F2 (x−f)ωa,c The OP’s (qn)nwith respect to ωF1,F2 aare eigenfunctions of a higher-order difference operator with −s=r=rF1+rF2 −1. Conjecture C: Let ωa,Nbe the Krawtchouk weight and consider (F1,F2finite) ωF1,F2 a,N=Y f∈F1 (x−f)Y f∈F2 (N−1−f−x)ωa,N The OP’s (qn)nwith respect to ωF1,F2 aare eigenfunctions of a higher-order difference operator with −s=r=rF1+rF2 −1.
Introduction Methodology Examples Conjectures (Dur´ an, 2012) For a finite set Fconsider rF=Pf∈Ff− nF(nF−1) 2+ 1, where nF= #(F). Conjecture A: Let ωabe the Charlier weight and consider (Ffinite) ωF a=Y f∈F (x−f)ωa The OP’s (qn)nwith respect to ωF aare eigenfunctions of a higher-order difference operator with −s=r=rF. Conjecture B: Let ωa,cbe the Meixner weight and consider (F1,F2finite) ωF1,F2 a,c=Y f∈F1 (x+c+f)Y f∈F2 (x−f)ωa,c The OP’s (qn)nwith respect to ωF1,F2 aare eigenfunctions of a higher-order difference operator with −s=r=rF1+rF2 −1. Conjecture C: Let ωa,Nbe the Krawtchouk weight and consider (F1,F2finite) ωF1,F2 a,N=Y f∈F1 (x−f)Y f∈F2 (N−1−f−x)ωa,N The OP’s (qn)nwith respect to ωF1,F2 aare eigenfunctions of a higher-order difference operator with −s=r=rF1+rF2 −1.
Introduction Methodology Examples Outline 1Introduction Classical discrete orthogonal polynomials Krall orthogonal polynomials 2Methodology D-operators Choice of polynomials Identifying the measure 3Examples Charlier, Meixner and Krawtchouk polynomials
Introduction Methodology Examples D-operators Let Abe an algebra of (differential or difference) operators and (pn)na family of polynomials such that there exists Dp∈ A with Dp(pn) = npn. Given a sequence of numbers (εn)n, let us consider the operator D(pn) = n X j=1 (−1)j+1εn· · · εn−jpn−j=εnpn−1−εnεn−1pn−2+· · · We say that Dis an D-operator associated with Aand (pn)nif D ∈ A. Laguerre:εn=−1⇒ D =d dx . Charlier:εn= 1 ⇒ D =∇. Meixner: ε1 n=a 1−a⇒ D1=a 1−a∆, ε2 n=1 1−a⇒ D2=1 1−a∇. Krawtchouk: ε1 n=1 1−a⇒ D1=1 1−a∇, ε2 n=−a 1−a⇒ D2=−a 1−a∆.
Introduction Methodology Examples D-operators GOAL: Extend the previous Theorem for the case that we consider a linear combination of m+ 1 consecutive pn’s: qn=pn+βn,1pn−1+βn,2pn−2+· · · +βn,mpn−m Let R1,R2,...,Rmbe marbitrary polynomials and mD-operators D1,D2,...,Dmdefined by the sequences (εh n)n,h= 1,...,m. Define the auxiliary functions ξh n,iby ξh n,i=εh nεh n−1· · · εh n−i+1 and assume that the following Casorati determinant never vanish (n≥0) Ω(n) = ξ1 n−1,m−1R1(n−1) ξ1 n−2,m−2R1(n−2) · · · R1(n−m) . . .. . ..... . . ξm n−1,m−1Rm(n−1) ξm n−2,m−2Rm(n−2) · · · Rm(n−m) 6= 0
Introduction Methodology Examples D-operators GOAL: Extend the previous Theorem for the case that we consider a linear combination of m+ 1 consecutive pn’s: qn=pn+βn,1pn−1+βn,2pn−2+· · · +βn,mpn−m Let R1,R2,...,Rmbe marbitrary polynomials and mD-operators D1,D2,...,Dmdefined by the sequences (εh n)n,h= 1,...,m. Define the auxiliary functions ξh n,iby ξh n,i=εh nεh n−1· · · εh n−i+1 and assume that the following Casorati determinant never vanish (n≥0) Ω(n) = ξ1 n−1,m−1R1(n−1) ξ1 n−2,m−2R1(n−2) · · · R1(n−m) . . .. . ..... . . ξm n−1,m−1Rm(n−1) ξm n−2,m−2Rm(n−2) · · · Rm(n−m) 6= 0
Introduction Methodology Examples D-operators Now consider the sequence of polynomials (qn)ndefined by qn(x) = pn(x)−pn−1(x)· · · (−1)mpn−m(x) ξ1 n,mR1(n)ξ1 n−1,m−1R1(n−1) · · · R1(n−m) . . .. . ..... . . ξm n,mRm(n)ξm n−1,m−1Rm(n−1) · · · Rm(n−m) Observation:qnis a linear combination of of m+ 1 consecutive pn’s. Define for h= 1,...,m,the following functions Mh(x) = m X j=1 (−1)h+jξh x,m−jdet ξl x+j−r,m−rRl(x+j−r)(l6=h r6=j) Observation:Mhare linear combinations of adjoint determinants of Ω(x). If we assume that Ω(x) and Mh(x) are polynomials in x, then ∃Dq∈ A with Dq(qn) = P(n)qnand P(x)−P(x−1) = Ω(x), where Dq=P(Dp) + m X h=1 Mh(Dp)DhRh(Dp)
Introduction Methodology Examples D-operators Now consider the sequence of polynomials (qn)ndefined by qn(x) = pn(x)−pn−1(x)· · · (−1)mpn−m(x) ξ1 n,mR1(n)ξ1 n−1,m−1R1(n−1) · · · R1(n−m) . . .. . ..... . . ξm n,mRm(n)ξm n−1,m−1Rm(n−1) · · · Rm(n−m) Observation:qnis a linear combination of of m+ 1 consecutive pn’s. Define for h= 1,...,m,the following functions Mh(x) = m X j=1 (−1)h+jξh x,m−jdet ξl x+j−r,m−rRl(x+j−r)(l6=h r6=j) Observation:Mhare linear combinations of adjoint determinants of Ω(x). If we assume that Ω(x) and Mh(x) are polynomials in x, then ∃Dq∈ A with Dq(qn) = P(n)qnand P(x)−P(x−1) = Ω(x), where Dq=P(Dp) + m X h=1 Mh(Dp)DhRh(Dp)
Introduction Methodology Examples D-operators Now consider the sequence of polynomials (qn)ndefined by qn(x) = pn(x)−pn−1(x)· · · (−1)mpn−m(x) ξ1 n,mR1(n)ξ1 n−1,m−1R1(n−1) · · · R1(n−m) . . .. . ..... . . ξm n,mRm(n)ξm n−1,m−1Rm(n−1) · · · Rm(n−m) Observation:qnis a linear combination of of m+ 1 consecutive pn’s. Define for h= 1,...,m,the following functions Mh(x) = m X j=1 (−1)h+jξh x,m−jdet ξl x+j−r,m−rRl(x+j−r)(l6=h r6=j) Observation:Mhare linear combinations of adjoint determinants of Ω(x). If we assume that Ω(x) and Mh(x) are polynomials in x, then ∃Dq∈ A with Dq(qn) = P(n)qnand P(x)−P(x−1) = Ω(x), where Dq=P(Dp) + m X h=1 Mh(Dp)DhRh(Dp)
Introduction Methodology Examples Choice of R1,R2,...,Rm GOAL: Make (qn)nbispectral (we already have Dq(qn) = λnqn). For that we have to make an appropriate choice of the arbitrary polynomials R1,R2,...,Rm. This choice is based on the following recurrence formula (h= 1,...,m): εh n+1an+1Rh j(n+ 1) −bnRh j(n) + cn εh n Rh j(n−1) = (ηhj+κh)Rh j(n),n∈Z where ηhand κhare real numbers independent of nand j, (an)n∈Z, (bn)n∈Z, (cn)n∈Zare the coefficients in the TTRR for the OP’s (pn)n, and (εh n)ndefines a D-operator for (pn)n. Classical discrete family D-operators Rj(x) Charlier: ca n,n≥0∇c−a j(−x−1), j≥0 Meixner: ma,c n,n≥0a 1−a∆m1/a,2−c j(−x−1), j≥0 1 1−a∇ma,2−c j(−x−1), j≥0 Krawtchouk: ka,N n,n≥01 1+a∇ka,−N j(−x−1), j≥0 −a 1+a∆k1/a,−N j(−x−1), j≥0
Introduction Methodology Examples Choice of R1,R2,...,Rm GOAL: Make (qn)nbispectral (we already have Dq(qn) = λnqn). For that we have to make an appropriate choice of the arbitrary polynomials R1,R2,...,Rm. This choice is based on the following recurrence formula (h= 1,...,m): εh n+1an+1Rh j(n+ 1) −bnRh j(n) + cn εh n Rh j(n−1) = (ηhj+κh)Rh j(n),n∈Z where ηhand κhare real numbers independent of nand j, (an)n∈Z, (bn)n∈Z, (cn)n∈Zare the coefficients in the TTRR for the OP’s (pn)n, and (εh n)ndefines a D-operator for (pn)n. Classical discrete family D-operators Rj(x) Charlier: ca n,n≥0∇c−a j(−x−1), j≥0 Meixner: ma,c n,n≥0a 1−a∆m1/a,2−c j(−x−1), j≥0 1 1−a∇ma,2−c j(−x−1), j≥0 Krawtchouk: ka,N n,n≥01 1+a∇ka,−N j(−x−1), j≥0 −a 1+a∆k1/a,−N j(−x−1), j≥0
Introduction Methodology Examples Choice of R1,R2,...,Rm GOAL: Make (qn)nbispectral (we already have Dq(qn) = λnqn). For that we have to make an appropriate choice of the arbitrary polynomials R1,R2,...,Rm. This choice is based on the following recurrence formula (h= 1,...,m): εh n+1an+1Rh j(n+ 1) −bnRh j(n) + cn εh n Rh j(n−1) = (ηhj+κh)Rh j(n),n∈Z where ηhand κhare real numbers independent of nand j, (an)n∈Z, (bn)n∈Z, (cn)n∈Zare the coefficients in the TTRR for the OP’s (pn)n, and (εh n)ndefines a D-operator for (pn)n. Classical discrete family D-operators Rj(x) Charlier: ca n,n≥0∇c−a j(−x−1), j≥0 Meixner: ma,c n,n≥0a 1−a∆m1/a,2−c j(−x−1), j≥0 1 1−a∇ma,2−c j(−x−1), j≥0 Krawtchouk: ka,N n,n≥01 1+a∇ka,−N j(−x−1), j≥0 −a 1+a∆k1/a,−N j(−x−1), j≥0
Introduction Methodology Examples Identifying the measure Given a set Gof mpositive integers, G={g1,...,gm}we then define the sequence of polynomials (qG n)nby qG n(x) = pn(x)−pn−1(x)· · · (−1)mpn−m(x) ξ1 n,mR1 g1(n)ξ1 n−1,m−1R1 g1(n−1) · · · R1 g1(n−m) . . .. . ..... . . ξm n,mRm gm(n)ξm n−1,m−1Rm gm(n−1) · · · Rm gm(n−m) (qG n)nwill be orthogonal w.r.t a Christoffel transform of ω(or several) ωF(x) = Y f∈F (x−f)ω(x) How is the set Grelated with the set F?:Gwill be identified by one of the following sets: I(F) = {1,2,...,fk}\{fk−f,f∈F}, Jh(F) = {0,1,2,...,fk+h−1}\{f−1,f∈F},h≥1 where fk= max Fand k= #(F).
Introduction Methodology Examples Identifying the measure Given a set Gof mpositive integers, G={g1,...,gm}we then define the sequence of polynomials (qG n)nby qG n(x) = pn(x)−pn−1(x)· · · (−1)mpn−m(x) ξ1 n,mR1 g1(n)ξ1 n−1,m−1R1 g1(n−1) · · · R1 g1(n−m) . . .. . ..... . . ξm n,mRm gm(n)ξm n−1,m−1Rm gm(n−1) · · · Rm gm(n−m) (qG n)nwill be orthogonal w.r.t a Christoffel transform of ω(or several) ωF(x) = Y f∈F (x−f)ω(x) How is the set Grelated with the set F?:Gwill be identified by one of the following sets: I(F) = {1,2,...,fk}\{fk−f,f∈F}, Jh(F) = {0,1,2,...,fk+h−1}\{f−1,f∈F},h≥1 where fk= max Fand k= #(F).
Introduction Methodology Examples Charlier polynomials: explicit example Let a= 1, F={1,3},G=I(F) = {1,3}. qG n Ω(n)=c1 n+βn,1c1 n−1+βn,2c1 n−2are orthogonal w.r.t ˜ωF 1= (x+ 3)(x+ 1)ω1(x+ 4) The difference operator (of order 8) satisfying Dq(qG n) = P(n)qG nis Dq=P(D1) + M1(D1)∇R1(D1) + M2(D1)∇R2(D1) where D1=−xs−1+ (x+ 1)s0−s1,D1(c1 n) = nc1 n,n≥0 R1(x) = −x,R2(x) = −1 6(x3+ 3x2+ 5x+ 2) M1(x) = x2+ 2x+ 2,M2(x) = −2 P(x) = −x 12(x3−2x2−x−2)
Introduction Methodology Examples Meixner polynomials In this case have two different D-operators. That means that we will have to consider two sets of positive integers F1,F2⊂N. Consider H=Jh(F1) = {h1,...,hm1},K=I(F2) = {k1,...,km2}and m=m1+m2,ωa,cthe Meixner measure and (ma,c n)n. If we define (qn)nby qn(x) = (1 −a)mma,c n(x) am −(1 −a)m−1ma,c n−1(x) am−1··· (−1)mma,c n−m(x) m1/a,2−c h1(−n−1) m1/a,2−c h1(−n)··· m1/a,2−c h1(−n+m−1) . . .. . ..... . . m1/a,2−c hm1(−n−1) m1/a,2−c hm1(−n)··· m1/a,2−c hm1(−n+m−1) ma,2−c k1(−n−1) am ma,2−c k1(−n) am−1··· ma,2−c k1(−n+m−1) . . .. . ..... . . ma,2−c km2(−n−1) am ma,2−c km2(−n) am−1··· ma,2−c km2(−n+m−1)
Introduction Methodology Examples Meixner polynomials In this case have two different D-operators. That means that we will have to consider two sets of positive integers F1,F2⊂N. Consider H=Jh(F1) = {h1,...,hm1},K=I(F2) = {k1,...,km2}and m=m1+m2,ωa,cthe Meixner measure and (ma,c n)n. If we define (qn)nby qn(x) = (1 −a)mma,c n(x) am −(1 −a)m−1ma,c n−1(x) am−1··· (−1)mma,c n−m(x) m1/a,2−c h1(−n−1) m1/a,2−c h1(−n)··· m1/a,2−c h1(−n+m−1) . . .. . ..... . . m1/a,2−c hm1(−n−1) m1/a,2−c hm1(−n)··· m1/a,2−c hm1(−n+m−1) ma,2−c k1(−n−1) am ma,2−c k1(−n) am−1··· ma,2−c k1(−n+m−1) . . .. . ..... . . ma,2−c km2(−n−1) am ma,2−c km2(−n) am−1··· ma,2−c km2(−n+m−1)
Introduction Methodology Examples Meixner polynomials Then the polynomials (qn)nare orthogonal with respect to the measure ˜ωF1,F2,h a,c=Y f∈F1 (x+c−f)Y f∈F2 (x+f2,M+1−f)ωa,c−f1,M−f2,M−h−1(x+f2,M+1) and they are eigenfunctions of a higher order difference operator Dqwith −s=r=X f∈F2 f−X f∈F1 f−nF1(nF1−1) 2−nF2(nF2−1) 2+nF1(f1,M+h)+1 where nFi= #(Fi) and fi,M= max Fi,i= 1,2. Proof of Conjecture B: Write ˜ F1={f1,M−f+ 1,f∈F1}, ˜c=c+f1,M+f2,M+ 2 and h= min F1. In particular Jh(˜ F1) = I(F1). Therefore we have ωF1,F2 a,c= (1 −a)c−˜c˜ω˜ F1,F2,h a,˜c(x−f2,M−1)
Introduction Methodology Examples Meixner polynomials Then the polynomials (qn)nare orthogonal with respect to the measure ˜ωF1,F2,h a,c=Y f∈F1 (x+c−f)Y f∈F2 (x+f2,M+1−f)ωa,c−f1,M−f2,M−h−1(x+f2,M+1) and they are eigenfunctions of a higher order difference operator Dqwith −s=r=X f∈F2 f−X f∈F1 f−nF1(nF1−1) 2−nF2(nF2−1) 2+nF1(f1,M+h)+1 where nFi= #(Fi) and fi,M= max Fi,i= 1,2. Proof of Conjecture B: Write ˜ F1={f1,M−f+ 1,f∈F1}, ˜c=c+f1,M+f2,M+ 2 and h= min F1. In particular Jh(˜ F1) = I(F1). Therefore we have ωF1,F2 a,c= (1 −a)c−˜c˜ω˜ F1,F2,h a,˜c(x−f2,M−1)
Introduction Methodology Examples Krawtchouk polynomials Again we have two different D-operators. Consider F1,F2⊂Nfinite and K=I(F1) = {k1,...,km2},H=Jh(F2) = {h1,...,hm1},m=m1+m2, ωa,Nthe Krawtchouk measure and (ka,N n)n. We assume that f1,M,f2,M<N/2 (so that F1∩ {N−1−f,f∈F2}=∅), where fi,M= max Fi. If we define (qn)nby qn(x) = (1 + a)mka,N n(x)−(1 + a)m−1ka,N n−1(x)··· (−1)mka,N n−m(x) ka,−N k1(−n−1) ka,−N k1(−n)··· ka,−N k1(−n+m−1) . . .. . ..... . . ka,−N km1(−n−1) ka,−N km1(−n)··· ka,−N km1(−n+m−1) (−a)mk1/a,−N h1(−n−1) (−a)m−1k1/a,−N h1(−n)··· k1/a,−N h1(−n+m−1) . . . . . ..... . . (−a)mk1/a,−N hm2(−n−1) (−a)m−1k1/a,−N hm2(−n)··· k1/a,−N hm2(−n+m−1)
Introduction Methodology Examples Krawtchouk polynomials Again we have two different D-operators. Consider F1,F2⊂Nfinite and K=I(F1) = {k1,...,km2},H=Jh(F2) = {h1,...,hm1},m=m1+m2, ωa,Nthe Krawtchouk measure and (ka,N n)n. We assume that f1,M,f2,M<N/2 (so that F1∩ {N−1−f,f∈F2}=∅), where fi,M= max Fi. If we define (qn)nby qn(x) = (1 + a)mka,N n(x)−(1 + a)m−1ka,N n−1(x)··· (−1)mka,N n−m(x) ka,−N k1(−n−1) ka,−N k1(−n)··· ka,−N k1(−n+m−1) . . .. . ..... . . ka,−N km1(−n−1) ka,−N km1(−n)··· ka,−N km1(−n+m−1) (−a)mk1/a,−N h1(−n−1) (−a)m−1k1/a,−N h1(−n)··· k1/a,−N h1(−n+m−1) . . . . . ..... . . (−a)mk1/a,−N hm2(−n−1) (−a)m−1k1/a,−N hm2(−n)··· k1/a,−N hm2(−n+m−1)
Introduction Methodology Examples Krawtchouk polynomials Then the polynomials (qn)nare orthogonal with respect to the measure ˜ωF1,F2,h a,N=Y f∈F1 (x+f1,M+1−f)Y f∈F2 (N−x−1+f)ωa,N+f1,M+f2,M+h+1(x+f1,M+1) and they are eigenfunctions of a higher order difference operator Dqwith −s=r=X f∈F1 f−X f∈F2 f−nF1(nF1−1) 2−nF2(nF2−1) 2+nF2(f2,M+h)+1 where nFi= #(Fi) and fi,M= max Fi,i= 1,2. Proof of Conjecture C: Write ˜ F1={f1,M−f+ 1,f∈F1}, ˜ F2={f1,M+f2,M−f+ 2,f∈F2},˜ N=N−f1,M−f2,M−2 and h= min F1. In particular Jh(˜ F2) = I(F2). Therefore we have ωF1,F2 a,N= ˜ω˜ F1,˜ F2,h a,˜ N(x−f1,M−1)
Introduction Methodology Examples Krawtchouk polynomials Then the polynomials (qn)nare orthogonal with respect to the measure ˜ωF1,F2,h a,N=Y f∈F1 (x+f1,M+1−f)Y f∈F2 (N−x−1+f)ωa,N+f1,M+f2,M+h+1(x+f1,M+1) and they are eigenfunctions of a higher order difference operator Dqwith −s=r=X f∈F1 f−X f∈F2 f−nF1(nF1−1) 2−nF2(nF2−1) 2+nF2(f2,M+h)+1 where nFi= #(Fi) and fi,M= max Fi,i= 1,2. Proof of Conjecture C: Write ˜ F1={f1,M−f+ 1,f∈F1}, ˜ F2={f1,M+f2,M−f+ 2,f∈F2},˜ N=N−f1,M−f2,M−2 and h= min F1. In particular Jh(˜ F2) = I(F2). Therefore we have ωF1,F2 a,N= ˜ω˜ F1,˜ F2,h a,˜ N(x−f1,M−1)