On characterizations of classical polynomials

ON CHARACTERIZATIONS OF CLASSICAL POLYNOMIALS R. ´ ALVAREZ-NODARSE Abs ac . I is well known ha he classical amilies o Jacobi, La- gue e, He mi e, and Bessel polynomials a e cha ac e ized as eigen ec- o s o a second o de linea di e en ial ope a o wi h polynomial co- e icien s, Rod igues o mula, e c. In his pape we p esen an uni ied s udy o he classical disc e e polynomials and q-polynomials o he q- Hahn ableau by using he di e ence calculus on linea - ype la ices. We ob ain in a s aigh o wa d way se e al cha ac e iza ion heo ems o he classical disc e e and q-polynomials o he q-Hahn ableau. Fi- nally, a de ailed discussion o he Ma celln e . al. cha ac e iza ion is p esen ed. 1. In oduc ion The classical polynomials ( hose o He mi e, Lague e, Jacobi, and Bessel) a e he mos impo an ins ances o o hogonal polynomials. One o he easons is because hey sa is y no only a h ee- e m ecu ence ela ion (TTRR) xPn(x) = αnPn+1(x) + βnPn(x) + γnPn−1(x), γn6= 0, P−1(x) = 0, P0(x) = 1,(1.1) bu also o he use ul p ope ies: hey a e he eigen ec o s o a second o de linea di e en ial equa ion wi h polynomial coe icien s, hei de i a i es also cons i u e an o hogonal amily, hei gene a ion unc ions can be gi en explici ly, among o he s (see o ins ances [1, 8, 24, 25] o he mo e ecen wo k [3]). Among all hese p ope ies he e a e e y impo an ones ha cha ac e ize he classical amilies. In ac no e e y p ope y cha ac e izes he classical polynomials. The simples example is he TTRR (1.1). I is well known (see e.g. [8]) ha he TTRR cha ac e izes he o hogonal polynomials i γn6= 0 o all n∈N. This is he so-called Fa a d Theo em ( o a e iew see [18]). Ne e heless he e exis se e al amilies ha sa is y he TTRR bu no a linea di e en ial equa ion wi h polynomial coe icien s, o a Rod igues- ype o mula. In ac only ew amilies o o hogonal polynomials sa is y hese p ope ies as we will show. Fo e iews on he cha ac e iza ion heo ems see [1, 3, 8]. 2000 Ma hema ics Subjec Classi ica ion. 33C45,33D45. Key wo ds and ph ases. classical polynomials, q-Hahn ableau, disc e e polynomials, cha ac e iza ion heo ems. 1 2 R. ´ ALVAREZ-NODARSE The oldes cha ac e iza ion is he so called Hahn cha ac e iza ion —unless his was i s ly obse ed and p o ed o he Jacobi, Lague e, and He mi e polynomials by N. Sonin in 1887—. In [11], Hahn p o ed he ollowing Theo em 1.1 (Sonin-Hahn [11, 19]).Gi en a sequence o o hogonal poly- nomials (Pn)n, i is a classical sequence i an only i he sequence o hei de i a i es (P0 n)nis an o hogonal sequence. In ac he ollowing heo em holds (see he nice su ey pape [1] and also [19, 20]) Theo em 1.2. The ollowing p ope ies a e equi alen : (1) (Pn)nis a classical o hogonal polynomial sequence (COPS), (2) The sequence o hei de i a i es (P0 n)nis an COPS1, (3) (Pn)nsa is ies he second o de linea di e en ial equa ion wi h poly- nomial coe icien s (Bochne [7]) σ(x)P00 n(x) + τ(x)P0 n(x) + λPn(x) = 0, whe e deg(σ)≤2,deg(τ) = 1, and a e independen o n, and λis a cons an independen o x. (4) (Pn)ncan be exp essed by he Rod igues o mula (T icomi [27] and C ye [9])Pn(x) = Bn ρ(x) dn dxn[σn(x)ρ(x)]. (5) The polynomials a e o hogonal wi h espec o a weigh unc ion ρ ha sa is ies he Pea son di e en ial equa ion [σ(x)ρ(x)]0=τ(x)ρ(x), whe e he polynomials σand τa e such ha deg(σ)≤2,deg(τ) = 1 (Hildeb and [14]). (6) The e exis h ee sequences (an)n,(bn)n,(cn)n, and a polynomial σ, deg(σ)≤2, such ha (Al-Salam & Chiha a [2]) σ(x)P0 n(x) = anPn+1(x) + bnPn(x) + cnPn−1(x), n ≥1.(1.2) (7) The e exis wo sequences ( n)nand (gn)nsuch ha he ollowing ela ion o he monic polynomials holds (Ma cell´an e al [19]) Pn(x) = P0 n+1(x) n+ 1 + nP0 n(x) + gnP0 n−1(x), n ≥1.(1.3) The p oo o his heo em can be ound in he appendix A. A na u al ex ension o he classical polynomials a e he so-called disc e e polynomials ( hose o Cha lie , Meixne , K a chuk, and Hahn, see e.g. [8, 24, 25]) and he q-polynomials (see e.g. [6, 24, 25]). In ac , Hahn in 1949 [13] posed he p oblem o inding all he o hogonal polynomial sequences ha sa is y he condi ions 2–5 om heo em 1.2 bu ins ead o using he 1No ice ha his is no he Hahn heo em. In he Hahn heo em he o hogonali y o bo h sequences i is impossed whe eas he e a mo e es ic i e condi ions is supposed: (Pn)no (P0 n)nis a classical amily. ON CHARACTERIZATIONS OF CLASSICAL POLYNOMIALS 3 de i a i es, he use he linea ope a o Lq,w Lq,w (x) = (qx +w)− (x) (q−1)x+w, q, w ∈R+. Hahn sol ed he p oblem o he case when q∈(0,1) and w= 0, ha leads o he q-Hahn ableau (see e.g. [16] and [5]). The case w=q= 1, leads o he classical disc e e polynomials o Cha lie , Meixne , K a chuk, and Hahn (see [8, 17, 24]). A comple e s udy o he cha ac e iza ion heo ems o hese wo cases has been pe o med using a unc ional app oach in he pape s [10] (disc e e case) and [21] (“q” case). The main aim o he p esen pape is wice: on one hand o p esen a e y simple and uni ied app oach o he a o e said wo cases using he heo y o di e ence equa ions on la ices p esen ed in [24, 25], and on he o he hand o comple e he s udy s a ed in [10, 20, 21]. The s uc u e o he pape is as ollows: In sec ion 2 we in oduce he “linea ” la ices x(s) and cha ac e ize hem. In sec ion 3 he cha ac e i- za ion heo em is p esen ed and p o ed o any linea - ype la ice and, as co olla ies, he co esponding heo ems o he uni o m la ice x(s) = sand he q-linea la ice x(s) = c1qs+c2a e ob ained. Finally, in Sec ion 4, we discuss each case in de ails as well as he classical case ( ha can be ob ained aking an app op ia e limi q→1−). In pa icula , some p oblems ela ed wi h he Ma cell´an e al. cha ac e iza ion [19] a e discussed. 2. The linea - ype la ices x(s) De ini ion 2.1. We say ha x(s)is a linea - ype la ice i x(s+ζ) = F(ζ)x(s) + G(ζ),∀s, ζ ∈C, F(ζ)6= 0.(2.1) Ob iously o he linea la ice x(s) = swe ha e F(ζ) = 1 and G(ζ) = ζ. Ano he impo an ins ance o he linea - ype la ice is he q-linea la ice, (q6={0,±1}), i.e., he unc ions o he o m x(s) = Aqs+B. In his case x(s+ζ) = F(ζ)x(s) + G(ζ), whe e F(ζ) = qζand G(ζ) = B(1 −qζ). P oposi ion 2.2. Le q6={0,±1}. The unc ion x(z)is a q-linea la ice o zi and only i i sa is ies x(z+ 1) = qx(z) + C. P oo . A s aigh o wa d compu a ion shows ha i x(z) is a q-linea unc- ion o n, i.e., x(z) = cqz+d hen i sa is ies he ecu ence o mula x(z+ 1) = qx(z) + C, whe e C=d(1 −q) is a cons an . Bu he gene al solu ion o he di e ence equa ion x(z+ 1) = qx(z) + Cis x(z) = Aqz+D, whe e Aand Da e, in gene al, non-ze o cons an s.  No ice ha o he linea - ype la ices, i Qm(x(s)) is a polynomial o deg ee min x(s), Qm(x(s+α)) is also a m− h deg ee polynomial in x(s), i.e., Qm(x(s+α)) = e Qm(x(s)). Mo eo e , o he linea - ype la ices we ha e he ollowing 4 R. ´ ALVAREZ-NODARSE Lemma 2.3. Le x(s)be a linea - ype la ice and Qm(x(s)) a polynomial o deg ee min x(s). Then ∆Qm(x(s+α)) ∆x(s+β)=Rm−1(x(s)),∀α, β ∈C, whe e Rm−1(x(s)) is again a polynomial in x(s)bu o deg ee m−1and ∆ (s) = (s+ 1) − (s). P oo . I is su icien o p o e he lema o he powe s xn(s). Since x(s) is a linea - ype la ice ∆xn(s+α) ∆x(s+β)=∆(F(α)x(s) + G(α))n F(β)∆x(s)= n X k=0 n kF(α)kG(α)n−k F(β) ∆xk(s) ∆x(s). Bu ∆xk(s)/∆x(s) is a polynomial o deg ee k−1 in x(s) and he e o e ∆xn(s+α)/∆x(s+β) also is.  To conclude his sec ion le poin ou he ollowing Rema k 2.4. F om P oposi ion 2.2 and De ini ion 2.1 i ollows ha he only linea - ype la ices a e hose co esponding o F(1) = 1 ( he linea la ice x(s) = C1s+C2) and he ones when F(1) = q6={0,±1}( he q-linea la ices x(s) = c1qs+c2). 3. The cha ac e iza ion heo em o classical polynomials In he sequel we will assume ha (Pn[x(s)])nis a sequence o o hogonal polynomials on a linea - ype la ice x(s). Fo sake o simplici y we will deno e Pn(s) := Pn[x(s)]. Since Pn(s) a e o hogonal hey sa is y he TRRR x(s)Pn(s) = αnPn+1(s) + βnPn(s) + γnPn−1(s), P−1(s) = 0, P0(s) = 1. (3.1) Le us poin ou ha i γn6= 0, o all n∈N, hen he abo e TTRR de ines an o hogonal polynomial sequence. Ne e heless he e a e se e al examples o which γn= 0 o some n0∈N(e.g. he Hahn and q-Hahn polynomials). In his case we ha e a ini e amily o o hogonal polynomials (see e.g. [8, 25]). In he i s case, i.e., when γn6= 0, o all n∈Nwe say ha i is a quasi-de ini e case [8] (also called he egula case) whe eas in he second one, we ge a weak-quasi-de ini e case o weak- egula case. He e we will deal wi h he “classical” polynomials and we will assume ha γn6= 0 o all n∈ N whe e by Nwe deno e he se N= 1,2,...,n0 o some n0∈No N:= N. He e we will use he no a ion o he heo y o di e ence calculus on non- uni o m la ices ( o mo e de ails see [25, §13] o [24, chap e 3]). Le s=a, a + 1, a + 2,.... We will de ine he o wa d and backwa d di e ences in x(s) by ∆y[x(s)] ∆x(s),∇y[x(s)] ∇x(s), ON CHARACTERIZATIONS OF CLASSICAL POLYNOMIALS 5 espec i ely, whe e ∇ (s) = (s)− (s−1), ∆ (s) = (s+ 1) − (s). Fo he ope a o ∆ we ha e ∆{ (s)g(s)}=g(s){∆ (s)}+ (s+ 1){∆g(s)}.(3.2) Thus he ollowing o mula o summa ion by pa s holds b X s=a (s)∆g(s) = (s)g(s) b+1 a − b X s=a∆ (s)g(s+ 1).(3.3) Also we de ine he k- h o wa d di e ence o a unc ion (s) by ∆(k) (s) := ∆ ∆xk−1(s) ∆ ∆xk−2(s)... ∆ ∆x(s) (s), xm(s) = xs+m 2. Rema k 3.1. No ice ha he di e ences ∆(k)Pn(s)can be w i en in he linea - ype la ice, up o a cons an ac o , as (∆/∆x(s))kPn(s). Mo eo e , he ope a o ∆/∆x(s) o he q-linea la ice x(s) = c1qsbecomes in o he classical Jackson ope a o Dqde ined by DςP(x) = P(ςx)−P(x) x(ς−1) , ς 6= 0,±1.(3.4) Nex we s a e he Hahn-Lesky heo em: Theo em 3.2. Gi en a sequence o o hogonal polynomials (Pn)n, i is a classical sequence i an only i •The sequence o hei ini e di e ences (∆Pn)nis an o hogonal se- quence [17, 10]. •The sequence o hei q-di e ences (DqPn)nis an o hogonal se- quence [13, 21]. No ice ha since we a e deal wi h linea la ices he s a emen o he heo em can be eplaced by he ollowing equi alen one: Theo em 3.2. A sequence o o hogonal polynomials (Pn)nis classical i and only i he sequence o hei ini e di e ences (∆/∆x(s)Pn)nis an o - hogonal sequence. The s anda d p oo o his heo em can be ound in [17] o he linea la ice x(s) = s, and in [10] using he unc ional echnique de eloped by Ma oni. Fo he q-linea la ice x(s) = qsi has been done by Hahn in [13] and using a unc ional app oach in [21]. We s a wi h he ollowing De ini ion 3.3. We say ha he sequence (Pn)nis a classical amily on he linea - ype la ice i hey a e o hogonal wi h espec o he disc e e measu e ρ(s)∇x1(s), i.e., b−1 X s=a Pn(s)Pm(s)ρ(s)∇x1(s) = δnmd2 n,∆s= 1,(3.5) 6 R. ´ ALVAREZ-NODARSE whe e ρis he solu ion o he Pea son- ype equa ion ∆ ∆x(s−1/2)[σ(s)ρ(s)] = τ(s)ρ(s),(3.6) and σand τa e ixed polynomials on x(s)o deg ee a mos 2 and exac ly 1. The unc ion ρis usually called he o hogonalizing weigh unc ion o he polynomial amily (Pn)n. Now we a e eady o enuncia e ou main esul : Theo em 3.4. Le x(s)be a linea - ype la ice and le σ(s)and ρ(s)be wo unc ions such ha akσ(a)ρ(a) = bkσ(b)ρ(b) = 0, o all k≤0. Then, he ollowing p ope ies a e equi alen (1) (Pn)nis a classical o hogonal polynomial sequence (COPS). (2) The sequence o hei di e ences ∆(1)Pnnalso is an COPS. (3) (Pn)nsa is ies he second o de linea di e ence equa ion wi h poly- nomial coe icien s σ(s)∆ ∆x(s−1/2) ∇Pn(s) ∇x(s)+τ(s)∆Pn(s) ∆x(s)+λPn(s) = 0,(3.7) whe e deg(σ)≤2,deg(τ) = 1, a e independen o nand λis a cons an independen o x. (4) (Pn)ncan be exp essed by he Rod igues- ype o mula2 Pn(s) = Bn ρ(s) ∇ ∇x1(s) ∇ ∇x2(s)··· ∇ ∇xn(s)[ρn(s)].(3.8) (5) The polynomials a e o hogonal wi h espec o a weigh unc ion ρ ha sa is ies he Pea son- ype di e ence equa ion (3.6), whe e deg(σ)≤2,deg(τ) = 1. (6) The e exis h ee sequences (an)n,(bn)n,(cn)n, and a polynomial φ, deg(φ)≤2, such ha φ(x)∆Pn(s) ∆x(s)=anPn+1(x) + bnPn(x) + cnPn−1(x), n ≥1. (7) The e exis h ee sequences (en)n,( n)n,(gn)nsuch ha he ollow- ing ela ion holds o all n≥1 Pn(x) = en ∆Pn+1(s) ∆x(s)+ n ∆Pn(s) ∆x(s)+gn ∆Pn−1(s) ∆x(s), en6= 0, gn6=γn, whe e γnis he co esponding coe icien o he TTRR (1.1). As a simple consequence o he abo e heo em we ha e he ollowing Co olla y 3.5 ([10, 21]).The disc e e polynomials on he linea la ice x(s) = sa e classical. The q-polynomials in he q-linea la ice (o expo- nen ial la ice) x(s) = c1qs+c2a e classical. 2The ope a o ∇ ∇x1(s) ∇ ∇x2(s)· · · ∇ ∇xn(s)in he linea ype la ices can be ew i en in he o m ∇n o he linea la ice and q−n(n+1)/2“∇ ∇x(s)”n o he q-linea ones. ON CHARACTERIZATIONS OF CLASSICAL POLYNOMIALS 7 P oo . I ollows om he ac ha x(s) = sand x(s) = c1qs+c2a e linea - ype la ices.  Le us p o e he Theo em 3.4. The idea o he p oo is summa ized in he nex igu e:  * HH Hj ? 6  1 2 3 4 5 HH Hj  * 6 6 6 7 We s a p o ing ha (1)→(2): P oposi ion 3.6. Le x(s)be a linea - ype la ice and le (Pn)nbe a clas- sical amily o hogonal wi h espec o a weigh unc ion ρ, solu ion o he Pea son- ype equa ion (3.6) and such ha 3 σ(a)ρ(a) = σ(b)ρ(b) = 0.(3.9) Then he sequence ∆(1)Pn(s)n, whe e ∆(1)Pn(s) = ∆Pn(s) ∆x(s), is also a clas- sical o hogonal amily wi h espec o he unc ion ρ1(s)∆x(s), whe e he weigh unc ion is ρ1(s) = σ(s+ 1)ρ(s+ 1). P oo . Le Qk(s) be an a bi a y k- h deg ee polynomial on x(s), k < n. The o hogonali y condi ions o (Pn)nyield, o all k < n, 0 = b−1 X s=a Pn(s)Qk−1(s)τ(s)ρ(s)∇x1(s) ( om (3.6)) = b−1 X s=a Pn(s)Qk−1(s)∆(σ(s)ρ(s)) ( om (3.3), (3.9)) =− b−1 X s=a ∆(Pn(s)Qk−1(s))σ(s+ 1)ρ(s+ 1) Applying he Leibniz ule (3.2) 0 = − b−1 X s=a (∆Pn(s))Qk−1(s)σ(s+ 1)ρ(s+ 1)+ b−1 X s=a Pn(s+ 1)(∆Qk−1(s))σ(s+ 1)ρ(s+ 1) (s→s−1, and (3.9)) 3This condi ion leads o he so-called disc e e o hogonal polynomials, i.e., polynomials wi h a disc e e o hogonali y o he o m (3.5). Fo he q-linea la ices (3.5) becomes in o he q-Jackson in eg al (see e.g. [5, 15, 16]). Fo he con inuous o hogonali y see [24, §3.10]. 8 R. ´ ALVAREZ-NODARSE =− b−2 X s=a∆Pn(s) ∆x(s)Qk−1(s)σ(s+ 1)ρ(s+ 1)∇x1(s+ 1/2)+ b X s=a+1 Pn(s) ∆Qk−1(s−1) ∆x(s−1/2) σ(s)ρ(s)∇x1(s) Nex , we use Lemma 2.3 as well as he condi ions (3.9), hen 0 = − b−2 X s=a∆Pn(s) ∆x(s)Qk−1(s)σ(s+ 1)ρ(s+ 1)∇x1(s+1/2)+ b−1 X s=a Pn(s)Rk−2(s)σ(s) |{z } deg ee ≤n ρ(s)∇x1(s) ( om (3.9), (3.5)) =− b−2 X s=a∆Pn(s) ∆x(s)Qk−1(s)σ(s+ 1)ρ(s+ 1)∇x1(s). Thus, ∆Pn(s)/∆x(s) is o hogonal wi h espec o ρ1(s)∇x1(s+ 1/2) = σ(s+ 1)ρ(s+ 1)∆x(s). We only need now o p o e ha ∆(1)Pn(s) is a classical amily. Fo doing his no ice ha he weigh unc ion ρ1(s) sa is y he Pea son ype equa ion (see e.g. [24, §3.2.2]) ∆ ∆x1(s−1/2) [σ(s)ρ1(s)] = τ1(s)ρ1(s) whe e τ1is a i s deg ee polynomial on x(s) gi en by τ1(s) = σ(s+ 1) −σ(s) + τ(s+ 1)∆x1(s) ∆x(s). Thus ρ1sa is ies a di e ence equa ion o he o m (3.6). This comple e he p oo .  In he same way, using induc ion we ha e Co olla y 3.7. Le x(s)be a linea - ype la ice and le (Pn)nbe a classical amily. Then, he sequence o hei k- h ini e di e ences ∆(k)Pn(s), whe e ∆(k):= ∆ ∆xk−1(s) ∆ ∆xk−2(s)... ∆ ∆x(s), also is a classical amily. Now we p o e ha (1)+(2)→(3): P oposi ion 3.8. Le x(s)be a linea - ype la ice. I he sequences (Pn)n and ∆(1)Pnna e classical, hen (Pn)nsa is ies he second o de linea di e ence equa ion o hype geome ic ype (3.7). ON CHARACTERIZATIONS OF CLASSICAL POLYNOMIALS 9 P oo . Le k < n. Then, using he o hogonali y o ∆(1)Pn, 0 = b−2 X s=a ∆Pn(s) ∆x(s) ∆Qk(s) ∆x(s)σ(s+ 1)ρ(s+ 1)∇x1(s+1/2) ( om (3.9)) = b−1 X s=a ∆Pn(s) ∆x(s)∆Qk(s)σ(s+ 1)ρ(s+ 1) ( om (3.3), (3.9)) =− b−1 X s=a Qk(s)∆∆Pn(s−1) ∆x(s−1) σ(s)ρ(s)(∆ (s) = ∇ (s+ 1)) =− b−1 X s=a Qk(s)∆∇Pn(s) ∇x(s)σ(s)ρ(s)( om (3.2)) =− b−1 X s=a Qk(s)σ(s)ρ(s)∆∇Pn(s) ∇x(s)+∇Pn(s+1) ∇x(s+1) ∆[σ(s)ρ(s)] ( om (3.6)) =− b−1 X s=a Qk(s) σ(s)∆ ∆x(s−1/2) ∇Pn(s) ∇x(s)+τ(s)∆Pn(s) ∆x(s)!ρ(s)∇x1(s). Bu , since he la ice x(s) is o he linea ype, Q(s) := σ(s)∆ ∆x(s−1/2) ∇Pn(s) ∇x(s)+τ(s)∆Pn(s) ∆x(s) is a polynomial o deg ee nin x(s). The e o e, i should be, up o a cons an ac o (in gene al depending on n) he polynomial Pn(s). Thus Q(s) = −λPn(s).  Rema k 3.9. The p oo o he las p oposi ion in he linea la ice x(s) = s can be ound in he i s Russian edi ion o he book [25]. The las p oposi ion is e y impo an because i gi es a e y simple me hod o inding he classical polynomials on he linea - ype la ice. In ac , i was he key in he p oo s o Hahn and Lesky o p o ing he Theo em 3.2. The solu ions o he di e ence equa ion (3.7) ha e been ex ensi ely s ud- ied (see e.g. [6, 24, 25]). In pa icula hey can be w i en by he Rod igues- ype o mula (3.8) [24, 25], so (3)→(4). Le us men ion ha om he Rod igues- ype o mula (3.8) one can ob ain an explici exp ession o he classical polynomials in e ms o he hype geome ic o basic hype geome ic se ies as i is shown in se e al p e ious wo ks (see e.g. [6, 24]). Ano he consequence o he Rod igues o mula is he ollowing: Pu ing n= 1 in (3.8) we ob ain P1(s) = B1 ρ(s) ∆ ∆x(s−1/2)[σ(s)ρ(s)] ⇒∆ ∆x(s−1/2)[σ(s)ρ(s)] = ρ(s)τ(s), i.e. he Pea son- ype equa ion (3.6) hus (4)→(5). 16 R. ´ ALVAREZ-NODARSE Bu now, using he exp ession (see e.g. [3, page 108]) cn=λnγn/n, we see ha o all n≥1, cn6= 0. The condi ion p+na 6= 0 o all n∈ N is he admissibili y condi ion in his case. Le us now analyze he s uc u e ela ion (4.12). In his case [3, page 109] gn=−(n−1)aγn p+(n−2)a, he e o e in he quasi-de ini e case gn6= 0. I γn=gn o all n, hen we ob ain ha p+ (2n−3)a= 0, o all nwhich is in con adic ion wi h he admissibili y condi ion. Rema k 4.3. In [10] he condi ion gn6= 0 o all n∈ N was imposed bu no he mo e es ic i e one gn6=γn, om whe e he i s one immedia ely ollows. Fo he disc e e case in [10] he admissibili y condi ion p+na 6= 0 i is assumed and he e o e gn6=γn o all n∈ N . F om he abo e discussion also ollows ha he classical disc e e polyno- mials a e comple ely cha ac e ized by he ela ion (4.12) wi h he es ic ion gn6=γn o all n∈ N. Mo eo e , i gn=γn o all n∈ N, hen he co e- sponding o hogonal polynomial sequence, i such a sequence exis s, is no a classical one. 4.3. The classical case. The classical case can be ob ained om he q-case aking he limi q→1−. Ne e heless he Theo em 1.2 can be p o en using he same scheme sec ion 3. The only di e ence is ha he e one uses he s anda d in eg al calculus and in eg a ion by pa s ins ead o he calculus wi h he di e ence ope a o . O pa icula in e es is he p oo o p ope y 7 so we will p o ide i he e: Taking de i a i es o he TTRR (1.1) and using (1.3), we ha e he exp ession xP0 n(x) = n n+ 1P0 n+1(x) + (βn− n)P0 n(x) + (γn−gn)P0 n−1(x),(4.13) om whe e, i gn6=γn,∀n∈ N, and using he Fa a d heo em he sequence (P0 n)nis an OPS, and he e o e by he Sonin-Hahn Theo em 1.1 Pnis a classical amily. No ice again ha he condi ion gn6=γnshould be imposed. Using he o mulas in [20] i is easy o see ha his condi ion is equi alen o he condi ion nσ00/2 + τ0= 0 which is no hing else ha he admissibili y condi ion o he classical polynomials [20]. Le us poin ou ha he mo e es ic i e condi ion γn6=gn o all n∈Nwas no conside ed in [19] ( hey conside ed only he egula case, i.e., γn6= 0). As in he cases al eady dis- cussed we conclude ha he classical con inuous polynomials a e comple ely cha ac e ized by he ela ion (1.3) wi h he es ic ion gn6=γn o all n∈N. Mo eo e , i gn=γn o n= 1,2,...,n0, hen he co esponding o hogonal polynomial sequence, i such a sequence exis s, is no a classical one. 4.4. The Ma cell´an e al. cha ac e iza ion. A his poin he ollowing ques ion a ises: wha happens i we do no impose he condi ion gn6=γn, ∀n= 1,2,...,n0? The e is any amily o o hogonal polynomials, necessa ily non classical, ha sa is ies he TTRR (1.1) whe e γn6= 0 o n∈ N, and he ela ion (1.3) wi h gn=γn o all n∈ N? i.e., ON CHARACTERIZATIONS OF CLASSICAL POLYNOMIALS 17 Pn(x) = P0 n+1(x) n+ 1 + nP0 n(x) + γnP0 n−1(x).(4.14) To answe his ques ion we can use (4.13) bu ew i en in he o m4 P0 n+1(x) = n+ 1 n(x−βn+ n)P0 n(x), ha leads o P0 n(x) = n n−1 Y j=1 (x−βj+ j), n ≥2. The e o e, subs i u ing he las exp ession in (4.14) we ge , deno ing ξj= βj− j, Pn(x) = [(x−ξn)(x−ξn−1) + n n(x−ξn−1) + (n−1)γn] n−2 Y j=1 (x−ξj). Bu his implies ha o n≥3, wo consecu i e polynomials ha e common ze os ha is a con adic ion. The e o e he e is no any amily o o hogonal polynomials ha sa is y (4.14). Fo he linea la ices x(s) = sand x(s) = qs he si ua ion is he same. We p esen he e he compu a ions only o he q-case, he o he case is analogous —in ac he inal exp ession o he polynomials Pncoincide wi h he one in he classical “con inuous” case. Fo he q-case we p oceed as be o e, i.e., we ake he q-de i a i es o he TTRR (3.1) and use he ela ion (4.3) whe e en= 1/[n]q,gn=γn,F(1) = 1, G(1) = 0, we ob ain DqPn+1(x) = [n+ 1]q [n]q (x−ξn/q), ξj=βj− j. Subs i u ing i in (4.3) when gn=γnwe ob ain he ollowing exp ession o he polynomials Pn Pn(x) = [(x−ξn/q)(x−ξn−1/q) + [n]q n(x−ξn−1/q) + [n−1]qγn] n−2 Y j=1 (x−ξj/q). As be o e, om his exp ession ollows ha o n≥3, wo consecu i e polynomials has common ze os, ha is in con adic ion wi h he ac ha hey cons i u es an o hogonal sequence. F om he abo e discussion ollows ha he s uc u e ela ion (3.11) when gn6=γn o all n∈ N comple ely cha ac e izes he classical o hogonal polynomials. 4As in sec ion 4.3 we will ake he de i a i e o he TTRR (1.1) bu now use (4.14). 18 R. ´ ALVAREZ-NODARSE Acknowledgemen s. The au ho hanks L. Ca doso, J.S. Dehesa, A. Du ´an, F. Ma cell´an, J. C. Medem, and J.C. Pe onilho o s imula ing discussions. This wo k was suppo ed by he Minis e io de Ciencia y Tecnolog´ıa o Spain unde he g an BFM-2003-06335-C03, and he Jun a de Andaluc´ıa unde g an FQM-262. Also he inancial suppo by Acciones In eg adas Hispano- Lusas HP2002-065 & E-6/03 is acknowledged. Appendix A. The classical polynomials In his appendix we will p esen he p oo o he Theo em 1.2. We will ollow he same scheme in Sec ion 3 (see igu e 1). As s a ing poin we will use he Pea son equa ion, i.e., we say ha he classical polynomials a e he polynomials o hogonal wi h espec o a con- inuous weigh unc ion ρsuppo ed in he in e al (a, b), solu ion o he Pea son equa ion [σ(x)ρ(x)]0=τ(x)ρ(x),(A.1) whe e σand τa e polynomials o deg ee a leas wo and exac ly one, e- spec i ely, and such ha he ollowing bounda y condi ions hold5σ(a)ρ(a) = σ(b)ρ(b) = 0. (1)→(2): Using he o hogonali y o he classical amily (Pn)nwi h espec o ρwe ha e ha o any polynomial o deg ee less han o equal o k−1, Qk−1, wi h k < n, 0 = Zb a Pn(x)Qk−1(x)τ(x) |{z } deg ee≤k<n ρ(x)dx =Zb a Pn(x)Qk−1(x)[σ(x)ρ(x)]0dx =Pn(x)Qk−1(x)σ(x)ρ(x)b a |{z } =0 −Zb a [Pn(x)Qk−1(x)]0σ(x)ρ(x)dx =−Zb a Pn(x) deg ee<n z}| { Q0 k−1(x)σ(x)ρ(x)dx | {z } =0 −Zb a P0 n(x)Qk−1(x)[σ(x)ρ(x)]dx. Thus P0 nis o hogonal o any polynomial o deg ee k−1< n −1, i.e., (P0 n)n is also an o hogonal amily. Fu he mo e, since he weigh unc ion o he sequence (P0 n)nis ρ1(x) = σ(x)ρ(x), we ha e ha hey sa is y he equa ion [σ(x)ρ1(x)]0= [τ(x) + σ0(x)]ρ1(x), i.e., a Pea son equa ion (A.1). 5These condi ions ollow om he ac ha o he classical amilies he momen s µn=Rb axnρ(x)dx,n≥0, o he measu e associa ed wi h ρ(x) a e be ini e. ON CHARACTERIZATIONS OF CLASSICAL POLYNOMIALS 19 (2)→(3): We use now ha (P0 n)nis an o hogonal amily wi h espec o he weigh unc ion ρ1(x) = σ(x)ρ(x). Thus 0 = Zb a P0 n(x)Q0 k(x)τ(x)ρ1(x)dx =P0 n(x)Qk(x)σ(x)ρ(x)b a |{z } =0 −Zb a [σ(x)ρ(x)P0 n(x)]0Qk(x)dx =−Zb a Qk(x){[σ(x)ρ(x)]0 |{z } =τ(x)ρ(x) P0 n(x) + σ(x)ρ(x)P00 n(x)} =Zb a Qk(x)[σ(x)P00 n(x) + τ(x)P0 n(x)]ρ(x)dx. Bu since he las in eg al anishes o e e y polynomial Qko deg ee k < n hen σ(x)P00 n(x)+τ(x)P00 n(x) should be p opo ional o Pn, i.e., σ(x)P00 n(x)+ τ(x)P00 n(x) = −λnPn, whe e λnis a cons an , in gene al depending on n. (3)→(4): The solu ion o he abo e di e en ial equa ion can be w i en in he ollowing compac o m (see e.g. [25, §2] o [24, §1.2]) usually called he Rod igues o mula Pn(x) = Bn ρ(x) dn dxn[σn(x)ρ(x)], whe e Bnis a cons an . (4)→(5): I ollows om he Rod igues o mula jus pu ing n= 1. (4)→(6): F om he Rod igues o mula he ollowing exp ession (see e.g. [25, Eq. (7) page 25]) immedia ely ollows σ(x)P0 n(x) = λn nτ0 nτn(x)Pn(x)−Bn Bn+1 Pn+1(x), τn(x) = τ(x) + nσ0(x), om whe e, using he h ee- e m ecu ence ela ion o he amily (Pn)n he s uc u e ela ion (1.2) ollows. (6)→(2): Suppose ha (1.2) holds whe e deg σ≤2 and (Pn)nis an o hog- onal amily. No ice ha he in eg al Zb a Qk(x)P0 n(x)σ(x)ρ(x)dx= Zb a Qk(x)ρ(x)[anPn+1(x)+bnPn(x)+cnPn−1(x)]dx anishes o all k < n −1. Then (P0 n)nis an o hogonal amily wi h espec o he weigh unc ion ρ1(x) = σ(x)ρ(x) and he e o e by he Sonin-Hahn Theo em 1.1 (Pn)nis a classical amily. (1)+(2)→(7): Fo p o ing his we suppose ha (Pn)nand (P0 n)na e o hog- onal wi h espec o ρ(x) and ρ1(x) = σ(x)ρ(x), espec i ely. I (Pn)nis a monic sequence hen Pn(x) = 1 n+ 1P0 n+1 + nP0 n(x) + gnP0 n−1(x) + n−2 X k=1 ck(n)P0 k(x). 20 R. ´ ALVAREZ-NODARSE Bu ck(n) = Rb aPn(x)P0 k(x)σ(x)ρ(x)dx Rb a[P0 k(x)]2σ(x)ρ(x)dx = 0, since deg P0 kσ≤k+ 1 < n −2 and (Pn)nis and o hogonal amily wi h espec o ρ(x). Finally he p oo (7)→(2) is p esen ed in sec ion 4.3. Re e ences [1] W. A. Al-Salam, Cha ac e iza ion heo ems o o hogonal polynomials. In: O hog- onal Polynomials: Theo y and P ac ice. P. Ne ai (Ed.) NATO ASI Se ies C, Vol. 294. Kluwe Acad. Publ., Do d ech , 1990, 1-24. [2] W. A. Al-Salam and T. S. Chiha a, Ano he cha ac e iza ion o he classical o hog- onal polynomials. SIAM J. Ma h. Anal. 3(1972) 65-70. [3] R. ´ Al a ez-Noda se, Polinomios hipe geom´e icos y q-polinomios. Monog a ´ıas del Semina io Ma em´a ico “Ga c´ıa de Galdeano” Vol. 26. 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