On q-polynomials and some of their applications
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On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse IMUS & Dpto. An´alisis Matem´atico, Universidad de Sevilla Selected Topics in Mathematical Physics In honor of Professor Natig Atakishiyev Instituto de Matem´aticas, Cuernavaca, UNAM, 28–30 November 2016 On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 1/37
Why to talk about q-polynomials today? This Conference was organized in honor to Natig Atakishiyev. Natig was born in Azerbaijan and in 1991 moved to M´exico when he is living now. He has 109 papers reviewed in mathscinet and more than 100 in Journals of the Citation Index. He is very well known for his mathematical works on SF and OP and specially for his important contributions to the theory of q-polynomials but also for his works related with different kind of harmonic quantum oscillators. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 2/37
Why to talk about q-polynomials today? This Conference was organized in honor to Natig Atakishiyev. Natig was born in Azerbaijan and in 1991 moved to M´exico when he is living now. He has 109 papers reviewed in mathscinet and more than 100 in Journals of the Citation Index. He is very well known for his mathematical works on SF and OP and specially for his important contributions to the theory of q-polynomials but also for his works related with different kind of harmonic quantum oscillators. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 2/37
Why to talk about q-polynomials today? This Conference was organized in honor to Natig Atakishiyev. Natig was born in Azerbaijan and in 1991 moved to M´exico when he is living now. He has 109 papers reviewed in mathscinet and more than 100 in Journals of the Citation Index. He is very well known for his mathematical works on SF and OP and specially for his important contributions to the theory of q-polynomials but also for his works related with different kind of harmonic quantum oscillators. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 2/37
On Special Functions ... Special Functions (SF) appear in (almost) all context of Mathematics and other Sciences. As Alberto Grunbaum one time said: “Special functions are to mathematics what pipes are to a house: nobody wants to exhibit them openly but nothing works without them”. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 3/37
q-Polynomials are special cases of OP⊂SF Definition 1 Given a sequence of normal pol. (Pn)nwe said that (Pn)nis an OPS w.r.t. µif ∀n6=m∈N, ZR Pn(x)Pm(x)dµ(x) = δn,mdn,dn6= 0 If µis a positive measure ⇒dn>0∀n,⇒we said that SOP is positive definite. IIf dµ(x) = ρ(x)dx ⇒ρis a continuous weight function IIf dµ(x) = Pkδ(x−xk)ρ(xk)dx ⇒ρis a discrete weight function On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 4/37
q-Polynomials are special cases of OP⊂SF Definition 1 Given a sequence of normal pol. (Pn)nwe said that (Pn)nis an OPS w.r.t. µif ∀n6=m∈N, ZR Pn(x)Pm(x)dµ(x) = δn,mdn,dn6= 0 If µis a positive measure ⇒dn>0∀n,⇒we said that SOP is positive definite. IIf dµ(x) = ρ(x)dx ⇒ρis a continuous weight function IIf dµ(x) = Pkδ(x−xk)ρ(xk)dx ⇒ρis a discrete weight function On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 4/37
q-Polynomials are special cases of OP⊂SF Definition 1 Given a sequence of normal pol. (Pn)nwe said that (Pn)nis an OPS w.r.t. µif ∀n6=m∈N, ZR Pn(x)Pm(x)dµ(x) = δn,mdn,dn6= 0 If µis a positive measure ⇒dn>0∀n,⇒we said that SOP is positive definite. IIf dµ(x) = ρ(x)dx ⇒ρis a continuous weight function IIf dµ(x) = Pkδ(x−xk)ρ(xk)dx ⇒ρis a discrete weight function On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 4/37
TTRR: A characterization of an OPS If ZR Pn(x)Pm(x)dµ(x) = δn,m⇒ ∃ (an)ny (bn)nsuch that xPn(x) = an+1Pn+1(x) + bnPn(x) + anPn−1(x),n≥0, ¿There exists a converse result? ¿There are any other characterizations? On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 5/37
Some characterizations of classical OP Bochner (1929): They are the only solution of σ(x)y00(x) + τ(x)y0(x) + λny(x) = 0,deg σ≤2,deg τ= 1. This is the hypergeometric equation! Tricomi (1955): They satisfy the Rodrigues Eq. Pn(x) = Bn ρ(x) dn dxn[ρ(x)σn(x)] ,n= 0,1,2,··· ρ(x)≥0(FR) Hildebrandt (1931): ρsatisfy the Pearson Eq.: [ρ(x)σ(x)]0=τ(x)ρ(x),deg(σ)≤2,deg(τ)=1 etc. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 7/37
Some characterizations of classical OP Bochner (1929): They are the only solution of σ(x)y00(x) + τ(x)y0(x) + λny(x) = 0,deg σ≤2,deg τ= 1. This is the hypergeometric equation! Tricomi (1955): They satisfy the Rodrigues Eq. Pn(x) = Bn ρ(x) dn dxn[ρ(x)σn(x)] ,n= 0,1,2,··· ρ(x)≥0(FR) Hildebrandt (1931): ρsatisfy the Pearson Eq.: [ρ(x)σ(x)]0=τ(x)ρ(x),deg(σ)≤2,deg(τ)=1 etc. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 7/37
Some characterizations of classical OP Bochner (1929): They are the only solution of σ(x)y00(x) + τ(x)y0(x) + λny(x) = 0,deg σ≤2,deg τ= 1. This is the hypergeometric equation! Tricomi (1955): They satisfy the Rodrigues Eq. Pn(x) = Bn ρ(x) dn dxn[ρ(x)σn(x)] ,n= 0,1,2,··· ρ(x)≥0(FR) Hildebrandt (1931): ρsatisfy the Pearson Eq.: [ρ(x)σ(x)]0=τ(x)ρ(x),deg(σ)≤2,deg(τ)=1 etc. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 7/37
The q−polynomials are extensions of the classical ones The first families: Markov 1884 Stieltjes 1898 in connection with the moment problem: Stieltjes-Wiegert OP The Hahn problem (1949): Let be the OPS (Pn)nand define the operator Θw q:R7→ R,Θw qp(x) = p(qx +w)−p(x) (q−1)x+w Find, if there exist, the OPS such that: 1. (Θw qPn(x))nis and OPS 2. σ(x)Θw qΘw q−1Pn(x) + τ(x)Θw qPn(x) + λPn(x)=0 (DE) 3. ∃π∈Pyρt.q. ρ(x)Pn(x) = [Θw q]n[π(x)ρ(x)] (RF) If w= 0 and q→1⇒Θw qf(x)→d dx : Clasical case! On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 8/37
The q−polynomials are extensions of the classical ones The first families: Markov 1884 Stieltjes 1898 in connection with the moment problem: Stieltjes-Wiegert OP The Hahn problem (1949): Let be the OPS (Pn)nand define the operator Θw q:R7→ R,Θw qp(x) = p(qx +w)−p(x) (q−1)x+w Find, if there exist, the OPS such that: 1. (Θw qPn(x))nis and OPS 2. σ(x)Θw qΘw q−1Pn(x) + τ(x)Θw qPn(x) + λPn(x)=0 (DE) 3. ∃π∈Pyρt.q. ρ(x)Pn(x) = [Θw q]n[π(x)ρ(x)] (RF) If w= 0 and q→1⇒Θw qf(x)→d dx : Clasical case! On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 8/37
The q−polynomials are extensions of the classical ones The first families: Markov 1884 Stieltjes 1898 in connection with the moment problem: Stieltjes-Wiegert OP The Hahn problem (1949): Let be the OPS (Pn)nand define the operator Θw q:R7→ R,Θw qp(x) = p(qx +w)−p(x) (q−1)x+w Find, if there exist, the OPS such that: 1. (Θw qPn(x))nis and OPS 2. σ(x)Θw qΘw q−1Pn(x) + τ(x)Θw qPn(x) + λPn(x)=0 (DE) 3. ∃π∈Pyρt.q. ρ(x)Pn(x) = [Θw q]n[π(x)ρ(x)] (RF) If w= 0 and q→1⇒Θw qf(x)→d dx : Clasical case! On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 8/37
The q−polynomials are extensions of the classical ones The first families: Markov 1884 Stieltjes 1898 in connection with the moment problem: Stieltjes-Wiegert OP The Hahn problem (1949): Let be the OPS (Pn)nand define the operator Θw q:R7→ R,Θw qp(x) = p(qx +w)−p(x) (q−1)x+w Find, if there exist, the OPS such that: 1. (Θw qPn(x))nis and OPS 2. σ(x)Θw qΘw q−1Pn(x) + τ(x)Θw qPn(x) + λPn(x)=0 (DE) 3. ∃π∈Pyρt.q. ρ(x)Pn(x) = [Θw q]n[π(x)ρ(x)] (RF) If w= 0 and q→1⇒Θw qf(x)→d dx : Clasical case! On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 8/37
The q−polynomials are extensions of the classical ones The first families: Markov 1884 Stieltjes 1898 in connection with the moment problem: Stieltjes-Wiegert OP The Hahn problem (1949): Let be the OPS (Pn)nand define the operator Θw q:R7→ R,Θw qp(x) = p(qx +w)−p(x) (q−1)x+w Find, if there exist, the OPS such that: 1. (Θw qPn(x))nis and OPS 2. σ(x)Θw qΘw q−1Pn(x) + τ(x)Θw qPn(x) + λPn(x)=0 (DE) 3. ∃π∈Pyρt.q. ρ(x)Pn(x) = [Θw q]n[π(x)ρ(x)] (RF) If w= 0 and q→1⇒Θw qf(x)→d dx : Clasical case! On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 8/37
“Discrete” polynomials and q−polynomials Case q= 1 and w= 1 ⇒“discrete” (Lesky, 1962) Θw qf(x)=∆f(x) := f(x+ 1) −f(x),∇f(x)=∆f(x−1) σ(x)∆∇Pn(x) + τ(x)∆Pn(x) + λPn(x) = 0 Case q∈(0,1) y w= 0 ⇒q’s (Hahn 1949, . . . ) Θw qf(x) := Θqf(x) = f(qx)−f(x) (q−1)x σ(x)ΘqΘq−1Pn(x) + τ(x)ΘqPn(x) + λPn(x) = 0 In the next years the q-polynomials appeared in several contexts. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 9/37
“Discrete” polynomials and q−polynomials Case q= 1 and w= 1 ⇒“discrete” (Lesky, 1962) Θw qf(x)=∆f(x) := f(x+ 1) −f(x),∇f(x)=∆f(x−1) σ(x)∆∇Pn(x) + τ(x)∆Pn(x) + λPn(x) = 0 Case q∈(0,1) y w= 0 ⇒q’s (Hahn 1949, . . . ) Θw qf(x) := Θqf(x) = f(qx)−f(x) (q−1)x σ(x)ΘqΘq−1Pn(x) + τ(x)ΘqPn(x) + λPn(x) = 0 In the next years the q-polynomials appeared in several contexts. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 9/37
The q-hypergemetric Eq. of NU eσy00(x) + eτy0(x) + λy(x)=0 ⇓ σ(s)∆ ∆x(s−1 2)∇y(s) ∇x(s)+τ(s)∆y(s) ∆x(s)+λny(s) = 0 ∇f(s) = f(s)−f(s−1),∆f(s) = f(s+ 1) −f(s) σ(s) = ˜σ(x(s)) −1 2˜τ(x(s))∆x(s−1 2), τ(s) = ˜τ(x(s)). There are nice polynomial solution for any function x(s)? x(s) = c1(q)qs+c2(q)q−s+c3(q) = c1(q)[qs+q−s−µ] + c3(q) Sufficient cond. NU (1983). Necessary cond. Atakishiyev, Rahman y Suslov Const. Appr. (1993). On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 14/37
The q-hypergemetric Eq. of NU eσy00(x) + eτy0(x) + λy(x)=0 ⇓ σ(s)∆ ∆x(s−1 2)∇y(s) ∇x(s)+τ(s)∆y(s) ∆x(s)+λny(s) = 0 ∇f(s) = f(s)−f(s−1),∆f(s) = f(s+ 1) −f(s) σ(s) = ˜σ(x(s)) −1 2˜τ(x(s))∆x(s−1 2), τ(s) = ˜τ(x(s)). There are nice polynomial solution for any function x(s)? x(s) = c1(q)qs+c2(q)q−s+c3(q) = c1(q)[qs+q−s−µ] + c3(q) Sufficient cond. NU (1983). Necessary cond. Atakishiyev, Rahman y Suslov Const. Appr. (1993). On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 14/37
The q-hypergemetric Eq. of NU eσy00(x) + eτy0(x) + λy(x)=0 ⇓ σ(s)∆ ∆x(s−1 2)∇y(s) ∇x(s)+τ(s)∆y(s) ∆x(s)+λny(s) = 0 ∇f(s) = f(s)−f(s−1),∆f(s) = f(s+ 1) −f(s) σ(s) = ˜σ(x(s)) −1 2˜τ(x(s))∆x(s−1 2), τ(s) = ˜τ(x(s)). There are nice polynomial solution for any function x(s)? x(s) = c1(q)qs+c2(q)q−s+c3(q) = c1(q)[qs+q−s−µ] + c3(q) Sufficient cond. NU (1983). Necessary cond. Atakishiyev, Rahman y Suslov Const. Appr. (1993). On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 14/37
Some basic properties IThe q-analogue of the Rodrigues formula Pn(s) = Bn ρ(s)∇ ∇x(s+1 2)··· ∇ ∇x(s+n 2) | {z } ∇(n) "ρ(s+n) n Y m=1 σ(s+m)# | {z } ρn(s) where ρ(s) if the sol. of ∆ [σ(x)ρ(s)] = τ(s)ρ(s)∆x(s−1 2), IDifferentiation or ladder-type formulas σ(s)∇Pn(x(s))q ∇x(s)=λn [n]q τn(s) τ0 n Pn(x(s))q−αnλ2n [2n]q Pn+1(x(s))q For linear-type lattices x(s+α) = A(α)x(s) + B(α) (q-Hahn Tableau) there is a complete study in Medem, et. al. JCAM (2001) and RAN, JCAM (2006). For the general case see Foupouagnigni et al. Integral Transforms Spec. Funct. (2011). On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 15/37
Some basic properties IThe q-analogue of the Rodrigues formula Pn(s) = Bn ρ(s)∇ ∇x(s+1 2)··· ∇ ∇x(s+n 2) |{z } ∇(n) "ρ(s+n) n Y m=1 σ(s+m)# | {z } ρn(s) where ρ(s) if the sol. of ∆ [σ(x)ρ(s)] = τ(s)ρ(s)∆x(s−1 2), IDifferentiation or ladder-type formulas τn(s) = τ0 nx(s+n 2) + τn(0) σ(s)∇Pn(x(s))q ∇x(s)=λn [n]q τn(s) τ0 n Pn(x(s))q−αnλ2n [2n]q Pn+1(x(s))q For linear-type lattices x(s+α) = A(α)x(s) + B(α) (q-Hahn Tableau) there is a complete study in Medem, et. al. JCAM (2001) and RAN, JCAM (2006). For the general case see Foupouagnigni et al. Integral Transforms Spec. Funct. (2011). On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 15/37
Some basic properties IThe q-analogue of the Rodrigues formula Pn(s) = Bn ρ(s)∇ ∇x(s+1 2)··· ∇ ∇x(s+n 2) |{z } ∇(n) "ρ(s+n) n Y m=1 σ(s+m)# | {z } ρn(s) where ρ(s) if the sol. of ∆ [σ(x)ρ(s)] = τ(s)ρ(s)∆x(s−1 2), IDifferentiation or ladder-type formulas τn(s) = τ0 nx(s+n 2)+τn(0) σ(s)∇Pn(x(s))q ∇x(s)=λn [n]q τn(s) τ0 n Pn(x(s))q−αnλ2n [2n]q Pn+1(x(s))q For linear-type lattices x(s+α) = A(α)x(s) + B(α) (q-Hahn Tableau) there is a complete study in Medem, et. al. JCAM (2001) and RAN, JCAM (2006). For the general case see Foupouagnigni et al. Integral Transforms Spec. Funct. (2011). On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 15/37
Some basic properties IThe q-analogue of the Rodrigues formula Pn(s) = Bn ρ(s)∇ ∇x(s+1 2)··· ∇ ∇x(s+n 2) |{z } ∇(n) "ρ(s+n) n Y m=1 σ(s+m)# | {z } ρn(s) where ρ(s) if the sol. of ∆ [σ(x)ρ(s)] = τ(s)ρ(s)∆x(s−1 2), IDifferentiation or ladder-type formulas τn(s) = τ0 nx(s+n 2)+τn(0) σ(s)∇Pn(x(s))q ∇x(s)=λn [n]q τn(s) τ0 n Pn(x(s))q−αnλ2n [2n]q Pn+1(x(s))q For linear-type lattices x(s+α) = A(α)x(s) + B(α) (q-Hahn Tableau) there is a complete study in Medem, et. al. JCAM (2001) and RAN, JCAM (2006). For the general case see Foupouagnigni et al. Integral Transforms Spec. Funct. (2011). On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 15/37
A corollary of the NU Eq. Let x(s) = c1(q)[qs+q−s−µ] + c3(q) The most general case of the NU Eq. corresponds to the choice: σ(s) = q−2s(qs−qs1)(qs−qs2)(qs−qs3)(qs−qs4). and the corresponding general polynomial solution can be expressed in term of basic hypergeometric series Pn(s) = 4φ3q−n,q2µ+n−1+s1+s2+s3+s4,qs1−s,qs1+s+µ qs1+s2+µ,qs1+s3+µ,qs1+s4+µq,q From the above solution we can obtain Askey-Wilson, q-Racah, q-duales de Hahn, q-Hahn, . . . NU Integral Transforms Spec. Funct. (1993); Atakishiyev, Rahman y Suslov Const. Appr. (1993). On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 16/37
The q-hypergemetric Eq. of NU: A final remark There are a series of interesting papers by Natig (some of then with other people) that further developed the theory initiated by NU: The study of the orthogonality of Askey-Wilson polynomials The moments of the weight functions of q-polynomials The study of the continuous orthogonality of the solutions of the NU Eq. including the discrete case. etc. In 1998 Koekoek and Swarttouw compiled in a report all known families of q-polynomials that was called the q-Askey Tableu. q-Askey ⊂Nikiforov-Uvarov Tableau All q-OP are in the q-Askey tableau? NO In RAN, Medem JCAM (2001) we found two new families within the q-Hahn tableau. One of then is a positive definite case that has been recently studied by Area et. al. (2016). On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 17/37
The q-hypergemetric Eq. of NU: A final remark There are a series of interesting papers by Natig (some of then with other people) that further developed the theory initiated by NU: The study of the orthogonality of Askey-Wilson polynomials The moments of the weight functions of q-polynomials The study of the continuous orthogonality of the solutions of the NU Eq. including the discrete case. etc. In 1998 Koekoek and Swarttouw compiled in a report all known families of q-polynomials that was called the q-Askey Tableu. q-Askey ⊂Nikiforov-Uvarov Tableau All q-OP are in the q-Askey tableau? NO In RAN, Medem JCAM (2001) we found two new families within the q-Hahn tableau. One of then is a positive definite case that has been recently studied by Area et. al. (2016). On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 17/37
Factorization method of Schr¨odinger 1940, Infeld and Hull 1951 Given a Hamiltonian, that is a 2oorder diff. operator Hϕn=λnϕn,Hde orden 2 To find 1oorder diff operators aand a+such that H=a+a,a+ϕn=αnϕn+1,aϕn=βnϕn−1,(a+)∗=a,a∗=a+. Interest: Solving aϕ0= 0, one gets ϕ0, and a+ϕngenerate the others On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 19/37
The typical example is the quantum harmonic oscillator HΨn(x) := −Ψ00 n(x) + x2Ψn(x)=(H−H++I)Ψn(x) = λnΨn(x). a:= H+=x+d dx ,a+:= H−=x−d dx m H+Ψ0= 0,H+Ψn=√2nΨn−1,H−Ψn=√2n+ 2 Ψn+1. How it works? xΨ0(x)+Ψ0 0(x)=0 ⇒Ψ0(x) = 1 4 √πe−x2/2, and [H−]nΨ0(x) = p(2n)!!Ψn(x), thus Ψn(x) = 1 π1 4p(2n)!![H−]ne−x2/2=1 π1 4p(2n)!! xI−d dx n e−x2/2. This the classical algebraic realization of the quantum oscillator. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 20/37
The typical example is the quantum harmonic oscillator HΨn(x) := −Ψ00 n(x) + x2Ψn(x)=(H−H++I)Ψn(x) = λnΨn(x). a:= H+=x+d dx ,a+:= H−=x−d dx m H+Ψ0= 0,H+Ψn=√2nΨn−1,H−Ψn=√2n+ 2 Ψn+1. How it works? xΨ0(x)+Ψ0 0(x)=0 ⇒Ψ0(x) = 1 4 √πe−x2/2, and [H−]nΨ0(x) = p(2n)!!Ψn(x), thus Ψn(x) = 1 π1 4p(2n)!![H−]ne−x2/2=1 π1 4p(2n)!! xI−d dx n e−x2/2. This the classical algebraic realization of the quantum oscillator. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 20/37
The typical example is the quantum harmonic oscillator HΨn(x) := −Ψ00 n(x) + x2Ψn(x)=(H−H++I)Ψn(x) = λnΨn(x). a:= H+=x+d dx ,a+:= H−=x−d dx m H+Ψ0= 0,H+Ψn=√2nΨn−1,H−Ψn=√2n+ 2 Ψn+1. How it works? xΨ0(x)+Ψ0 0(x)=0 ⇒Ψ0(x) = 1 4 √πe−x2/2, and [H−]nΨ0(x) = p(2n)!!Ψn(x), thus Ψn(x) = 1 π1 4p(2n)!![H−]ne−x2/2=1 π1 4p(2n)!! xI−d dx n e−x2/2. This the classical algebraic realization of the quantum oscillator. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 20/37
Factorization of the NU equation Bankerezako (Askey-Wilson Case, JCAM 1999) Lorente (Continuous and discrete classical pols., JPA 2001) RAN, Costas Santos (NU Eq. JPA 2001): ϕn(s) = sρ(s) d2 n Pn(x(s))q,H(s,n)ϕn(s)=0 H(s,n)≡√σ(−s−µ+1)σ(s) ∇x(s)e−∂s+√σ(−s−µ)σ(s+1) ∆x(s)e∂s− σ(−s−µ) ∆x(s)+σ(s) ∇x(s)−λn∆x(s−1/2)I. Main properties: 1The orthonormal functions ϕnsatisfy a 2odiff Eq. & TTRR 2There exist tow ladder oerators: L+(s,n)ϕn(s) = Anϕn+1(s) and L−(s,n)ϕn(s) = Bnϕn−1(s) On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 21/37
Factorization of the NU equation Bankerezako (Askey-Wilson Case, JCAM 1999) Lorente (Continuous and discrete classical pols., JPA 2001) RAN, Costas Santos (NU Eq. JPA 2001): ϕn(s) = sρ(s) d2 n Pn(x(s))q,H(s,n)ϕn(s)=0 H(s,n)≡√σ(−s−µ+1)σ(s) ∇x(s)e−∂s+√σ(−s−µ)σ(s+1) ∆x(s)e∂s− σ(−s−µ) ∆x(s)+σ(s) ∇x(s)−λn∆x(s−1/2)I. Main properties: 1The orthonormal functions ϕnsatisfy a 2odiff Eq. & TTRR 2There exist tow ladder oerators: L+(s,n)ϕn(s) = Anϕn+1(s) and L−(s,n)ϕn(s) = Bnϕn−1(s) On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 21/37
Factorization of the NU equation H(s,n)≡pσ(−s−µ+ 1)σ(s)1 ∇x(s)e−∂s+pΘ(s)σ(s+ 1) 1 ∆x(s)e∂s− σ(−s−µ) ∆x(s)+σ(s) ∇x(s)−λn∆x(s−1/2)I. Theorem: The operator H(s,n) admits the following factorization u(s+ 1,n)H(s,n) = L−(s,n+ 1)L+(s,n)−h∓(n)I, u(s,n)H(s,n+ 1) = L+(s,n)L−(s,n+ 1) −h∓(n)I, respectively, where h±(n) = λ2n−2 [2n−2]q λ2n [2n]qαn−1γn,u(s,n) = λn [n]q τn(s) τ0 n−σ(s) ∇x(s) where αand γare the coeff. of the TTRR. This is not a good solution to the problem. Why? On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 22/37
Factorization of the NU equation H(s,n)≡pσ(−s−µ+ 1)σ(s)1 ∇x(s)e−∂s+pΘ(s)σ(s+ 1) 1 ∆x(s)e∂s− σ(−s−µ) ∆x(s)+σ(s) ∇x(s)−λn∆x(s−1/2)I. Theorem: The operator H(s,n) admits the following factorization u(s+ 1,n)H(s,n) = L−(s,n+ 1)L+(s,n)−h∓(n)I, u(s,n)H(s,n+ 1) = L+(s,n)L−(s,n+ 1) −h∓(n)I, respectively, where h±(n) = λ2n−2 [2n−2]q λ2n [2n]qαn−1γn,u(s,n) = λn [n]q τn(s) τ0 n−σ(s) ∇x(s) where αand γare the coeff. of the TTRR. This is not a good solution to the problem. Why? On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 22/37
Motivation Going further [a,a+] := aa+−a+a=I In fact we have: [a,H] = a, [a+,H] = −a+ Thus a,a+,Hand Iform a closed Lie algebra: ⇒ THE DYNAMICAL ALGEBRA of the HO and the SODE The problem of build such dynamical algebra from the NU Eq. was proposed by Atakishiyev in 2002 motivated by the previous works of MacFarlane 1989, Biedenharn 1989, Atakishiyev et at 1991, 1994, 1996, ... J+=α√a+a a+J−=βa√a+a J0=γH On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 23/37
Motivation Going further [a,a+] := aa+−a+a=I In fact we have: [a,H] = a, [a+,H] = −a+ Thus a,a+,Hand Iform a closed Lie algebra: ⇒ THE DYNAMICAL ALGEBRA of the HO and the SODE The problem of build such dynamical algebra from the NU Eq. was proposed by Atakishiyev in 2002 motivated by the previous works of MacFarlane 1989, Biedenharn 1989, Atakishiyev et at 1991, 1994, 1996, ... J+=α√a+a a+J−=βa√a+a J0=γH On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 23/37
Motivation Going further [a,a+] := aa+−a+a=I In fact we have: [a,H] = a, [a+,H] = −a+ Thus a,a+,Hand Iform a closed Lie algebra: ⇒ THE DYNAMICAL ALGEBRA of the HO and the SODE The problem of build such dynamical algebra from the NU Eq. was proposed by Atakishiyev in 2002 motivated by the previous works of MacFarlane 1989, Biedenharn 1989, Atakishiyev et at 1991, 1994, 1996, ... J+=α√a+a a+J−=βa√a+a J0=γH α= 1 β= 1 γ= 1 [J0,J±] = ±J±[J+,J−] = −2J0SU(1,1) On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 23/37
The q-wave functions ϕnand the q-Hamiltonian Hq Hq(s)ϕn(s) = λnϕn(s), Hq(s) := 1 ∇x1(s)A(s)Hq(s)1 A(s), ϕn(s) = A(s)pρ(s) dn Pn(s;q), where Hq(s) := −pσ(−s−µ+1)σ(s) ∇x(s)e−∂s−pσ(−s−µ)σ(s+ 1) ∆x(s)e∂s +σ(−s−µ) ∆x(s)+σ(s) ∇x(s)I, The next step is to find two operators a(s) and b(s) such that Hq(s) = b(s)a(s) We will follow an original idea by Atakishiyev: On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 24/37
The q-wave functions ϕnand the q-Hamiltonian Hq Hq(s)ϕn(s) = λnϕn(s), Hq(s) := 1 ∇x1(s)A(s)Hq(s)1 A(s), ϕn(s) = A(s)pρ(s) dn Pn(s;q), where Hq(s) := −pσ(−s−µ+1)σ(s) ∇x(s)e−∂s−pσ(−s−µ)σ(s+ 1) ∆x(s)e∂s +σ(−s−µ) ∆x(s)+σ(s) ∇x(s)I, The next step is to find two operators a(s) and b(s) such that Hq(s) = b(s)a(s) We will follow an original idea by Atakishiyev: On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 24/37
The α-operators In order to factorize an arbitrary difference equation, one should express it explicitly in terms of the shift operators exp(ad ds ), defined as exp(ad ds )f(s) = f(s+a),a∈C. Let α∈Rand A(s) and B(s) are continuous functions. We define a family of α-down and α-up operators by a↓ α(s):= B(s) p∇x1(s)e−α∂s e∂ssσ(s) ∇x(s)−sσ(−s−µ) ∆x(s)!1 A(s), a↑ α(s):= A(s) ∇x1(s) sσ(s) ∇x(s)e−∂s−sσ(−s−µ) ∆x(s)!eα∂sp∇x1(s) B(s). Hq(s) = a↑ α(s)a↓ α(s),∀α∈R,and B(s). On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 25/37
The α-operators In order to factorize an arbitrary difference equation, one should express it explicitly in terms of the shift operators exp(ad ds ), defined as exp(ad ds )f(s) = f(s+a),a∈C. Let α∈Rand A(s) and B(s) are continuous functions. We define a family of α-down and α-up operators by a↓ α(s):= B(s) p∇x1(s)e−α∂s e∂ssσ(s) ∇x(s)−sσ(−s−µ) ∆x(s)!1 A(s), a↑ α(s):= A(s) ∇x1(s) sσ(s) ∇x(s)e−∂s−sσ(−s−µ) ∆x(s)!eα∂sp∇x1(s) B(s). Hq(s) = a↑ α(s)a↓ α(s),∀α∈R,and B(s). On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 25/37
The Dynamical Algebra Definition: Let ςbe a complex number, and let a(s) and b(s) be two operators. We define the ς-commutator of aand bas [a(s),b(s)]ς=a(s)b(s)−ςb(s)a(s), ς =qγ6= 0. Why we are interested in this? Proposition: Let Hq(s) be an operator, such that ∃a(s),b(s) and ς, Λ∈C, that Hq(s) = b(s)a(s), and [a(s),b(s)]ς= Λ. Then, if Hq(s)Φ(s) = λΦ(s)⇒ Hq(s){a(s)Φ(s)}=ς−1(λ−Λ) {a(s)Φ(s)},a(s) lowering op. Hq(s){b(s)Φ(s)}= (Λ + ςλ){b(s)Φ(s)}.b(s) raising op. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 26/37
The Dynamical Algebra Definition: Let ςbe a complex number, and let a(s) and b(s) be two operators. We define the ς-commutator of aand bas [a(s),b(s)]ς=a(s)b(s)−ςb(s)a(s), ς =qγ6= 0. Why we are interested in this? Proposition: Let Hq(s) be an operator, such that ∃a(s),b(s) and ς, Λ∈C, that Hq(s) = b(s)a(s), and [a(s),b(s)]ς= Λ. Then, if Hq(s)Φ(s) = λΦ(s)⇒ Hq(s){a(s)Φ(s)}=ς−1(λ−Λ) {a(s)Φ(s)},a(s) lowering op. Hq(s){b(s)Φ(s)}= (Λ + ςλ){b(s)Φ(s)}.b(s) raising op. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 26/37
The Dynamical Algebra Definition: Let ςbe a complex number, and let a(s) and b(s) be two operators. We define the ς-commutator of aand bas [a(s),b(s)]ς=a(s)b(s)−ςb(s)a(s), ς =qγ6= 0. Why we are interested in this? Proposition: Let Hq(s) be an operator, such that ∃a(s),b(s) and ς, Λ∈C, that Hq(s) = b(s)a(s), and [a(s),b(s)]ς= Λ. Then, if Hq(s)Φ(s) = λΦ(s)⇒ Hq(s){a(s)Φ(s)}=ς−1(λ−Λ) {a(s)Φ(s)},a(s) lowering op. Hq(s){b(s)Φ(s)}= (Λ + ςλ){b(s)Φ(s)}.b(s) raising op. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 26/37
The Dynamical Algebra Definition: Let ςbe a complex number, and let a(s) and b(s) be two operators. We define the ς-commutator of aand bas [a(s),b(s)]ς=a(s)b(s)−ςb(s)a(s), ς =qγ6= 0. Why we are interested in this? Proposition: Let Hq(s) be an operator, such that ∃a(s),b(s) and ς, Λ∈C, that Hq(s) = b(s)a(s), and [a(s),b(s)]ς= Λ. Then, if Hq(s)Φ(s) = λΦ(s)⇒ Hq(s){a(s)Φ(s)}=ς−1(λ−Λ) {a(s)Φ(s)},a(s) lowering op. Hq(s){b(s)Φ(s)}= (Λ + ςλ){b(s)Φ(s)}.b(s) raising op. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 26/37
The Dynamical Algebra Definition: Let ςbe a complex number, and let a(s) and b(s) be two operators. We define the ς-commutator of aand bas [a(s),b(s)]ς=a(s)b(s)−ςb(s)a(s), ς =qγ6= 0. Why we are interested in this? Proposition: Let Hq(s) be an operator, such that ∃a(s),b(s) and ς, Λ∈C, that Hq(s) = b(s)a(s), and [a(s),b(s)]ς= Λ. Then, if Hq(s)Φ(s) = λΦ(s)⇒ Hq(s){a(s)Φ(s)}=ς−1(λ−Λ) {a(s)Φ(s)}, a(s) lowering op. Hq(s){b(s)Φ(s)}= (Λ + ςλ){b(s)Φ(s)}.b(s) raising op. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 26/37
The Dynamical Algebra The operators e b(s), e b+(s), and N(s) lead to the dynamical algebra suq(1,1) with the generators (β−1=q+q−1.) K0(s) = 1 2(N(s)+1/2) ,K+(s) = β(e b+(s))2,K−(s) = βe b2(s), [K0(s),K±(s)] = ±K±(s),[K−(s),K+(s)] = [2K0(s)]q2, of the algebra suq(1,1). Similarly we can derive the dynamical algebra suq(2). On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 28/37
The Dynamical Algebra The operators e b(s), e b+(s), and N(s) lead to the dynamical algebra suq(1,1) with the generators (β−1=q+q−1.) K0(s) = 1 2(N(s)+1/2) ,K+(s) = β(e b+(s))2,K−(s) = βe b2(s), [K0(s),K±(s)] = ±K±(s),[K−(s),K+(s)] = [2K0(s)]q2, of the algebra suq(1,1). Similarly we can derive the dynamical algebra suq(2). On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 28/37
Dynamical Algebra: Problem 1 Let us now pose the problem 1: Given the operator Hq(s) To find two operators a(s)and b(s)and ς∈Csuch that: 1) the Hamiltonian Hq(s) = b(s)a(s)and 2) [a(s),b(s)]ς=I. For the first question we already have the answer: the operators b(s) = a↑ α(s) and a(s) = a↓ α(s) factorize the Hamiltonian Hq(s) = a↑ α(s)a↓ α(s). The question is under which conditions they also satisfy the commutation relation. In the following we assume that A(s) = B(s). ⇒ On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 29/37
Dynamical Algebra: Problem 1 Let us now pose the problem 1: Given the operator Hq(s) To find two operators a(s)and b(s)and ς∈Csuch that: 1) the Hamiltonian Hq(s) = b(s)a(s)and 2) [a(s),b(s)]ς=I. For the first question we already have the answer: the operators b(s) = a↑ α(s) and a(s) = a↓ α(s) factorize the Hamiltonian Hq(s) = a↑ α(s)a↓ α(s). The question is under which conditions they also satisfy the commutation relation. In the following we assume that A(s) = B(s). ⇒ On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 29/37
Dynamical Algebra: Problem 1 Let us now pose the problem 1: Given the operator Hq(s) To find two operators a(s)and b(s)and ς∈Csuch that: 1) the Hamiltonian Hq(s) = b(s)a(s)and 2) [a(s),b(s)]ς=I. For the first question we already have the answer: the operators b(s) = a↑ α(s) and a(s) = a↓ α(s) factorize the Hamiltonian Hq(s) = a↑ α(s)a↓ α(s). The question is under which conditions they also satisfy the commutation relation. In the following we assume that A(s) = B(s). ⇒ On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 29/37
Dynamical Algebra: Problem 1 Let us now pose the problem 1: Given the operator Hq(s) To find two operators a(s)and b(s)and ς∈Csuch that: 1) the Hamiltonian Hq(s) = b(s)a(s)and 2) [a(s),b(s)]ς=I. For the first question we already have the answer: the operators b(s) = a↑ α(s) and a(s) = a↓ α(s) factorize the Hamiltonian Hq(s) = a↑ α(s)a↓ α(s). The question is under which conditions they also satisfy the commutation relation. In the following we assume that A(s) = B(s). ⇒ On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 29/37
Dynamical Algebra: Problem 1 Let us now pose the problem 1: Given the operator Hq(s) To find two operators a(s)and b(s)and ς∈Csuch that: 1) the Hamiltonian Hq(s) = b(s)a(s)and 2) [a(s),b(s)]ς=I. For the first question we already have the answer: the operators b(s) = a↑ α(s) and a(s) = a↓ α(s) factorize the Hamiltonian Hq(s) = a↑ α(s)a↓ α(s). The question is under which conditions they also satisfy the commutation relation. In the following we assume that A(s) = B(s). ⇒ On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 29/37
Dynamical Algebra: Problem 1 Let us now pose the problem 1: Given the operator Hq(s) To find two operators a(s)and b(s)and ς∈Csuch that: 1) the Hamiltonian Hq(s) = b(s)a(s)and 2) [a(s),b(s)]ς=I. For the first question we already have the answer: the operators b(s) = a↑ α(s) and a(s) = a↓ α(s) factorize the Hamiltonian Hq(s) = a↑ α(s)a↓ α(s). The question is under which conditions they also satisfy the commutation relation. In the following we assume that A(s) = B(s). ⇒ On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 29/37
The Dynamical Algebra: Problem 1 Theorem 1: NECESSARY CONDITION Let (ϕn)nthe eigenfunctions of Hq(s) corresponding to the eigenvalues (λn)nand suppose that the problem 1 has a solution for Λ6= 0. Then, the eigenvalues λnof the NU q-equation are q-linear or q−1-linear functions of n, i.e., λn=C1qn+C3or λn=C2q−n+C3, respectively. λn=−[n]qqn−1 2+q−n−1 2 2eτ0+ [n−1]q eσ00 2=C1qn+C2q−n+C3 Impossible to solve: general Askey-Wilson, q-Racah, big and little q-Jacobi polynomials. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 30/37
The Dynamical Algebra: Problem 1 Theorem 1: NECESSARY CONDITION Let (ϕn)nthe eigenfunctions of Hq(s) corresponding to the eigenvalues (λn)nand suppose that the problem 1 has a solution for Λ6= 0. Then, the eigenvalues λnof the NU q-equation are q-linear or q−1-linear functions of n, i.e., λn=C1qn+C3or λn=C2q−n+C3, respectively. λn=−[n]qqn−1 2+q−n−1 2 2eτ0+ [n−1]q eσ00 2 =C1qn+C2q−n+C3 Impossible to solve: general Askey-Wilson, q-Racah, big and little q-Jacobi polynomials. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 30/37
The Dynamical Algebra: Problem 2 To find two operators a(s)and b(s)and a constant ςsuch that the Hamiltonian Hq(s) = b(s)a(s)and [a(s),b(s)]ς=Iand such that a(s)and b(s)are the lowering and raising operators, i.e., a(s)ϕn(s) = DnΦn−1(s) and b(s)ϕn(s) = UnΦn+1(s). Answer: λnshould be aς-linear function, i.e., λnhas the form λn=Aςn+Dand this is again a necessary condition When the α-operators are mutually adjoint? Answer: It depends of the “scalar product”. E.g. Discrete case: α= 0 is a sufficient condition On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 32/37
Example 1: The Al-Salam & Carlitz I The q-Hamiltonian is, in this case, (x=qs). Hq(s)=−q2√a(x−1)(x−a) (q−1)2x2e−∂s−√a(1−qx)(a−qx) x2e∂s+√q(q(x−1)x+a(1+q−qx)) (q−1)2x2I Then, Hq(s)ϕn(s) = q3 21−q−n (1−q)2ϕn(s) and the operators a↓(s)≡a↓ 0(s) = q1 4x−1 q1 2−q−1 2p(x−1/q) (x−a/q)e∂s−√a I, a↑(s)≡a↑ 0(s) = q1 4x−1 q1 2−q−1 2p(x−1) (x−a)e−∂s−pa/q I, are such that On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 33/37
Example 1: The Al-Salam & Carlitz I a↑(s)a↓(s) = Hq(s),and [a↓(s),a↑(s)]q−1=1 kq . Notice that since this is a discrete case when α= 0 the operators a↑(s) and a↓(s) are mutually adjoint. Other related cases: the Al-Salam & Carlitz I polynomials with a=−1 are the discrete q-Hermite I hn(x;q) Changing qby q−1, we obtain the factorization and the dynamical algebra for the Al-Salam & Carlitz functions II from where putting a=−1 and x→ix follows the solution for the discrete Hermite q-polynomials e hn(x;q) On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 34/37
Example 1: The Al-Salam & Carlitz I a↑(s)a↓(s) = Hq(s),and [a↓(s),a↑(s)]q−1=1 kq . Notice that since this is a discrete case when α= 0 the operators a↑(s) and a↓(s) are mutually adjoint. Other related cases: the Al-Salam & Carlitz I polynomials with a=−1 are the discrete q-Hermite I hn(x;q) Changing qby q−1, we obtain the factorization and the dynamical algebra for the Al-Salam & Carlitz functions II from where putting a=−1 and x→ix follows the solution for the discrete Hermite q-polynomials e hn(x;q) On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 34/37
Example 1: The Al-Salam & Carlitz I a↑(s)a↓(s) = Hq(s),and [a↓(s),a↑(s)]q−1=1 kq . Notice that since this is a discrete case when α= 0 the operators a↑(s) and a↓(s) are mutually adjoint. Other related cases: the Al-Salam & Carlitz I polynomials with a=−1 are the discrete q-Hermite I hn(x;q) Changing qby q−1, we obtain the factorization and the dynamical algebra for the Al-Salam & Carlitz functions II from where putting a=−1 and x→ix follows the solution for the discrete Hermite q-polynomials e hn(x;q) On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 34/37
Example 1: The Al-Salam & Carlitz I a↑(s)a↓(s) = Hq(s),and [a↓(s),a↑(s)]q−1=1 kq . Notice that since this is a discrete case when α= 0 the operators a↑(s) and a↓(s) are mutually adjoint. Other related cases: the Al-Salam & Carlitz I polynomials with a=−1 are the discrete q-Hermite I hn(x;q) Changing qby q−1, we obtain the factorization and the dynamical algebra for the Al-Salam & Carlitz functions II from where putting a=−1 and x→ix follows the solution for the discrete Hermite q-polynomials e hn(x;q) On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 34/37
Other examples in the q-Askey Tableau x(s)Pn(s)qσ(s) + τ(s)∇x1(s)σ(s)λn qsU(a) n(x;q)a(x−1)(x−a)q3 21−q−n (1−q)2 q−sV(a) n(x;q) (1 −x)(a−x)a q 1 21−qn (1−q)2 qshn(x;q)−1x2−1q3 21−q−n (1−q)2 qse hn(x;q) 1 + x21q1 21−qn (1−q)2 q−svµ n(x;q)µ(1 −1/q)(µ−q/qs)q3 21−qn (1−q)2 qsSn(x;q)x2q−1x q 1 21−qn (1−q)2 qspn(x;a|q)−ax q−1x(x−1) q1 21−q−n (1−q)2 qsLα n(x;q)ax(x+ 1) q−1x q 1 2a1−qn (1−q)2 q−sCn(x;a;q)x(x−1) q−1ax q 1 21−qn (1−q)2 Al-Salam & Carlitz I, II, discrete q-Hermite I, II, q-Charlier-type, Stieltjes-Wigert, Wall polynomials, discrete q-Laguerre, q-Charlier. The Askey-Wilson case: Only for some special cases. continuous q-Laguerre and continuous q-Hermite polynomials. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 35/37
Other examples in the q-Askey Tableau x(s)Pn(s)qσ(s) + τ(s)∇x1(s)σ(s)λn qsU(a) n(x;q)a(x−1)(x−a)q3 21−q−n (1−q)2 q−sV(a) n(x;q) (1 −x)(a−x)a q 1 21−qn (1−q)2 qshn(x;q)−1x2−1q3 21−q−n (1−q)2 qse hn(x;q) 1 + x21q1 21−qn (1−q)2 q−svµ n(x;q)µ(1 −1/q)(µ−q/qs)q3 21−qn (1−q)2 qsSn(x;q)x2q−1x q 1 21−qn (1−q)2 qspn(x;a|q)−ax q−1x(x−1) q1 21−q−n (1−q)2 qsLα n(x;q)ax(x+ 1) q−1x q 1 2a1−qn (1−q)2 q−sCn(x;a;q)x(x−1) q−1ax q 1 21−qn (1−q)2 Al-Salam & Carlitz I, II, discrete q-Hermite I, II, q-Charlier-type, Stieltjes-Wigert, Wall polynomials, discrete q-Laguerre, q-Charlier. The Askey-Wilson case: Only for some special cases. continuous q-Laguerre and continuous q-Hermite polynomials. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 35/37
Open problems What about the lowering and raising properties of the α-down and α-up operators? What happen for the general Askey-Wilson case and for the general big q-Jacobi? There exists a more general dynamical algebra? Of which kind? Last but not least ... Some other relevant results related with the q-polynomials by Natig: Classical-type integral transform formulas: Mellin transforms, Fourier-Gauss transforms, etc. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 36/37
Open problems What about the lowering and raising properties of the α-down and α-up operators? What happen for the general Askey-Wilson case and for the general big q-Jacobi? There exists a more general dynamical algebra? Of which kind? Last but not least ... Some other relevant results related with the q-polynomials by Natig: Classical-type integral transform formulas: Mellin transforms, Fourier-Gauss transforms, etc. On q-polynomials and some of their applications Renato ´ Alvarez-Nodarse 36/37