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.
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E-mail add ess:[email p o ec ed]