Modi ica ions o quasi-de ini e linea unc ionals ia
addi ion o del a and de i a i es o del a Di ac unc ions
R. ´
Al a ez-Noda se1∗
, J. A es´u2†
, and F. Ma cell´an2‡
1Depa amen o de An´alisis Ma em´a ico, Uni e sidad de Se illa.
Apdo. 1160, E-41080 Se illa, Spain and
Ins i u o Ca los I de F´ısica Te´o ica y Compu acional,
Uni e sidad de G anada. E-18071, G anada, Spain
2Depa amen o de Ma em´a icas, E.P.S., Uni e sidad Ca los III de Mad id.
A e. Uni e sidad 30, E-28911, Legan´es, Mad id, Spain
To appea in Indaga iones Ma hema icae N.S.
Key wo ds and ph ases: Hype geome ic unc ions, quasi-de ini e linea unc ional,
classical polynomials, ke nels, o hogonal polynomials
AMS (MOS) subjec classi ica ion: 33C45, 33A65, 42C05.
Abs ac
We conside he gene al heo y o he modi ica ions o quasi-de ini e linea unc ionals
by adding disc e e measu es. We analyze he exis ence o he co esponding o hogonal
polynomial sequences wi h espec o such linea unc ionals. The h ee- e m ecu ence
ela ion, lowe ing and aising ope a o s as well as he second o de linea di e en ial
equa ion ha he sequences o monic o hogonal polynomials sa is y when he linea
unc ional is semiclassical a e also es ablished. A ele an example is conside ed in de ails.
1 In oduc ion.
Le

be a linea unc ional in he ec o space Po polynomials wi h complex coe icien s.
This unc ional is said o be quasi-de ini e [8] i he p incipal subma ices o he in ini e Hankel
ma ix associa ed wi h he sequence o he momen s (un)no he linea unc ional

a e non-
singula . No ice ha un=h

, xniwhe e h·,·i deno es he duali y b acke .
∗E-mail: [email p o ec ed]
†E-mail: ja esu@ma h.uc3m.es
‡E-mail: pacoma [email protected]
1
Assuming

is a quasi-de ini e linea unc ional, he e exis s a sequence o monic polynomials
(Pn)nwi h deg Pn=n, such ha
h

, PnPmi=knδn,m, kn6= 0.
We in oduce a new linea unc ional
e

=

+
M
X
i=1
Aiδ(x−ai)−
N
X
j=1
Bjδ0(x−bj),(1.1)
whe e (Ai)M
i=1 and (Bj)N
j=1 a e eal numbe s, δ(x−y) and δ0(x−y) mean he Di ac linea
unc ional and i s de i a i e, espec i ely, de ined by
hδ(x−y), p(x)i=p(y),hδ0(x−y), p(x)i=−p0(y),∀p∈P.
Such a kind o linea unc ionals ha e been s udied in ensi ely in he las en yea s. Fo
ins ance, i Bi= 0, i= 1,2,...,N and

is a classical linea unc ional (He mi e, Lague e,
Jacobi, and Bessel) necessa y and su icien condi ions in o de o
e

be quasi-de ini e a e gi en in
[3]. No ice ha in he Lague e case (wi h M= 1 and a1= 0), Jacobi (wi h M= 2 and a1= 1,
a2=−1), Bessel case (wi h M= 1 and a1= 0) we ge he so-called K all- ype o hogonal
polynomials which a e ela ed o he spec al analysis o linea di e en ial ope a o s (see e.g.
[10, 15]) and also appea in he Da boux ans o ma ion on he Jacobi ma ix associa ed wi h
he h ee- e m ecu ence ela ion which such polynomials sa is y (see [12, 13], as well as [14]
o some i e a ion o he Da boux ans o ma ion).
I some Biis nonze o, hen a gene al app oach o he quasi-de ini eness o he linea unc-
ional
e

is p esen ed in [6]. Fo he case o classical unc ionals we ha e s udied he co e-
sponding sequences o o hogonal polynomials in se e al pape s: Fo he Lague e unc ional
see [1, 2], o he Bessel linea unc ional see [4] and, inally, o he Jacobi linea unc ional see
[5]. No ice ha in hese cases he poin s whe e he Di ac linea unc ionals a e suppo ed a e
in ima ely ela ed o he end poin s o he suppo o

.
Ve y ecen ly and ollowing a mul iple in eg al ep esen a ion o Heine o Hankel ma ices
as well as he co esponding sequences o o hogonal polynomials, in [9] a simila s udy has
been done. Su p isingly, he au ho s do no e e he abo e con ibu ions and e en, hey do
no discuss he quasi-de ini e cha ac e o he linea unc ional
e

. Basically hey ob ain ou
o iginal esul s using a di e en app oach.
The aim o his con ibu ion is o analyze he linea unc ional
e

in oduced in (1.1) and o
ind necessa y and su icien condi ions o i s quasi-de ini eness. Nex , we deduce an explici
ep esen a ion o he co esponding sequence o monic o hogonal polynomials ( e
Pn)n. This
cons i u es he con en s o he sec ion 2. In sec ion 3 we ob ain he coe icien s o he h ee-
e m ecu ence ela ion. In sec ion 4 we assume

is a semiclassical linea unc ional and hen
we deduce a aising as well as a lowe ing ope a o associa ed wi h
e

. In sec ion 5 we deduce a
second o de linea di e en ial equa ion om he abo e esul s. Finally, in sec ion 6 an example
ela ed o he He mi e polynomials is wo ked ou .
2
2 De ini ion o he unc ional
e

.
Conside he linea unc ional
e

gi en in (1.1), i.e.,
e

=

+
M
X
i=1
Aiδ(x−ai)−
N
X
j=1
Bjδ0(x−bj).
I
e

is quasi-de ini e hen he e exis s a sequence o monic polynomials ( e
Pn)no hogonal wi h
espec o
e

and he e o e we can conside he Fou ie expansion
e
Pn(x) = Pn(x) +
n−1
X
k=0
λn,kPk(x).(2.1)
Then, o 0 ≤k≤n−1,
λn,k =h

,e
Pn(x)Pk(x)i
h

, P2
k(x)i=−
M
X
i=1
Aie
Pn(ai)Pk(ai) +
N
X
j=1
Bje
Pn(bj)P0
k(bj) +
N
X
j=1
Bje
P0
n(bj)Pk(bj)
h

, P2
k(x)i
=−
M
X
i=1
Aie
Pn(ai)Pk(ai)
h

, P2
k(x)i−
N
X
j=1
Bje
Pn(bj)P0
k(bj)
h

, P2
k(x)i−
N
X
j=1
Bje
P0
n(bj)Pk(bj)
h

, P2
k(x)i.
Thus, (2.1) becomes
e
Pn(x) = Pn(x)−
M
X
i=1
Aie
Pn(ai)Kn−1(x, ai)−
N
X
j=1
Bje
Pn(bj)K(0,1)
n−1(x, bj)−
N
X
j=1
Bje
P0
n(bj)Kn−1(x, bj),
(2.2)
whe e, as usual, we deno e
K(i,j)
n(x, y) =
n
X
l=0
P(i)
l(x)P(j)
l(y)
h

, P2
l(x)i, P(i)
l(x) := di
dxiPl(x), i, j ∈N.
Fo a sake o simplici y K(0,0)
n(x, y) := Kn(x, y) deno es he ep oducing ke nel associa ed wi h
he linea unc ional

. I is e y well known ha h

,Kn(x, y)p(x)i=p(y) o e e y polynomial
p(x) o deg ee less han o equal o n.
I we e alua e (2.2) o x=ak,k= 1,2,...,M we ge
e
Pn(ak) = Pn(ak)−
M
X
i=1
Aie
Pn(ai)Kn−1(ak, ai)−
N
X
j=1
Bje
Pn(bj)K(0,1)
n−1(ak, bj)
−
N
X
j=1
Bje
P0
n(bj)Kn−1(ak, bj).
(2.3)
3
A simila e alua ion in (2.2) o x=bk,k= 1,2,...,N, yields
e
Pn(bk) = Pn(bk)−
M
X
i=1
Aie
Pn(ai)Kn−1(bk, ai)−
N
X
j=1
Bje
Pn(bj)K(0,1)
n−1(bk, bj)
−
N
X
j=1
Bje
P0
n(bj)Kn−1(bk, bj).
(2.4)
Finally, aking de i a i es in (2.2) and e alua ing he esul ing exp ession o x=bk,k=
1,2,...,N, we ob ain
e
P0
n(bk) = P0
n(bk)−
M
X
i=1
Aie
Pn(ai)K(1,0)
n−1(bk, ai)−
N
X
j=1
Bje
Pn(bj)K(1,1)
n−1(bk, bj)
−
N
X
j=1
Bje
P0
n(bj)K(1,0)
n−1(bk, bj).
(2.5)
Thus we ge a sys em o M+2Nlinea equa ions in he a iables ( e
Pn(ak))M
k=1, ( e
Pn(bk))N
k=1 and
(e
P0
n(bk))N
k=1.
In o de o simpli y he abo e exp essions we in oduce he ollowing no a ions (ATis he
anspose o A):
Pn(~z) = (Pn(z1), Pn(z2),...,Pn(zk))T,P0
n(~z) = (P0
n(z1), P0
n(z2), . . . , P0
n(zk))T,
whe e ~z = (z1, z2,···, zk)T. Also we in oduce he ma ices K(i,j)
n−1(~z, ~y)∈Cp×qwhose (m, n)
en y is K(i,j)
n−1(zm, yn). He e ~z = (z1, z2,...,zp) and ~y = (y1, y2, . . . , yq). Finally, we in-
oduce he ma ices associa ed wi h he mass poin s A= diag (A1, A2, . . . , AM) and B=
diag (B1, B2,...,BN).
Wi h hese no a ions he linea sys em o equa ions (2.3), (2.4) and (2.5) becomes

e
Pn(~a)
e
Pn(~
b)
e
P0
n(~
b)
=

Pn(~a)
Pn(~
b)
P0
n(~
b)
−Kn−1D
e
Pn(~a)
e
Pn(~
b)
e
P0
n(~
b)
,(2.6)
whe e ~a = (a1, a2,...,aM)T,~
b= (b1, b2,...,bN)T, and
D = 

A0 0
0 0 B
0B0
,Kn−1=

Kn−1(~a,~a)Kn−1(~a,~
b)K(0,1)
n−1(~a,~
b)
Kn−1(~
b,~a)Kn−1(~
b,~
b)K(0,1)
n−1(~
b,~
b)
K(1,0)
n−1(~
b,~a)K(0,1)
n−1(~
b,~
b)K(1,1)
n−1(~
b,~
b)
,
o , equi alen ly,

e
Pn(~a)
e
Pn(~
b)
e
P0
n(~
b)
=

Pn(~a)
Pn(~
b)
P0
n(~
b)
−

Kn−1(~a,~a)Ae
Pn(~a) + K(0,1)
n−1(~a,~
b)Be
Pn(~
b) + Kn−1(~a,~
b)Be
P0
n(~
b)
Kn−1(~
b,~a)Ae
Pn(~a) + K(0,1)
n−1(~
b,~
b)Be
Pn(~
b) + Kn−1(~
b,~
b)Be
P0
n(~
b)
K(1,0)
n−1(~
b,~a)Ae
Pn(~a) + K(1,1)
n−1(~
b,~
b)Be
Pn(~
b) + K(0,1)
n−1(~
b,~
b)Be
P0
n(~
b)
.
4
No ice ha all he in ol ed block ma ices ha e he app op ia e dimensions. Thus, i he
ma ix I+Kn−1D, whe e I is he iden i y ma ix, is non singula , hen we ge he exis ence and
uniqueness o he solu ion o (2.6).
In such a case (2.2) becomes
e
Pn(x) = Pn(x)−(KT
n−1(x,~a),KT
n−1(x,~
b),K(0,1)
n−1
T
(x,~a)) D(I + Kn−1D)−1

Pn(~a)
Pn(~
b)
P0
n(~
b)
.(2.7)
On he o he hand, aking in o accoun he quasi-de ini e cha ac e o
e

, we ge
06=h
e

,e
P2
n(x)i=h
e

,e
Pn(x)Pn(x)i=h

,e
Pn(x)Pn(x)i+
M
X
i=1
Aie
Pn(ai)Pn(ai)
+
N
X
j=1
Bje
P0
n(bj)Pn(bj) + e
Pn(bj)P0
n(bj).
Using he p e ious no a ions, as well as h

,e
Pn(x)Pn(x)i=h

, P2
n(x)i, he abo e exp ession
becomes
h
e

,e
P2
n(x)i=h

, P2
n(x)i+ ( PT
n(~a),PT
n(~
b),P0
nT(~
b))D
e
Pn(~a)
e
Pn(~
b)
e
P0
n(~
b)
(2.8)
=h

, P2
n(x)i+ ( PT
n(~a),PT
n(~
b),P0
nT(~
b) ) D (I + Kn−1D)−1

Pn(~a)
Pn(~
b)
P0
n(~
b)
.
Thus, as a conclusion, we ha e p o ed ha i
e

is a quasi-de ini e linea unc ional hen he
ollowing condi ions hold:
1. The ma ix I + Kn−1D is nonsingula o e e y n∈N, i.e.
de (I + Kn−1D) 6= 0,∀n∈N.(2.9)
2. Fo all n∈N
h
e

,e
P2
n(x)i=h

, P2
n(x)i+( PT
n(~a),PT
n(~
b),P0
nT(~
b) )D (I+Kn−1D)−1

Pn(~a)
Pn(~
b)
P0
n(~
b)
6= 0.(2.10)
Con e sely, (2.9) and (2.10) a e also su icien condi ions o he quasi-de ini e cha ac e o
e

.
In o de o p o e i we p oceed as ollows. Le e
Pnbe he polynomial gi en by (2.7). Then, o
0≤j≤n−1
h
e

,e
Pn(x)Pj(x)i=h

,e
Pn(x)Pj(x)i+ ( PT
j(~a),PT
j(~
b),P0
jT(~
b) ) D (I + Kn−1D)−1

Pn(~a)
Pn(~
b)
P0
n(~
b)
.
5

Now, aking in o accoun he ep oducing p ope y o he ke nel and h

, Pj(x)K(0,1)
n−1(x, b)i=
P0
j(b), we ge
h

,e
Pn(x)Pj(x)i=h

, Pn(x)Pj(x)i−(PT
j(~a),PT
j(~
b),P0
jT(~
b) ) D (I + Kn−1D)−1

Pn(~a)
Pn(~
b)
P0
n(~
b)
.
F om he abo e wo o mulas one has h
e

,e
Pn(x)Pj(x)i=h

, Pn(x)Pj(x)i= 0, o 0 ≤j≤n−1.
On he o he hand,
h
e

,e
Pn(x)Pn(x)i=h

,e
Pn(x)Pn(x)i+ ( PT
n(~a),PT
n(~
b),P0
nT(~
b) ) D (I + Kn−1D)−1

Pn(~a)
Pn(~
b)
P0
n(~
b)

=h

, P2
n(x)i+ ( PT
n(~a),PT
n(~
b),P0
nT(~
b) ) D (I + Kn−1D)−1

Pn(~a)
Pn(~
b)
P0
n(~
b)
6= 0,
om (2.10). Thus, e
Pnis he n- h monic polynomial o hogonal wi h espec o he linea
unc ional
e

. Then we ha e p o ed
Theo em 1 The linea unc ional
e

gi en by (1.1) is a quasi-de ini e linea unc ional i and
only i
(i) The ma ix I + Kn−1Dis nonsingula o e e y n∈N.
(ii) h

, P2
n(x)i+ ( PT
n(~a),PT
n(~
b),P0
nT(~
b) ) D (I + Kn−1D)−1

Pn(~a)
Pn(~
b)
P0
n(~
b)
6= 0, o e e y n∈N.
In such a case, he co esponding sequence (e
Pn)no monic o hogonal polynomials is gi en by
e
Pn(x) = Pn(x)−(KT
n−1(x,~a),KT
n−1(x,~
b),K(0,1)
n−1
T
(x,~a))D(I + Kn−1D)−1

Pn(~a)
Pn(~
b)
P0
n(~
b)
.
Rema ks:
1. I he en ies o he ma ix D a e nonze o, hen D is an he mi ian ma ix which is non-
singula . Then
D(I + Kn−1D)−1= (D−1+Kn−1)−1= Mn−1,
whe e D−1+Kn−1is an he mi ian ma ix. Thus (2.10) means ha
1 + εn(b
PT
n(~a),b
PT
n(~
b),b
P0
nT(~
b))Mn−1
b
Pn(~a)
b
Pn(~
b)
b
P0
n(~
b)
6= 0,b
Pn(ai) = Pn(ai)
p|h

, P2
n(x)i|,
and εn= sign (h

, P2
n(x)i).
6
2. I some en ies o D a e ze o, hen we can educe he size o ou sys em. The same
si ua ion holds when (ai)M
i=1 T(bj)N
j=1 6=∅. In such a case, he co esponding equa ions in
(2.3) a e educed because o he epe i ion in (2.4).
3. Taking in o accoun he Ch is o el-Da boux o mula we ha e
Kn−1(x, y) = 1
knPn(x)Pn−1(y)−Pn(y)Pn−1(x)
x−y, kn=h

, P2
n(x)i,
as well as
K(0,1)
n−1(x, y) = 1
knPn(x)P0
n−1(y)−P0
n(y)Pn−1(x)
x−y+Pn(x)Pn−1(y)−Pn(y)Pn−1(x)
(x−y)2.
Inse ing his wo exp essions in (2.7) and deno ing φ(x) = QM
i=1(x−ai)QN
j=1(x−bj)2,
we deduce
φ(x)e
Pn(x) = A(x;n)Pn(x) + B(x;n)Pn−1(x),(2.11)
whe e A(x;n) and B(x;n) a e polynomials o deg ee independen o nand a mos 2N+M
and 2N+M−1, espec i ely.
On he o he hand, om he h ee- e m ecu ence ela ion ha he sequence (Pn)n
sa is ies
xPn(x) = Pn+1(x) + βnPn(x) + γnPn−1(x),(2.12)
and aking in o accoun (2.11) we ge , o n≥1
φ(x)e
Pn−1(x) = C(x;n)Pn(x) + D(x;n)Pn−1(x),(2.13)
whe e
C(x;n) = −B(x;n−1)
γn−1
, D(x;n) = A(x;n−1) + x−βn−1
γn−1
B(x;n−1).
4. An in e se p ocess can be done in o de o eco e he linea unc ional

in e ms o
e

(we need o add o
e

he same masses bu wi h opposi e sign). In such a way we can
deduce he exis ence o polynomials A(x;n) and B(x;n) wi h deg ees independen o n
such ha
φ(x)Pn(x) = A(x;n)e
Pn(x) + B(x;n)e
Pn−1(x),(2.14)
as well as he coun e pa
φ(x)Pn−1(x) = C(x;n)e
Pn(x) + D(x;n)e
Pn−1(x).(2.15)
7
3 A Th ee-Te m Recu ence Rela ion o (
e
Pn)n.
In he ollowing we assume ha
e

is quasi-de ini e. Then, he sequence ( e
Pn)no monic poly-
nomials o hogonal wi h espec o
e

sa is ies a h ee- e m ecu ence ela ion (TTRR)
xe
Pn(x) = e
Pn+1(x) + e
βne
Pn(x) + eγne
Pn−1(x), n ∈N,(3.1)
wi h he ini ial condi ions e
P−1(x) = 0, e
P0(x) = 1.
Ou aim is o ob ain he coe icien s e
βnand eγno he TTRR (3.1) o he polynomials e
Pn
o hogonal wi h espec o
e

, in e ms o he coe icien s βnand γno he TTRR (2.12) o he
monic polynomials o hogonal wi h espec o

.
To do i we p oceed as ollows. By de ini ion
eγn=h
e

,e
P2
n(x)i
h
e

,e
P2
n−1(x)i
.(3.2)
Taking in o accoun (2.10) as well as ema k 1 o heo em 1, we ge , o n > 1
eγn=γn
1 + εn(b
PT
n(~a),b
PT
n(~
b),b
P0
nT(~
b))Mn−1(b
Pn(~a),b
Pn(~
b),b
P0
n(~
b))T
1 + εn−1(b
PT
n−1(~a),b
PT
n−1(~
b),b
P0
n−1T(~
b))Mn−2(b
Pn−1(~a),b
Pn−1(~
b),b
P0
n−1(~
b) )T,
as well as, o n= 1
eγ1=h
e

,e
P2
1(x)i
h
e

,e
P2
0(x)i
=h
e

,e
P2
1(x)i
h

,e
P2
0(x)i+PM
i=1 Ai
=γ1
1 + ε1(b
PT
1(~a),b
PT
1(~
b),b
P0
1T(~
b))M0(b
P1(~a),b
P1(~
b),b
P0
1(~
b) )T
1 + PM
i=1 Ai/u0
,
whe e u0=h

,1iis he i s momen o he unc ional

.
On he o he hand, e
βn=h
e

, x e
P2
n(x)i/h
e

,e
P2
n(x)i. Ne e heless, i is be e o compu e e
βnin
a di e en way. I ebndeno e he coe icien o xn−1 o e
Pnand bn he co esponding coe icien
o xn−1 o Pnwe ha e, e
βn=ebn−ebn+1. To ob ain ebnwe use he Eq. (2.7) which yields
ebn=bn−εnεn−1|γn|1/2(b
PT
n−1(~a),b
PT
n−1(~
b),b
P0
n−1T(~
b))Mn−1
b
PT
n(~a)
b
PT
n(~
b)
b
P0
nT(~
b)
.
Thus, o n≥1,
e
βn=βn+εnεn+1|γn+1|1/2(b
PT
n(~a),b
PT
n(~
b),b
P0
nT(~
b))Mn
b
PT
n+1(~a)
b
PT
n+1(~
b)
b
P0
n+1T(~
b)

−εnεn−1|γn|1/2(b
PT
n−1(~a),b
PT
n−1(~
b),b
P0
n−1T(~
b))Mn−1
b
PT
n(~a)
b
PT
n(~
b)
b
P0
nT(~
b)
.
8
Finally, o n= 0 we ha e
e
β0=h
e

, xi
h
e

,1i=u1+PM
i=1 aiAi+PN
j=1 Bj
u0+PM
i=1 Ai
.
4 Raising and lowe ing ope a o s o (
e
Pn)n.
In he ollowing we assume ha he linea unc ional

is semiclassical, i.e., he e exis polyno-
mials ψand ν, wi h deg ν≥1, such ha
D(ψ

) = ν

, D =d
dx.(4.1)
He e we use he dis ibu ional no a ion in he sense ha o a polynomial πwe de ine he linea
unc ional π

in such a way ha
hπ

, pi:= h

, πpi,hD

, pi:= −h

, p0i,∀p∈P.
P oposi ion 1 I

is a semiclassical linea unc ional, hen he linea unc ional
e

in oduced
in (1.1) is also a semiclassical unc ional.
P oo : Taking in o accoun he ac ha o he polynomial φ(x) = QM
i=1(x−ai)QN
j=1(x−bj)2
we ge φ

=φ
e

, we can conside
D(φ2ψ
e

) = D(φ2ψ

) = φ2D(ψ

)+2φφ0ψ

=φ2ν

+2φ0ψφ

= (νφ+2φ0ψ)φ

= (νφ+2φ0ψ)φ
e

,
i.e., he e exis polynomials e
ψ=φ2ψand eν= (νφ + 2φ0ψ)φsuch ha D(e
ψ
e

) = eν
e

.
No ice ha he choice o e
ψand eνis no , in gene al, op imal. Fo ins ance, i
D(φψ
e

) = D(φψ

) = φD(ψ

) + φ0ψ

= (φν +φ0ψ)

,
and we assume φ0ψis a mul iple o φ, i.e., φ0ψ=ηφ, whe e ηis a polynomial hen he abo e
equa ion yields
D(φψ
e

) = φ(ν+η)

=φ(ν+η)
e

,
and hus e
ψ=φψ and eν=φ(ν+η). This is he eason why he s udy o he cases when
he se (ai)M
i=1 S(bj)N
j=1 is educed o he se o ze os o ψallows o educe subs an ially he
compu a ion.
P oposi ion 2 (Ma oni [16]) I

is a semiclassical linea unc ional, hen he e exis polyno-
mials M1(x;n),N1(x;n)wi h deg ee independen o n, such ha
ψ(x)P0
n(x) = M1(x;n)Pn(x) + N1(x;n)Pn−1(x).(4.2)
9
and
e
λ(x, 2m) = 3x4+ 4mx6+ (6x2+ (−2 + 4m)x4)a2(m) + 4mx2(−3 + 2x2+ 4mx2)b2(m)
−6xc(m) + 6x3c(m)−12mx3c(m) + 8mx5c(m)−16mc2(m) + 16m2x2c2(m)
+xb(m) (−3x+ 6x3+ 8mx5−36mc(m) + 8mx2c(m) + 32m2x2c(m))
+a(m)[7x3−2x5+ 8mx5+ 2x(5x2+ 6mx2−2x4+ 4mx4−3) b(m)−14c(m)
+14x2c(m)−4x4c(m) + 8mx4c(m)].
Fo he odd case, α(x, 2m+1) = x2+d(m)x+e(m), β(x, 2m+1) = (m)x, and λ2m+1 = 4m+2,
hen
eσ(x, 2m+ 1) e
H00
2m+1(x) + eτ(x, 2m+ 1) e
H0
2m+1(x) + eλ(x, 2m+ 1) e
H2m+1(x) = 0,
whe e
eσ(x, 2m+ 1) = x2hx2d2(m) + e2(m) + e(m) (2x2+ (m) + 2x2 (m)) + x2(x2− (m))
+x2(2x2 (m) + 2 2(m) + 4m 2(m)) + 2xd(m) (e(m) + x2(1 + (m))) i,
eτ(x, 2m+ 1) = −xh2x2(x2−1) d2(m)+2 (x2−2) e2(m) + e(m) (4x4−4x2+ 5 (m)−2x2 (m)
+4x4 (m)) + d(m) (4x5−2x3+ (4x3−6x)e(m) + 6x (m)−2x3 (m) + 4x5 (m))
+x2(2x4+5 (m)−2x2 (m)+4x4 (m)+2 2(m)+4m 2(m)+4x2 2(m)+8mx2 2(m)) i,
and
e
λ(x, 2m+ 1) = (6x2+ 4mx4)d2(m) + (11 −2x2+ 4mx2)e2(m) + xd(m) (7x2+ 2x4+ 8mx4
+ (15 −2x2+ 8mx2)e(m)−6 (m) + 16x2 (m) + 12mx2 (m) + 8mx4 (m))
+e(m) (6x2+ 8mx4−7 (m) + 22x2 (m) + 24mx2 (m)−4x4 (m) + 8mx4 (m))
+x2(3x2+ 2x4+ 4mx4−3 (m) + 6x2 (m) + 4x4 (m) + 8mx4 (m)−6 2(m)
−12m 2(m) + 8x2 2(m) + 24mx2 2(m) + 16m2x2 2(m))
Using he abo e SODE we can s udy he dis ibu ion o ze os o he e
Hnpolynomials by means
o he cen al momen s [7] and he WKB algo i hm [20] in a simila way as i was done in [3].
6.4 The aising and lowe ing ope a o s
Fo he aising and lowe ing ope a o s we ollow he algo i hm desc ibed in sec ion 4. We need
o ob ain he ela ions (2.14) and (2.15). Since (6.1) we ha e h

, pi=h
e

, pi−Ap(0) −Bp0(0),
i.e., he classic He mi e unc ional can be ob ained om he unc ional
e

ia he addi ion o
he same masses Aand Bbu wi h a di e en sign. Then we ha e he ollowing ep esen a ion
(compa e wi h (6.3))
Hn(x) = e
Hn(x) + AHn(0)e
Kn−1(x, 0) + BHn(0)e
K(0,1)
n−1(x, 0) + BH0
n(0)e
Kn−1(x, 0),(6.14)
16

whe e e
Knand e
K(0,1)
na e he ke nel polynomials co esponding o he amily e
Hn. Then using
he Ch is o el-Da boux o mula o he e
Hnpolynomials we ob ain
x2H2m(x) = [x2+ea(m)x+eb(m)] e
H2m(x) + [ec(m)x+e
d(m)] e
H2m−1(x), m ≥1
x2H2m+1(x) = [x2+ee(m)x]e
H2m+1(x) + e
(m)xe
H2m(x), m ≥0,
(6.15)
whe e
ea(m) = H2m(0)
h
e

,e
H2m−1i[Ae
H2m−1(0) + Be
H0
2m−1(0)],eb(m) = BH2m(0) e
H2m−1(0)
h
e

,e
H2m−1i,
ec(m) = −H2m(0)
h
e

,e
H2m−1i[Ae
H2m(0) + Be
H0
2m(0)],e
d(m) = −BH2m(0) e
H2m(0)
h
e

,e
H2m−1i,
ee(m) = B(2m+ 1)H2m(0) e
H2m(0)
h
e

,e
H2mi,e
(m) = −B(2m+ 1)H2m(0) e
H2m+1(0)
h
e

,e
H2mi.
F om he las equa ion in (6.15) i we change mby m−1 and use he TTRR o he polynomials
e
Hnwe ob ain he ollowing exp ession o he polynomials H2m−1(x)
x2H2m−1(x) = "x2+ee(m−1)x+x−e
β2m−1
eγ2m−1
xe
(m−1)#e
H2m−1(x)−xe
(m)
eγ2m−1e
H2m(x).(6.16)
Nex , we ew i e he i s equa ion in (6.10)
x2e
H2m(x) = α(x, 2m)H2m(x) + 2mβ(x, 2m)H2m−1(x).(6.17)
Thus A(x, 2m) = α(x, 2m) and B(x, 2m) = 2mβ(x, 2m). Using he TTRR o he He mi e
polynomials he second equa ion can be easily ans o med as ollows
x2e
H2m−1(x) = −2β(x, 2m−1)H2m(x) + [α(x, 2m−1) + 2xβ(x, 2m−1)]H2m−1(x).(6.18)
Thus C(x, 2m) = −2β(x, 2m−1) and D(x, 2m) = α(x, 2m−1) + 2xβ(x, 2m−1).
F om (6.15) and (6.16) we ha e
A(x, 2m) = x2+ea(m)x+eb(m), B(x, 2m) = ec(m)x+e
d(m),
C(x, 2m) = −xe
(m)
eγ2m−1
,D(x, 2m) = x2+ee(m−1)x+x−e
β2m−1
eγ2m−1
xe
(m−1).
Finally, om he p ope ies o he He mi e polynomials in o mulas (2.11) and (2.13) we ge
M1(x, 2m) = 0, N1(x, 2m) = 2m, M2(x, 2m−1) = 2x, N2(x, 2m−1) = −2.
Subs i u ing all he abo e o mulas in (4.5) and (4.6) we ob ain he lowe ing ope a o , and in
(4.7) he aising ope a o . The same can be pe o med o he odd case.
17
Be o e concluding his sec ion le us poin ou ha he e exis he lowe ing-like and aising-
like ope a o s. Fo he sake o comple eness we will show how hey can be ob ained o he
e en case. The odd case in comple ely simila . I we ew i e (6.12) in he o m
x3e
H0
2m(x) = γ(x, 2m)H2m(x) + 2mδ(x, 2m)H2m−1(x),
and use he equa ions (6.17) and (6.18) we ge he lowe ing-like ope a o
e
ψ(x; 2m)e
H0
2m(x) = M1(x; 2m)e
H2m(x) + N1(x; 2m)e
H2m−1(x),(6.19)
whe e
e
ψ(x, 2m) = x[α(x, 2m)(α(x, 2m−1) + 2xβ(x, 2m−1)) + 4mβ(x, 2m−1)β(x, 2m)],
M1(x; 2m) = γ(x, 2m)[α(x, 2m−1) + 2xβ(x, 2m−1)] + 4mβ(x, 2m−1)δ(x, 2m),
N1(x; 2m) = 2m[α(x, 2m)δ(x, 2m)−β(x, 2m)γ(x, 2m)].
To ob ain he aising-like ope a o we can use he TTRR o he polynomials e
Hn o subs i-
u e e
H2m−1(x) in (6.19) ha leads o
e
ψ(x; 2m)e
H0
2m(x) = M2(x; 2m)e
H2m(x) + N2(x; 2m)e
H2m+1(x),(6.20)
M2(x;n) = M1(x;n) + x−e
βn
eγn
N1(x;n), N2(x;n) = −N1(x;n)
eγn
.
The odd case can be ob ained in a simila way.
Acknowledgemen s: This wo k is pa ially suppo ed by Di ecci´on Gene al de In es igaci´on
(Minis e io de Ciencia y Tecnolog´ıa) o Spain BFM 2000-0206-C04, Jun a de Andaluc´ıa FQM-
0262, and INTAS no2000-272.
Re e ences
[1] R. ´
Al a ez-Noda se and F. Ma cell´an: A Gene aliza ion o he Classical Lague e Polynomials.
Rend. Ci c. Ma . Pale mo. Se ie II,44, (1995), 315-329.
[2] R. ´
Al a ez-Noda se and F. Ma cell´an: A Gene aliza ion o he Classical Lague e Polynomials:
Asymp o ic P ope ies and ze os. Appl. Analysis 62, (1996), 349-366.
[3] R. ´
Al a ez-Noda se, F. Ma cell´an, and J. Pe onilho: WKB app oxima ion and K all- ype o -
hogonal polynomials. Ac a Applicandae Ma hema icae,54, (1998), 27-58.
[4] J. A es´u, R. ´
Al a ez-Noda se, F. Ma cell´an, and K.H. Kwon: Some ex ension o he Bessel- ype
o hogonal polynomials. In eg al T ans. and Special Func ., (7) 3-4, (1998), 191-214.
[5] J. A es´u, R. ´
Al a ez-Noda se, and F. Ma cell´an: On a modi ica ion o he Jacobi linea unc-
ional: Asymp o ic p ope ies and ze os o he co esponding o hogonal polynomials. Ac a Ap-
plicandae Ma hema icae 71 (2002), 127-158.
18
[6] S. L. Belmehdi and F. Ma cell´an: O hogonal polynomials associa ed o some modi ica ions o a
linea unc ional. Appl. Analysis 46(1), (1992), 1-24.
[7] E. Buend´ıa, J. S. Dehesa and F. J. G´al ez: The dis ibu ion o he ze os o he polynomial
eigen unc ion o o dina y di e en ial ope a o s o a bi a y o de . In O hogonal Polynomials and
hei Applica ions. M.Al a o e al. Eds. Lec u e No es in Ma hema ics. Vol. 1329, Sp inge -
Ve lag, Be lin, 1988, 222-235.
[8] T.S. Chiha a: An In oduc ion o O hogonal Polynomials. (Go don and B each, New Yo k, 1978).
[9] Y. Chen and J. G i in: K all- ype polynomials ia he Heine o mula. J. Phys A: Ma h. Gen.
35, (2002), 637-656.
[10] W. N. E e i , K.H. Kwon, L. L. Li lejohn, and R. Wellman: O hogonal polynomial solu ions
o linea o dina y di e en ial equa ions. J. Compu . Appl. Ma h. 133, (2001), 85-109.
[11] A. Ga ido, J. A es´u, and F. Ma cell´an: An elec os a ic in e p e a ion o he ze os o he
F eud- ype o hogonal polynomials. Elec on. T ans. Nume . Anal. (2003) in p ess.
[12] F. A. G ¨unbaum and L. Haine: O hogonal polynomials sa is ying di e en ial equa ions: he ole
o he Da boux ans o ma ions. CRM P oceedings, Lec . No es 9, Ame . Ma h. Soc. P o idence,
RI 1996, 143-154.
[13] F. A. G ¨unbaum and L. Haine: Bispec al Da boux ans o ma ion: An ex ension o he K all
polynomials. In e na . Ma h. Res. No . 8, (1997), 359-392.
[14] F. A. G ¨unbaum, L. Haine, and E. Ho ozo : Some unc ions ha gene alize he K all-Lague e
polynomials. J. Compu . Appl. Ma h. 106, (1999), 271-297.
[15] K.H. Kwon, G.J. Yoon, and L.L. Li lejohn: Bochne -K all o hogonal polynomials. In Special
Func ions C. Dunkl, M. Ismail, and R. Wong Eds. Wo ld Scien i ic, Singapo e, 2000, 181-193.
[16] P. Ma oni, Une h´eo ie alg´eb ique des polynˆomes o hogonaux: Applica ion aux polynˆomes o -
hogonaux semiclassiques. In C. B ezinski e al. (Eds.) O hogonal Polynomials and hei Appli-
ca ions. IMACS Ann. Compu . Appl. Ma h. 9, Bal ze , Basel, 1991, 95-130.
[17] M.E.H. Ismail and J. Wimp: On di e en ial equa ions o o hogonal polynomials. Me hods Appl.
Anal. 5, (1998), 439-452.
[18] M.E.H. Ismail: An elec os a ic model o ze os o gene al o hogonal polynomials. Paci ic J.
Ma h. 193, (2000), 355-369.
[19] M.E.H. Ismail: Mo e on elec os a ic model o ze os o o hogonal polynomials. Nume . Func .
Anal. Op im. 21, (2000), 191-204.
[20] A.Za zo and J.S. Dehesa: Spec al P ope ies o solu ions o hype geome ic- ype di e en ial
equa ions. J. Compu . Appl. Ma h. 50, (1994), 613-623.
19