ON THE PROPERTIES FOR MODIFICATIONS OF CLASSICAL
ORTHOGONAL POLYNOMIALS OF DISCRETE VARIABLES.
1
R.

Al a ez-No da se, A.G.Ga ca and F. Ma cellan
Depa amen o de Ingenie a. Escuela Poli ecnica Supe io .
Uni e sidad Ca los III de Mad id.
Bu a que 15, 28911,Leganes,Mad id.
Key wo ds and ph ases: Meixne ,Cha lie and K a chuk p olynomials, disc e e measu es,
hyp e geome ic unc ions, asso cia ed p olynomials.
AMS (MOS) sub jec classica ion:
33A65
Abs ac
We conside a mo dica ion o momen unc ionals o some classical p olynomial s
o a disc e e a iable by adding a mass p oin a
x
= 0. We ob ain he esul ing o -
hogonal p olynomial s, iden i y hem as hyp e geome ic unc ions and de i e he second
o de die ence equa ion which hese p olynomial s sa is y. The co esp onding idiagonal
ma ices and asso cia ed p olynomials we e also s udied.
x
1 In o duc ion.
The s udy o o hogonal p olynomials wi h esp ec o a mo dica ion o a linea unc-
ional in he linea space o p olynomials wi h eal co ecien s ia he addi ion o one o wo
del a Di ac measu es has b een p e o med by se e al au ho s. In pa icula , Chiha a [5] has
conside ed some p op e ies o such p olynomials in e ms o he lo ca ion o he mass p oin
wi h esp ec o he supp o o a p osi i e measu e. Mo e ecen ly Ma cellan and Ma oni [10]
analyzed a mo e gene al si ua ion o egula ( quasi-deni e ) linea unc ionals, i.e., such
ha he p incipal subma ices o he co esp onding inni e Hankel ma ices asso cia ed wi h
he momen sequences a e nonsingula .
A sp ecial emphasis is gi en o he mo dica ions o classical linea unc ionals (He mi e,
Lague e, Jacobi and Bessel). Ko o nwinde [9] conside ed a sys em o p olynomials o hogonal
wi h esp ec o he classical weig h unc ion o Jacobi p olynomials wi h wo ex a p oin
masses added a
x
=

1 and
x
= 1. Fo gene alized Lague e p olynomials
L
;A
n
(
x
)
g
1
n
=0
ha a e o hogonal on [0
;
1
) wi h esp ec o he linea unc ional
C
on he linea space o
p olynomials wi h eal co ecien s dened as
<
C
; P >
=
Z
1
0
P
(
x
)
x

e

x
dx
+
AP
(0)
;  >

1
; A

0
;
Ko eko ek and Ko eko ek [8] ound a die en ial equa ion o he o m
A
1
X
i
=0
a
i
(
x
)
y
(
i
)
(
x
) +
xy
00
(
x
) + (

+ 1

x
)
y
0
(
x
) +
ny
= 0
;
whe e he co ecien s
a
i
(
x
)
; i
2
1
;
2
;
3
; :::
g
, a e indep enden o n and
a
0
dep ends on
n
bu
is indep enden o
x
. In he ab o e pap e , ep esen a ion o mulas o he new o hogonal
1
Oc ob e 29, 1996
1
p olynomial sequences, as well as he second o de die en ial equa ion ha such p olynomials
sa is y we e deduced.
In he op en p oblem sec ion o he P o ceedings o he Thi d In e na ional Symp osium
on O hogonal Polynomials and hei Applica ions held in E ice (I aly), R. Askey aised he
ollowing ques ion [1]:
Conside he Meixne Polynomials
M
 ;
n
(
x
)
, add o subs ac a mass poin a
x
= 0
and
nd he esul ing polynomials. Iden i y hem as hype geome ic unc ions and show ha hese
polynomials sa is y a die ence equa ion in
x
.
In [3] Ba inck and an Hae ingen ga e he solu ion o he p oblem o nding he second
o de die ence equa ion o gene alized Meixne p olynomials, as well as he inni e o de
die ence equa ion which hese p olynomials sa is y. Fo gene alized Cha lie p olynomials
Ba inck and Ko eko ek [4] ound he co esp onding inni e o de die ence equa ion.
In he pap e [2] we ob ained he ep esen a ion o such gene alized Meixne p olynomials
as an hyp e geome ic unc ion
3
F
2
, as well as he co esp onding second o de die ence
equa ion. Now we gene alize his esul o he K a chuk and Cha lie p olynomials and
con inue he algeb aic app oach p esen ed by Go doy, Ma cellan, Sal o, Za zo ( see [7] ) in
he amewo k o a mo e gene al heo y based in he addi ion o a del a Di ac measu e
o a disc e e semiclassical linea unc ional. We analyze he ela ion b e ween idiagonal
ma ices o he
pe u bed o gene alized
P
A
n
(
x
) and classical
P
n
(
x
) p olynomials, as well as
he asso cia ed p olynomials co esp onding o he sequence
P
A
n
(
x
)
g
1
n
=0
.
The s uc u e o he pap e is as ollows. In Sec ion 2, we p o ide he basic p op e ies
o he classical o hogonal p olynomials o disc e e a iable which will b e needed, as well
as he main da a o he Meixne , K a chuk and Cha lie p olynomials. In Sec ion 3 we
deduce exp essions o he gene alized Meixne , K a chuk and Cha lie p olynomials and i s
 s die ence de i a i es, as well as hei ep esen a ion as hyp e geome ic unc ions in
he di ec ion aised by Askey. In Sec ion 4, we nd he second o de die ence equa ion
which hese gene alized p olynomials sa is y. In Sec ion 5, om he h ee e m ecu ence
ela ion (TTRR) o he classical o hogonal p olynomials we nd he TTRR which sa is y
he p e u b ed ones. In Sec ion 6, om he ela ion o he p e u b ed p olynomials
P
A
n
(
x
)
as a linea combina ion o he classical ones, we nd he idiagonal ma ices asso cia ed
wi h he p e u b ed monic o hogonal p olinomial sequence (PMOPS)
P
A
n
(
x
)
g
1
n
=0
as a ank-
one p e u ba ion o he idiagonal ma ices asso cia ed wi h he classical monic o hogonal
p olynomial sequence (CMOPS)
P
n
(
x
)
g
1
n
=0
. Finally, in Sec ion 7 we nd he asso cia ed
p olynomials
P
(1)
;A
n
(
x
) co esp onding o
P
A
n
(
x
)
g
1
n
=0
in e ms o he asso cia ed p olynomials
P
(1)
n
(
x
) co esp onding o
P
n
(
x
)
g
1
n
=0
and he classical ones.
x
2 Some P elimina Resul s.
Fi s ly, we enclose some o mulas o he classical Meixne , K a chuk and Cha lie p oly-
nomials which a e use ul in o de o ob ain he gene alized p olynomials o hogonal wi h
esp ec o he linea unc ional
U
dened as a mo dica ion o he  s ones ough ou he
addi ion o a mass p oin . All he o mulas and o he p op e ies o he classical Meixne ,
K a chuk and Cha lie p olynomials can b e ound in a lo o b o oks ( see o ins ance he
excellen monog aph
O hogonal Polynomials in Disc e e Va iables
by A.F. Niki o o , S. K.
Suslo , V. B. U a o [11], Chap e 2.)
We will use monic p olynomials, i.e., p olynomials wi h leading co ecien equal o 1. The
classical o hogonal p olynomials o a disc e e a iable in he uni o m la ice
x
(
s
) =
s
a e he
2
p olynomial solu ion o a second o de linea die ence equa ion o hyp e geome ic yp e

(
x
)
4 5
P
n
(
x
) +

(
x
)
4
P
n
(
x
) +

n
P
n
(
x
) = 0
;
(1)
whe e
5
(
x
) =
(
x
)

(
x

1)
;
4
(
x
) =
(
x
+ 1)

(
x
)
:
He e

(
x
) and

(
x
) a e p olynomials in x o deg ee a mos 1 and 2, esp ec i ely, and

n
is a
cons an .
These p olynomials a e o hogonal wi h esp ec o he linea unc ional
L
on he linea
space o p olynomials wi h eal co ecien s dened as
<
L
; P >
=
X
x
2
IN

(
x
)
P
(
x
)
;
IN
=
0
;
1
;
2
; :::
g
;
(2)
whe e

(
x
) is some non-nega i e unc ion ( weigh unc ion ) supp o ed in a coun able se o
he eal line and such ha
4
[

(
x
)

(
x
)] =

(
x
)

(
x
)
:
The o hogonali y ela ion is
X
x
2
IN
P
n
(
x
)
P
m
(
x
)

(
x
) =

nm
d
2
n
;
(3)
whe e
d
2
n
deno es he squa e o he no m o hese classical p olynomials.
The p olynomial solu ions o equa ion (1) a e uniquely de e mined, up o a no malized
ac o (
R
n
), by he die ence analog o he Ro d igues o mula (see [11] page 24 Eq.(2.2.7)):
P
n
(
x
) =
R
n

(
x
)
5
n
"

(
x
+
n
)
n
Y
k
=1

(
x
+
k
)
#
:
(4)
They sa is y a h ee e m ecu ence ela ion o he o m
xP
n
(
x
) =
P
n
+1
(
x
) +

n
P
n
(
x
) +

n
P
n

1
(
x
)
; n

0
P

1
(
x
) = 0 and
P
0
(
x
) = 1
:
(5)
and he Ch is oel-Da b oux o mula
n

1
X
m
=0
P
m
(
x
)
P
m
(
y
)
d
2
m
=
1
x

y
P
n
(
x
)
P
n

1
(
y
)

P
n
(
y
)
P
n

1
(
x
)
d
2
n

1
n
= 1
;
2
;
3
; ::: :
(6)
We will conside he ollowing h ee classical monic o hogonal p olynomials (CMOP)
which a e solu ions o he die ence equa ion (1).
I.
The Meixne p olynomials, o hogonal wi h esp ec o he weigh unc ion

(
x
) sup-
p o ed on [0
;
1
), wi h

(
x
) =
x; 
(
x
) =
 

x
(1


)
;
0
<  <
1
;  >
0
; 
n
=
n
(1


)
;
and
R
n
=
1
(


1)
n
; 
(
x
) =

x
(

+
x
)
(

)(1 +
x
)
; d
2
n
=
n
!(

)
n

n
(1


)

+2
n
:
I I.
The K a chuk p olynomials, o hogonal wi h esp ec o he weigh unc ion

(
x
) sup-
p o ed on [0
; N
], wi h
n

N

(
x
) =
x; 
(
x
) =
N p

x
1

p
;
0
< p <
1
; 
n
=
n
1

p
;
3
and
R
n
= (
p

1)
n
; 
(
x
) =
p
x
N
!(1

p
)
N

x
(
N
+ 1

x
)(1 +
x
)
; d
2
n
=
n
!
N
!
p
n
(1

p
)
n
(
N

n
)!
:
I I I.
The Cha lie p olynomials, o hogonal wi h esp ec o he weigh unc ion

(
x
) sup-
p o ed on [0
;
1
), wi h

(
x
) =
x; ; 
(
x
) =


x ;  >
0
; 
n
=
n;
and
R
n
= (

1)
n
; 
(
x
) =

x
e


(1 +
x
)
; d
2
n
=
n
!

n
:
They sa is y he so called s uc u e ela ions
x
n
5
M
 ;
n
(
x
) =

(1



n
)


1
M
 ;
n

1
(
x
) +
M
 ;
n
(
x
)
;
(7)
x
n
5
K
p
n
(
x
) =
p
(
N

n
+ 1)
K
p
n

1
(
x
) +
K
p
n
(
x
)
;
(8)
x
n
5
C

n
(
x
) =
C

n

1
(
x
) +
C

n
(
x
)
:
(9)
These classical p olynomials can b e ep esen ed as hyp e geome ic unc ions (see [11] page
49, sec ion
2.7
)
M
 ;
n
(
x
) = (

)
n

n
(


1)
n
2
F
1


n;

x





1

1


;
(10)
K
p
n
(
x
) =
(

p
)
n
N
!
(
N

n
)!
2
F
1


n;

x

N




1
p

;
(11)
C

n
(
x
) = (


)
n
2
F
0


n;

x






1


;
(12)
whe e he hyp e geome ic unc ion is dened by
p
F
q

a
1
;a
2
;:::;a
p
b
1
;b
2
;:::;b
q
j
x
) =
1
X
k
=0
(
a
1
)
k
(
a
2
)
k
  
(
a
p
)
k
(
b
1
)
k
(
b
2
)
k

(
b
q
)
k
x
k
k
!
;
(
a
)
0
:= 1
;
(
a
)
k
:=
a
(
a
+ 1)(
a
+ 2)

(
a
+
k

1)
; k
= 1
;
2
;
3
; :::
As a consequence o hese ep esen a ions we can deduce
M
 ;
n
(0) =

n
(


1)
n
(
n
+

)
(

)
; K
p
n
(0) =
(

p
)
n
N
!
(
N

n
)!
; C

n
(0) = (


)
n
:
(13)
We ha e p o ed (see [2] o mula (26)) he ollowing p op e y o he ke nels o he Meixne
p olynomials.
K e
M
n

1
(
x;
0)

n

1
X
m
=0
M
 ;
m
(
x
)
M
 ;
m
(0)
d
2
m
=
(

1)
n

1
(1


)
n
+


1
n
!
5
M
 ;
n
(
x
)
:
(14)
I is s aigh o wa d o show ha he ke nels o
K
p
n
(
x
) and
C

n
(
x
) e i y he ollowing
ela ions:
K e
K
n

1
(
x;
0)

n

1
X
m
=0
K
p
m
(
x
)
K
p
m
(0)
d
2
m
=
(
p

1)
1

n
n
!
5
K
p
n
(
x
)
;
(15)
K e
C
n

1
(
x;
0)

n

1
X
m
=0
C

m
(
x
)
C

m
(0)
d
2
m
=
(

1)
n

1
n
!
5
C

n
(
x
)
:
(16)
4
x
3 The deni ion, o hogonal ela ion and ep esen a ion
as hyp e geome ic se ies.
Conside he linea unc ional
U
on he linea space o p olynomials wi h eal co ecien s
dened as
<
U
; P >
=
<
L
; P >
+
AP
(0)
; x
2
IN
; A

0
;
(17)
whe e
L
is a classical momen unc ional (2) asso cia ed o some classical p olynomials o a
disc e e a iable.
We will de e mine he monic p olynomials
P
A
n
(
x
) which a e o hogonal wi h esp ec o
he unc ional
U
and p o e ha hey exis o all p osi i e
A
(see (22) om b elow).
To ob ain his, we can w i e he Fou ie expansion o such gene alized p olynomials
P
A
n
(
x
) =
P
n
(
x
) +
n

1
X
k
=0
a
n;k
P
k
(
x
)
;
(18)
whe e
P
n
deno es he classical monic o hogonal p olynomial (CMOP) o deg ee
n
.
In o de o nd he unknown co ecien s
a
n;k
we will use he o hogonali y o he p oly-
nomials
P
A
n
(
x
) wi h esp ec o
U
, i.e.,
<
U
; P
A
n
(
x
)
P
k
(
x
)
>
= 0
8
k < n:
Now pu ing (18) in (17) we nd:
<
U
; P
A
n
(
x
)
P
k
(
x
)
>
=
<
L
; P
A
n
(
x
)
P
k
(
x
)
>
+
AP
A
n
(0)
P
k
(0)
:
(19)
I we use he decomp osi ion (18) and aking in o accoun he o hogonali y o he classical
o hogonal p olynomials wi h esp ec o he linea unc ional
L
, hen he co ecien s
a
n;k
a e
gi en by:
a
n;k
=

A
P
A
n
(0)
P
k
(0)
d
2
k
:
(20)
Finally he equa ion (18) p o ides us he exp ession
P
A
n
(
x
) =
P
n
(
x
)

AP
A
n
(0)
n

1
X
k
=0
P
k
(0)
P
k
(
x
)
d
2
k
:
(21)
F om (21) we can conclude ha he ep esen a ion o
P
A
n
(
x
) exis s o any p osi i e alue
o he mass
A
. To ob ain his i is enough o e alua e (21) in
x
= 0,
1 +
A
n

1
X
k
=0
(
P
k
(0))
2
d
2
k
!
P
A
n
(0) =
P
n
(0)
6
= 0
;
(22)
and use he ac ha
1 +
A
n

1
X
k
=0
(
P
k
(0))
2
d
2
k
>
0
n
= 1
;
2
;
3
; :::
F om (22) we can deduce he alues o
P
A
n
(0) as ollows
5

P
A
n
(0) =
P
n
(0)
1 +
A
n

1
X
k
=0
(
P
k
(0))
2
d
2
k
:
(23)
Now in o de o ob ain an explici exp ession o hese p olynomials we need some p op-
e ies o he ke nels o he CMOP. In [2] we sol ed his p oblem o he classical Meixne
p olynomials. In his wo k we will p o e a simila esul o Cha lie and K a chuk p olyno-
mials.
Doing some algeb aic calcula ions in (21) and aking in o accoun o mulas (14)-(16) we
ob ain he ollowing h ee exp essions o he gene alized p olynomials :
Fo Meixne p olynomials:
M
 ;;A
n
(
x
) =
M
 ;
n
(
x
)

AM
 ;;A
n
(0)
(

1)
n

1
(1


)
n
+


1
n
!
5
M
 ;
n
(
x
)
:
(24)
Fo K a chuk p olynomials:
K
p;A
n
(
x
) =
K
p
n
(
x
)

AK
p;A
n
(0)
(
p

1)
n

1
n
!
5
K
p
n
(
x
)
:
(25)
Fo Cha lie p olynomials:
C
;A
n
(
x
) =
C

n
(
x
)

AC
;A
n
(0)
(

1)
n

1
n
!
5
C

n
(
x
)
:
(26)
In he ab o e o mula he alues o he p olynomials in
x
= 0 could b e deduced om
(23). Then, we ob ain he ollowing analy ic exp ession o he Pe u b ed Monic O hogonal
Polynomials (PMOP)
P
A
n
(
x
) o
x
6
= 0 (when
x
= 0 we can use (23) )
M
 ;;A
n
(
x
) =
M
 ;
n
(
x
) +
B
n
5
M
 ;
n
(
x
) = (
I
+
B
n
5
)
M
 ;
n
(
x
)
;
(27)
K
p;A
n
(
x
) =
K
p
n
(
x
) +
A
n
5
K
p
n
(
x
) = (
I
+
A
n
5
)
K
p
n
(
x
)
;
(28)
C
;A
n
(
x
) =
C

n
(
x
) +
D
n
5
C

n
(
x
) = (
I
+
D
n
5
)
C

n
(
x
)
;
(29)
whe e
A
n
; B
n
and
D
n
a e cons an s gi en by:
B
n
=
A

n
(1


)


1
(

)
n
n
!(1 +
AK e
M
n

1
(0
;
0))
;
A
n
=
A
N
!
n
!(
N

n
)!
p
n
(1

p
)
1

n
(1 +
AK e
K
n

1
(0
;
0))
;
D
n
=
A

n
n
!(1 +
AK e
C
n

1
(0
;
0))
:
Rema k
: Using he Ro d igues o mula (4), some ex ension o i ollows in a s aigh o wa d
way.
M
 ;;A
n
(
x
) = (
I
+
B
n
5
)


n
(1


)
n
(
x
+ 1)

x
(

+
x
)
5
(
n
)

x
(
x
+

+
n
)
(
x
+ 1)

;
K
p;A
n
(
x
) = (
I
+
A
n
5
)

(

1)
n
(
x
+ 1)(
N

x
+ 1)
p
x
(1

p
)

x
5
(
n
)
p
x
(1

p
)

x
(
x
+ 1)(
N

x

n
+ 1)

;
6
C
;A
n
(
x
) = (
I
+
D
n
5
)

(

1)
n
(
x
+ 1)

x
5
(
n
)

x
+
n
(
x
+ 1)

:
Now we can es ablish he ollowing ep esen a ion as hyp e geome ic unc ions o he
gene alized p olynomials:
P op osi ion 1
The o hogonal polynomials
M
 ;;A
n
(
x
)
,
K
p;A
n
(
x
)
and
C
;A
n
(
x
)
a e, up o a
cons an ac o , gene alized hype geome ic unc ions. Mo e p ecisely
M
 ;;A
n
(
x
) = (

)
n

n
(


1)
n
3
F
2


n;

x;
1+
xB

1
n
 ; xB

1
n




1

1


;
(30)
K
p;A
n
(
x
) =
n
!(

p
)
n
N
!
n
!(
N

n
)!
3
F
2


n;

x;
1+
xA

1
n

N ; xA

1
n




1
p

;
(31)
C
;A
n
(
x
) = (


)
n
3
F
1


n;

x;
1+
xD

1
n
xD

1
n





1


:
(32)
Ske ch o he P o o : The p o o o his P op osi ion is simila o he p o o o he Meixne
case (see [2]). To ob ain he desi ed esul we need o pu he hyp e geome ic ep esen a ion
o hese p olynomials in o mulas (27), (28) and (29) and do some algeb aic calcula ions.
He e he co ecien s
xA

1
n
,
xB

1
n
and
xC

1
n
a e eal numb e s. In he case when hey a e
nonp osi i e in ege s we need o ake he analy ic con inua ion o he hyp e geome ic se ies
(30), (31) and (32).
I is s aigh o wa d o show ha o
A
= 0 he hyp e geome ic unc ions (30), (31) and
(32) yield o classical p olynomials (10), (11) and (12).
x
4 A second o de die ence equa ion.
In [2] we p o ed ha he Meixne p olynomials sa is y a second o de die ence equa ion.
To p o e his esul we only used ha in he die ence equa ion o hyp e geome ic yp e o
Meixne p olynomials he unc ion

(
x
) is equal o x. Taking in o accoun ha , o Cha lie
and K a chuk p olynomials,

(
x
) =
x
, hen he ollowing Theo em holds:
Theo em 1
The polynomials
M
 ;;A
n
(
x
)
; K
p;A
n
(
x
)
and
C
;A
n
(
x
)
sa is y a second o de linea
die ence equa ion
[
x
+
C
(
C 
n
+

n


(
x
))](
x

1)
4 5
P
A
n
(
x
) + (
x

1)

(
x
)
4
P
A
n
(
x
)+
+
C
[(

(
x
)

C 
n
)(

n

1


(
x

1)) +

n
(

n
+
C
) + (
x
+
C 
n
)
4

(
x
)]
4
P
A
n
(
x
)
+(
x

1)

n
P
A
n
(
x
) +
C 
n
[

n

1


(
x

1) +
C
(
4

(
x
) +

n
)]
P
A
n
(
x
) = 0
;
(33)
whe e
x
= 0
;
1
;
2
; :::;
5
(
x
) =
(
x
)

(
x

1)
;
4
(
x
) =
(
x
+ 1)

(
x
)
;
and by
P
A
n
we deno e he gene alized Meixne , K a chuk o Cha lie polynomials and C is
he cons an
B
n
; A
n
o
D
n
, espec i ely, which is a unc ion o
n
(see Sec ion 3 ).
The p o o o his Theo em o he Meixne case was gi en in [2]. He e we p o ide he
p o o o b o h h ee cases.
7
P o o : We will s a om he ep esen a ions (27), (28) and (29) o he gene alized p olyno-
mials
P
A
n
(
x
) =
P
n
(
x
) +
C
5
P
n
(
x
). Mul iplying his exp ession by
x
, using he second o de
die ence equa ion ha hese classical p olynomials sa is y
x
4 5
P
n
(
x
) +

(
x
)
4
P
n
(
x
) +

n
P
n
(
x
) = 0
;
(34)
and using he iden i y
5
P
n
(
x
) =
4
P
n
(
x
)
45
P
n
(
x
) we ob ain
xP
A
n
(
x
) = (
x
+
C 
n
)
P
n
(
x
) +
C
(
x
+

(
x
))
4
P
n
(
x
)
:
(35)
Now i we apply he op e a o
4
o (35), om (34) he equa ion
x
4
P
A
n
(
x
) = [
x

C 
(
x
)]
4
P
n
(
x
)

C 
n
P
n
(
x
) (36)
ollows. In he same way i we apply in (36) he op e a o
5
and using (34) we nd
x
(
x

1)
4 5
P
A
n
(
x
) =
=

[(
x

1)

(
x
) +
C 
(
x
)(

n


(
x

1)

1) +
C x
(

n
+
4

(
x
))]
4
P
n
(
x
)


[
x

1 +
C
(

n


(
x

1)

1)]

n
P
n
(
x
)
:
(37)
Now om (35),(36) and (37) he ollowing de e minan anishes






xP
A
n
(
x
)
a
(
x
)
b
(
x
)
x
4
P
A
n
(
x
)
c
(
x
)
d
(
x
)
x
(
x

1)
4 5
P
A
n
(
x
)
e
(
x
)
(
x
)






= 0
;
(38)
whe e
a
(
x
) = (
x
+
C 
n
)
; b
(
x
) =
C
(
x
+

(
x
))
; c
(
x
) =

C 
n
;
d
(
x
) =
x

C 
(
x
)
; e
(
x
) =

[
x

1 +
C
(

n

1


(
x

1))]

n
;
(
x
) =

[(
x

1)

(
x
) +
C
[

(
x
)(

n

1


(
x

1)) +
x
(

n
+
4

(
x
))]
:
Expanding he de e minan in (38) by he  s column and di iding by
x
2
;
he Theo em
ollows.
The die ence equa ion o he p e ious heo em (33) akes he o m:
Meixne case
M
 ;;A
n
(
x
)
x
+
B
n
[(1


)(
x
+
n
+
nB
n
)

 
]
g
(
x

1)
4 5
M
 ;;A
n
(
x
)+
(
x

1)[
 

x
(1


)]
4
M
 ;;A
n
(
x
)+
+
B
n
(1


)[
 
(
n
+
nB
n
+ 2
x

1) + (1


)(
x
+
n
2

(
x
+
nB
n
)(
x
+
n
))+
+2
nB
n
]

 
(1 +
 
)
g 4
M
 ;;A
n
(
x
) + (
x

1)
n
(1


)
M
 ;;A
n
(
x
)+
nB
n
(1


)[(1


)(
x
+
n
+
nB
n

B
n

1)

1

 
]
M
 ;;A
n
(
x
) = 0
:
(39)
8
K a chuk case
K
p;A
n
(
x
)
x
+
A
n
[
x
+
n
+
nA
n
1

p

N p
1

p
]
g
(
x

1)
4 5
K
p;A
n
(
x
)+
(
x

1)[
N p

x
1

p
]
4
K
p;A
n
(
x
)+
+
A
n
1
1

p
[
N p
1

p
(
n
+
nA
n
+ 2
x

1) +
x
+
n
2

(
x
+
nA
n
)(
x
+
n
)
1

p
+
+2
nA
n
]

N p
1

p
(1 +
N p
1

p
)
g 4
K
p;A
n
(
x
) +
n
1

p
(
x

1)
K
p;A
n
(
x
)+
nA
n
1

p
[
x
+
n
+
nA
n

A
n

1

N p
1

p

1]
K
p;A
n
(
x
) = 0
:
(40)
Cha lie case
C
;A
n
(
x
)
[
x
+
D
n
(
x
+
n
+
nD
n


)](
x

1)
4 5
C
;A
n
(
x
)+
(
x

1)(


x
)
4
C
;A
n
(
x
)+
+
D
n
[
n
2
+
x

2

+ 2
nD
n

(
n

x


)(
x


+
nD
n
)]
4
C
;A
n
(
x
)+
+(
x

1)
nC
;A
n
(
x
) +
nD
n
(
x
+
n



2 +
nD
n

D
n
)
C
;A
n
(
x
) = 0
:
(41)
x
5 Th ee Te m Recu ence Rela ions.
The gene alized p olynomials sa is y a h ee e m ecu ence ela ion (TTRR) o he o m
xP
A
n
(
x
) =
P
A
n
+1
(
x
) +

A
n
P
A
n
(
x
) +

A
n
P
A
n

1
(
x
)
; n

0
P
A

1
(
x
) = 0 and
P
A
0
(
x
) = 1
:
(42)
This is a simple consequence o hei o hogonali y wi h esp ec o a p osi i e unc ional
(see [6] o [11]). To ob ain he explici o mula o he ecu ence co ecien s we can com-
pa e he co ecien s o
x
n
in he wo sides o (42). Le
b
A
n
b e he co ecien o
x
n

1
in he
expansion
P
A
n
(
x
) =
x
n
+
b
A
n
x
n

1
+
:::
, hen :

A
n
=
b
A
n

b
A
n
+1
. To calcula e

A
n
is sucien
o e alua e (42) in
x
= 0 and ema k ha
P
A
n
(0)
6
= 0.
In o de o ob ain a gene al exp ession o he co ecien

A
n
we can use he o mulas (27),
(28) and (29) o he gene alized p olynomials
P
A
n
(
x
) =
P
n
(
x
) +
C
5
P
n
(
x
), whe e
C
=
B
n
; A
n
o
D
n
esp ec i ely. Doing some algeb aic calcula ions we nd ha
b
A
n
=
b
n
+
nC
, whe e
b
n
de-
no es he co ecien o he
n

1 p owe in he classical p olynomials
P
n
(
x
) =
x
n
+
b
n
x
n

1
+
:::
.
Using hese o mulas and he main da a [11] o classical p olynomials we ob ain o gen-
e alized Meixne , K a chuk and Cha lie p olynomials he ollowing TTRR co ecien s :
I
Meixne p olynomials:
b
n
=
n


1
(

+
n

1
2

+ 1

)
; 
n
=
n
+

(
n
+

)
1


;
b
A
n
=
n


1
(

+
n

1
2

+ 1

) +
A

n
(1


)


1
(

)
n
(
n

1)!(1 +
AK e
M
n

1
(0
;
0))
;
9