On the properties for modifications of classical orthogonal polynomials of discrete variables
Abstract
We consider a modi cation of moment functionals for some classical polynomials of a discrete variable by adding a mass point at x = 0. We obtain the resulting orthogonal polynomials, identify them as hypergeometric functions and derive the second order di erence equation which these polynomials satisfy. The corresponding tridiagonal matrices and associated polynomials were also studied.
Full text
ON THE PROPERTIES FOR MODIFICATIONS OF CLASSICAL
ORTHOGONAL POLYNOMIALS OF DISCRETE VARIABLES.
1
R.
Al a ez-No da se, A.G.Ga ca and F. Ma cellan
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,Leganes,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 classica ion:
33A65
Abs ac
We conside a mo dica 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 die 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 dica ion o a linea unc-
ional in he linea space o p olynomials wi h eal co ecien 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 cellan and Ma oni [10]
analyzed a mo e gene al si ua ion o egula ( quasi-deni e ) linea unc ionals, i.e., such
ha he p incipal subma ices o he co esp onding inni 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 dica 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 ecien s dened 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 die 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 ecien 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 die 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 die 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 die ence equa ion o gene alized Meixne p olynomials, as well as he inni e o de
die 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 inni e o de die 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 die 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 cellan, 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 die 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 die 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
dened as a mo dica 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 ecien 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 die 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 ecien s dened 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 die 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 oel-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 die 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 dened 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 deni 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 ecien s
dened 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 ecien 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 ecien 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 ecien 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 die ence equa ion.
In [2] we p o ed ha he Meixne p olynomials sa is y a second o de die ence equa ion.
To p o e his esul we only used ha in he die 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
die 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
die 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 die 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 ecien s we can com-
pa e he co ecien s o
x
n
in he wo sides o (42). Le
b
A
n
b e he co ecien 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 sucien
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 ecien
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 ecien 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 ecien 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