ETNA
Electronic Transactions on Numerical Analysis.
Volume 27, pp. 113-123, 2007.
Copyright  2007, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
ORTHOGONAL POLYNOMIALS AND RECURRENCE EQUATIONS, OPERATOR
EQUATIONS AND FACTORIZATION
WOLFRAM KOEPF
Abstract. This article surveys the classical orthogonal polynomial systems of the Hahn class, which are solutions of second-order differential, difference or -difference equations.
Orthogonal families satisfy three-term recurrence equations. Example applications of an algorithm to determine
whether a three-term recurrence equation has solutions in the Hahn class—implemented in the computer algebra
system Maple—are given.
Modifications of these families, in particular associated orthogonal systems, satisfy fourth-order operator equations. A factorization of these equations leads to a solution basis.
Key words. orthogonal polynomials, Hahn class, differential equations, difference equations, -difference equations, hypergeometric functions, factorization of operator polynomials, computer algebra, Maple
AMS subject classifications. 33C45, 33C20, 33D45, 33D15, 39A70
1. Classical Orthogonal Polynomials. To define families of orthogonal polynomials
on the real line, one uses a scalar product
!
!
%$%&
with nonnegative measure
and support in a real interval " #
(which can be infinite in
one or both directions).
As special cases one considers
( )+*(
'
absolutely continuous
measures
,
*
'
discrete measures
supported
in
, ,
*!
'
and discrete
measures
supported in -/. for some base -1032 .
!
A family 465
of polynomials
(1.1)
4 5
)87
5:9
5
7<; 5=?>
5
is called orthogonal w. r. t. the measure
46J
465
9
7; ; 5@=AB9
5
!
LK
7<; ; ; 5@=CB9+DEDED F7
8I
5HG
5
, if
I
A
8I
G
5
if
if
M
M
ON
G
O
N
P
The classical orthogonal polynomials can be defined as the common polynomial solutions
(1.1) of a differential equation of the type
Q
(1.2)
NVXW
!
4 5
; ; ! 9SR 4 5
; ! 9UT
R !
5 4 5
!)OI
P
R YZ(
9O[ %
\I
G
The case
shows
that
must be a!first
order polynomial:
,
N]_^
^
Q
whereas
because
of
the
function
turns
out
to
be
a
polynomial
of
degree
`
:
A 9
$b 9Vc
5
edfN Ngd
T
Q !a
#
. Considering the coefficient of , one finally gets 5
#
Wh 9 i
.
A complete characterization of the solution families of the differential equation (1.1) can
be given and leads to the following scheme ([2], 1929):
j
Received January 18, 2003. Accepted for publication March 04, 2003. Recommended by F. Marcellán.
Department of Mathematics, University of Kassel, Heinrich-Plett-Str. 40, D-34132 Kassel
(koepf@mathematik.uni-kassel.de).
113
ETNA
Kent State University
etna@mcs.kent.edu
114
WOLFRAM KOEPF
Q
!)+I
Q
!)kW
Q
!)O
Q
!)O A
Q
!)O A
5
powers
Hermite polynomials
Laguerre polynomials
powers, Bessel polynomials
Jacobi polynomials
dlW
All other solutions of (1.1) are translations of the above systems.1 It turns out that—besides
the powers—all these polynomial systems are orthogonal. However, the weight function of
the Bessel polynomials is not defined in a real interval but in the complex plane. However,
in both the Jacobi and Bessel case, for specific values of the parameters finite real orthogonal
families arise [1, 13, 15, 20].
*!
The weight function
corresponding to the system satisfies Pearson’s differential
equation
*oH
Q
nm
R !*!
from which it follows that
*)
p
Q
[/q:wrtsu%v
sxuyv%zb{
!
P
Further details can be found in [17].
2. Classical Discrete Families. The classical discrete orthogonal polynomials can be
defined as the polynomial solutions of the difference equation
!|~}
Q
(2.1)
|~!aZ!
9
! 9SR |
45
WhBd
465
9T
5<45
)8I~
}~!€L!Bdd+Wh
where
and
denote the forward and
backward difference operators, respectively.
_I
Nk„W
R !‚ƒ 9][ %
Q !
G
Again, from (2.1)N‡it8follows
that
(using
) and
A 9 $ 9†c
^
5
ˆ
f
d
N
!
3
N
l
d
h
W
T
9 i
#
#
(using
). The coefficient of
yields also 5
.
The classical discrete systems can be classified according to the scheme ([11]–[12], 1962,
[18], 1991):
Q
!)8I
Q
!)ˆW
Q
!)+
ŒŠŽ Q
!B^
5
+!…dW‰ DEDŠD !…d‹N
9
W‰
falling factorials
translated Charlier polynomials
falling factorials, Charlier, Meixner, Krawtchouk polynomials
Hahn polynomials
Again, all these families—besides the falling factorials—form orthogonal polynomial families.
*!
The weight function
corresponding to the system satisfies Pearson’s difference
equation
|
from which it follows that
m
*
*oH
Q
9
*!
Wh
Q
R !*!
! 9SR Q
9
Wh
P
1 If ‘“’(” is constant, one gets translated Hermite polynomials, if <‘•’(” has degree 1, then translated Laguerre
polynomials result, and if the degree of <‘•’(” is 2, then Jacobi polynomials or Bessel polynomials follow depending
on whether or not ‘“’(” has two different zeros (giving the interval bounds – and — if they are real) or one double
zero.
ETNA
Kent State University
etna@mcs.kent.edu
ORTHOGONAL POLYNOMIALS, RECURRENCE EQUATIONS AND FACTORIZATION
115
3. Hypergeometric Functions. The power series
¢
whose coefficients §
ž
P PŠP # ˜
# > E
¡ Ÿ
˜(™6š›
$ b$
š œ
U
P PEP
¢t£?¤¦¥
> Š
œ
œ
œ
¢
¢
œ
have rational term
ratio
¥
¢t¨ >
§
§
¥
¢
¢t¨ >
¢t¨ >
¢
7
¢
9
¥

DEDŠD 7
# >
$ E
7
D DŠD ©
>
9
©7
¢
¢
9
# ˜
$ š
9
7

is called the generalized hypergeometric
function. The summand
7

hypergeometric term w. r. t. .
Hence, because of
*
W‰
9
*!
Q 9SR !
Q
 ¢
§
¢
¢
¥
is called a

W‰
9
Wh
9
*!
R and since Q
and
are polynomials, the weight functions
of the classical discrete
orthogonal polynomials form hypergeometric terms w. r. t. the variable .
For the coefficients of the generalized hypergeometric function one obtains the formula
ž
P PŠP # ˜
# > E
$ b$
P PŠP
š œ
†
> E
˜ª™š›
¢£?¤
œ
9
Wh DŠDED 9
¡ Ÿ
¢
# >
©$ > ¢
¢
DŠDED ¢
# ˜
$
DEDED
¢
š
7?«
œ

71dW‰ œ
œydenotes

where # ¢
# #
#
the Pochhammer symbol or shifted factorial.
Simple examples of hypergeometric functions are the exponential function
[‰¬
the sine function
¤ ™ ¤
­y®•¯
­
­y®•¯
D ¤
™ > ›
¯
¯
¯
t%µª¶
t%µª¶
µ


d
W

9
°i± ^
ž
A
d
²
œ
œ
œ
  ¶œ¹E
œ
yº¼»“½¾ 5
³
³t·
asµ well
as ³E´
,¸
, . . . , but for example not
·
.
From the difference equation, one can determine a hypergeometric representation; see [9,






18]. As an example, the Hahn polynomials are given by2

¿
5
!À
§
Á
dfNÊdfÄN
%ÂHB
C ™ A ›
§
W
9
W(Šd¼Â
9
9
§
9
ž
Á
œ
W
P
œ
œ
œ
œ
4. q -Orthogonal Polynomials. To define orthogonal polynomials
on the lattice -(. , we
need some more notation.
The operator ([6], 1949)
Å
š
!
!dS
Wfd
-
-
2 In the Russian literature the parameters Æ and Ç are interchanged, È
is replaced by ÈÉnÊ , and the standardization is different; see [18, p. 54]. The given definition is the one of the American school; see [7].
ETNA
Kent State University
etna@mcs.kent.edu
116
WOLFRAM KOEPF
is called Hahn’s q-difference operator, and the - -brackets are defined by
"
7i&
Wfd
š
¢
-
Wfd
9ODŠDED‰9
-
-
¢ =Ë>
P
Å
š¸ Í
by de l’Hospital’s rule, the
limit
7
- -equivalent of the term
since
9
-
®“Ì
Since
kW
W
-Î
®“Ì
š¸ Í
8 ; !t
!
š
>
yields the continuous case. The - -brackets are the
"
>
7i&
7
š
P
The - -orthogonal polynomials of the Hahn class can be defined as the polynomial solutions
Å
Å
Å
of the - -difference equation
Q
š
>yÏ š 4 5
! 9SR 9UT
š 4 5
R !B8(
5 4 5
9U[ y
!B+I
OI
!)
P
A
9
$b
G
Analogously
to the
classical
case, one gets
,Q
#
kd
N&
N3dWE&
dÐ N&
T
š
š .
5
#"
"
>Ï š "
The classical - -systems can be classified according to the scheme ([19], 1993)3
Q
B+I
Q
BkW
Q
BO
ŒŠŽ Q
*
and
powers and - -Pochhammer symbols (5.1)
discrete - -Hermite II polynomials
- -Charlier-, - -Laguerre-, - -Meixner polynomials
- -Hahn polynomials, Big - -Jacobi polynomials
!tB^
The weight function
tion
9Uc
corresponding to the system satisfies the - -Pearson difference equaÅ
*oH
š m Q
R !*!
P
Hence we have
*
*
! 9
Q
-
Q
dW‰ R !
P
-
5. Basic Hypergeometric Series. ÒInstead
of considering series whose coefficients ¢
7<
±
¥
¢
0OÑ
have rational term ratio ¢¨ >
,
we
can also consider such series whose coeffi ¢ ±
¥
¥
¢ 03Ñ
cients ¢ have term ratio ¢t¨ >
w. r. t. some base -Y0g2 .
¥
¥
¥
This
leads to the - -hypergeometric
(or basic hypergeometric) series
Now the
ž
¡ Ÿ
P PŠP # Ó
# > E
À<
¢
¢
ӏÔÃÕ¦›
P
$ %$
Õ
S
œ
P
E
Š
P
P
¢t£?¤¥
>
œ
œ
œ
œ
coefficients—that are called - -hypergeometric
terms—are
¢
¥
À
¢
#> $ À ¢
> -
DEDED À
¢
¢
# Ó DEDŠD $ À ¢
Õ -
À
-
¢
m
d€Wh ¢
-
given by the formula
¨ Õ = Ó
©Ö×y
o >
3 Historically many more systems were introduced that fit in this list. A complete classification that boils down
to essentially seven different types can be found in [14], see also [16].
ETNA
Kent State University
etna@mcs.kent.edu
117
ORTHOGONAL POLYNOMIALS, RECURRENCE EQUATIONS AND FACTORIZATION
where
(5.1)
À
#
-
¢ Ë
= >
Ø
W¼d
¢
Ù #i-
Ù £?¤
denotes the - -Pochhammer symbol. Since
*
*!
Q
9
-
Q
dlWh R !
-
*!
is rational, the
weight
of a - -orthogonal system of the Hahn tableau is a - -hypergeometric
¢
3
- .
term w. r. t.
Classical orthogonal systems have (generally several) - -hypergeometric equivalents. For
example, the Big - -Jacobi polynomials are - -equivalents of the Jacobi polynomials and are
given by
465
!À
#
%$/ c À
-
C Ô A ›
-
=5 b$‰
#
c
#@-
ž
5 ¨ > y
-
À
-
œ
-
P
-
œ
œ
œ
œ
The families of the Hahn class that we considered in this article—with
absolutely continuous,
arithmetically discrete and geometrically discrete weights—can be generated by suitable limit
procedures from the Big - -Jacobi polynomials.
6. Computing the Difference Equation from a Recurrence Equation. From the differential
or ( - )-difference equation one can
determine
the three-term recurrence equation for
R !
4 5
in terms of the coefficients of Q
and
. Using a computer algebra system like
Maple one can easily compute how the three-term recurrence
equation corresponding to a
%$‰ c y
[
given system can be expressed by the five parameters #
and .
As an example case, we consider the discrete situation.
This defines the forward and backward difference operators
Ú
Delta:=(f,x)->subs(x=x+1,f)-f:
Ú
nabla:=(f,x)->f-subs(x=x-1,f):
We consider the three highest coefficients of the orthogonal polynomial
Ú
p:=k[n]*xˆn+kprime[n]*xˆ(n-1)+kprimeprime[n]*xˆ(n-2);
Û
Ü87
5
5
9lÝbÞßtàâáäã
5
» 5@=Ë> ¾ 9lÝ%Þßtàâá~ãÞßtàâáäã
5
R !
» 5@=A ¾
and
we define the polynomials Q
and
with symbolically given coefficients
%$‰ c y¦ [
#
:
Ú
Ånå
sigma:=a*xˆ2+b*x+c:
Ú
tau:=d*x+e:
8I
The polynomial 4 5
satisfies the difference equation
with:
Ú
DE:=sigma*Delta(nabla(p,x),x)+tau*Delta(p,x)+lambda[n]*p;
æèç
dS^
Üé
7
5
#
5
9lÝ%Þßtàâá~ã
A
9
$
5
ÝbÞßtàâáäã
5
ndW‰ » 5=?>
9
5
Ý%Þßtàâá~ãÞßtàâáäã
W‰ » 5=?>
9lÝ%Þßtàâá~ã
d
9Uc E7
dS^
5
» 5@=A
!
We replace the powers
9
Wh 5 9lÝ%Þßtàâá~ã
5
5
» @
S
d
^
ÝbÞßtàâáäãÒÞßtàâá~ã
5 =Ë> ¾
5
ndW‰
¾ 9lÝbÞßtàâáäãÒÞßtàâá~ã
5
9
W‰
¾ 9lÝbÞßtàâáäãÒÞßtàâá~ã
5
©7
5 9lÝbÞßtàâáäã
¾ 9T
5
5
5
‰
W
!…dW‰ 5
9
5
and
9
Wh » 5@=?> ¾ 9lÝbÞßtàâáäãÒÞßtàâá~ã
» 5@=A ¾ 9
7
» 5=A ¾ 9
» 5=A ¾ dS7
5
5
!ndlWh 5
5
9U[ ©7
d
5
ÝbÞßtàâáäã
9
» 5@=Ë> ¾ 9lÝ%Þßtàâá~ãÞßtàâáäã
by the binomial theorem
5
5
5
9
W‰ 5
» 5=?> ¾
» 5=A ¾ Wh » 5@=A ¾
ETNA
Kent State University
etna@mcs.kent.edu
118
WOLFRAM KOEPF
Ú
DE:=subs(
ê
(x+1)ˆn=xˆn+n*xˆ(n-1)+n*(n-1)/2*xˆ(n-2),
Ú
(x+1)ˆ(n-1)=subs(n=n-1,xˆn+n*xˆ(n-1)+n*(n-1)/2*xˆ(n-2)),
Ú
(x+1)ˆ(n-2)=subs(n=n-2,xˆn+n*xˆ(n-1)+n*(n-1)/2*xˆ(n-2)),
Ú
(x-1)ˆn=xˆn-n*xˆ(n-1)+n*(n-1)/2*xˆ(n-2),
Ú
(x-1)ˆ(n-1)=subs(n=n-1,xˆn-n*xˆ(n-1)+n*(n-1)/2*xˆ(n-2)),
Ú
(x-1)ˆ(n-2)=subs(n=n-2,xˆn-n*xˆ(n-1)+n*(n-1)/2*xˆ(n-2)) ë ,
Ú
DE):
and collect coefficients:
Ú
Ú
de:=collect(simplify(DE/xˆ(n-4)),x);
܈
ì ã
^
9
d
7
N A
5
Ý%Þßtàâá~ã
#
9
#
ÝbÞßtàâáäã
9Uc 7
W
9
^
5
N A
NÄ C
9Uï
d
5
c 7
ÝbÞßtàâáäã
d
°
[ÃÝ%Þßtàâá~ã
^
N A
5
5
°
d
^
5
ð
^
N
W
9
^
N A
5
d
°
[ÃÝ%Þßtàâá~ãÞßtàâáäã
5
N A
5
5
5
5
N
N
#
N
5
^
9
5
5
9
5
$ ÝbÞßtàâáäã
ÝbÞßtàâáäãÒÞßtàâá~ã
5
W
[ 7
A
B7
^
N A
5
^
Ý%Þßtàâá~ã
5
N‚d
9†[ÃÝ%Þßtàâá~ã
W
d
^
N
N
5
9
°
9
5
[ÝbÞßtàâáäãÒÞßtàâá~ã
^
[ÝbÞßtàâáäã
^
9U[ÝbÞßtàâáäã
cÄÝbÞßtàâáäãÒÞßtàâá~ã
N
5
N A
5
W
5
5
[ 7
5
N
5
9
5
5
N A
5
[ÝbÞßtàâáäãÞßtàâáäã
5
N A
5
N
Equating the highest coefficient gives the already mentioned identity for
Ú
9UT
W
N
W
9
N A
5
ÝbÞßtàâáäã
[ÃÝ%Þßtàâá~ãÞßtàâáäã
9Uï
5
°
$ ÝbÞßtàâáäã
°
°
9
$?7
9
ÝbÞßtàâáäãÒÞßtàâá~ã
#
$ ÝbÞßtàâáäãÒÞßtàâá~ã
N A cÄÝbÞßtàâáäãÒÞßtàâá~ã
d
d
Ý%Þßtàâá~ãÞßtàâáäã
#
N‚dS^
dSð
5
N A
N A
5
5
9
5
ÝbÞßtàâáäãÞßtàâáäã
5
Ý%Þßtàâá~ãÞßtàâáäã
ÝbÞßtàâáäã
ÝbÞßtàâáäã
cÄÝ%Þßtàâá~ã
Ý%Þßtàâá~ãÞßtàâáäã
dSð
dSð
5
9lT
5
5
9
ÝbÞßtàâáäã
#
dÐ
$ ÝbÞßtàâáäã
ÝbÞßtàâáäãÞßtàâáäã
ÝbÞßtàâáäã
5
9
N A
N
5
9
5
@îí
5
B7
^
$ ÝbÞßtàâáäãÒÞßtàâá~ã
ð
^
5
9
7
5
W
N
cÄÝ%Þßtàâá~ã
N‚d
9
N
9UT
ÝbÞßtàâáäãÞßtàâáäã
^
9
5
9UcÄÝbÞßtàâáäãÒÞßtàâá~ã
d
N
5
9†[ 7
N‚dS^Ã
N A
5
5
7
#
ÝbÞßtàâáäãÞßtàâáäã
9U[ÃÝ%Þßtàâá~ãÞßtàâáäã
9UcÄÝbÞßtàâáäã
N
9
#
5
$ ÝbÞßtàâáäãÒÞßtàâá~ã
ï
Nd
5
dS$?7
5
5
[ÃÝ%Þßtàâá~ã
B7
9
T
:
5
rule1:=lambda[n]=solve(coeff(de,x,4),lambda[n]);
ßtñò“ãŠó
T
ôdfNä€d
5
9
#
NÄ
#
This result can be substituted into the differential equation:
Ú
de:=expand(subs(rule1,de)):
7 ;
7
Equating the second highest coefficient gives 5 as rational multiple of
Ú
5
rule2:=kprime[n]=solve(coeff(de,x,3),kprime[n]);
ßtñò•ãEõ
Ü
ÝbÞßtàâáäã
5
^
W
7
5
Nä
and equating the third highest coefficient gives
Ú
Ú
N3dÐ^
:dS^
7 ;;
5
$
^
#
9
9
^
$N…dÐ
#
9
^ [ N
as rational multiple of
7
5
:4
rule3:=kprimeprime[n]=solve(coeff(subs(rule2,de),x,2),
kprimeprime[n]);
4 Of course, taking into consideration the fourth highest coefficients yields ö/ø÷ ÷ ÷ as rational multiple of ö ø , and
so forth.
ETNA
Kent State University
etna@mcs.kent.edu
119
ORTHOGONAL POLYNOMIALS, RECURRENCE EQUATIONS AND FACTORIZATION
Ü
ßtñò“ãEù
c N A
ú
9
Ý%Þßtàâá~ãÞßtàâáäã
#
ðà A N
9
dlW ï
[ A N
²
9
$ A N A
5
d
²
N
5
d
[ N:$ 9
^(I)$ A N3d
9
dÐ^Ã A
7
ú
[ N A $BdÐ^(I
ú
9
W
[ A
$ A
ú
N
dlW ï
[
)N A $
c N€€dlW ï)c N
²
W‰^
9
$ A N C
²
9
ú
$ [
d
$
ú
ú
9
A N C ± ydû^
9
[ N A
²
9
#
c
d
#
d
#
^
9
#
$N C ²
9
#
[ N
ú
c ²
N
^(I
9
²
9
@?^
9
#
#
N:$ 9
[
d
A N A
²
[ ²
9
#
N3d
[ N A ²
°
9
#
iy
We consider the monic case, hence
Ú
k[n]:=1;
7
ôW
5
and therefore
Ú
rule2;
Ý%Þßtàâá~ã
Ú
Nä
5
N3dS^
$ 9
^
^¼€dS^
$N3dü
^
9
#
^ [ 9
NÄ
#
rule3;
ÝbÞßtàâáäãÒÞßtàâá~ã
dW ï
d
$N A c ²
dÐ^Ã A
d
9
ú
9
²
c
[ A
+N
5
#
ú
²
dlW ï
$ A N A
^ªI)$)N
$ A N C
²
9
d
A N C
$N A [9
d
$ A
ú
A N A ± ú
[ N A
:dÐ^
d
²
#
^(I)$ A Nd
9
9
#
^
9
Now we would like to find the coefficients
(see (7.1)):
Ú
ðà A N
9
Á
5
$
NÄ?d
#
#
[ Nۏd
ú
ú
c N A
ú
9
W‰^
9
°
9
#
[ N
ú
$ [)9
c Nè:dW ï
²
9
²
[ ^
9
²
9
#
c N
9 ² $N C #
N
:
S
d
ª
^
I
[
[ N:$
A
²
9
[
#
#
NÄy
²
9
[ A N
å
8I
and ý5 in the recurrence equation þ
RE:=P(n+1)-(x-beta[n])*P(n)+gamma[n]*P(n-1);
ÿûç
Ú
Ü
ûN
9
Wh6dV!…d‹Á
5
ûNÄ 9
ûN‚dlWh
ý 5
RE:=subs(
ê
P(n)=p,P(n+1)=subs(n=n+1,p),P(n-1)=subs(n=n-1,p) ë ,RE);
Ú
O » 5 ¨ > ¾ 9ÝbÞßtàâáäã
5 9ÝbÞßtàâáäãÞßtàâáäã
» 5@=Ë> ¾
5 ¨ >
5 ¨ >
dlndüÁ
? 5 9lÝ%Þßtàâá~ã
» 5@?
» 5@=A ¾ = > ¾ 9ÝbÞßtàâáäãÒÞßtàâá~ã
5
5
5
! » 5@=?> ¾ 9ÝbÞßtàâáäã
» 5@
» 5@=C ¾ 9
= A ¾ 9ÝbÞßtàâáäãÒÞßtàâá~ã
ý5
5@=Ë>
5 =?>
@
ÿûç
We substitute the already known formulas:
Ú
RE:=subs(
ê
rule2,subs(n=n+1,rule2),subs(n=n-1,rule2),
Ú
rule3,subs(n=n+1,rule3),subs(n=n-1,rule3) ë ,RE):
and get a highly complicated expansion
Ú
Ú
re:=simplify(numer(normal(RE))/xˆ(n-3)):
Á
%$‰ c
Equating the highest coefficient gives 5 as rational function of #
Ú
rule4:=beta[n]=
factor(solve(coeff(re,x,3),beta[n]));
Ú
ßtñò•ã
Ü+Á
5
ôd
)N A
#
9
^
$N A
#
dü
N3dÐ^)$
#
#
:dS^
9 ^
#
#
N‚dÐ^
NÄ?
and equating the second highest coefficient yields finally ý
Ú
Ú
5
rule5:=gamma[n]=factor(subs(rule4,
solve(coeff(re,x,2),gamma[n])));
[
9
#
A N
9
^
y¦ [
#
9
N
^
and :
$)N
9U[ NÄ
as rational function, too:
ETNA
Kent State University
etna@mcs.kent.edu
120
WOLFRAM KOEPF
Ü
ßtñò•ã
W ï
9
d
c #
dlW ï
dû^
N
#
9
#
c ^
N A
²
9
A
#
N
²
$ A
@ A
9
 ²
#
[ A
#
9
#
N€$ A
²
9
#
N‚dS^
9
#
N $ A :
:
d
²
9
[ N A
ú
ˆ
ý 5
d
N
^
9
#
d
A [
#
d ^N
S
° ^ÃN
$ A :d
²
dlW ï
ú
A
#
[ 9
A #
c A
²
:d
N A
9
W ï
$ [ èd
²
A c
#
A
#
d
9
A
#
ú N
$ A
²
A c)9
#
²
9
$ A
d
$ A
#
N A
N
ú
#
C Ä
N N ± ²
9
dü C
N
ú
N…d
#
NÄ
#
$èd
°
9
#
$
#
$:dW ï
#
#
^
²
9
N A
#
[ #
ú
9
[ #
A c
@
Starting from a given three-term recurrence equation, one can use these identities in the
opposite direction to find the corresponding differential or ( - )-difference equation by solving
a quadratic system of equations; see [10].
Example 1: Let the recurrence equation
!dlndüN3dlWh
465 ¨ A
(6.1)
be given. The computations
Ú
read "hsum9.mpl";
ã
%Þ<ãEß
%Þ /ßtà
Ú
ã /áäã ©ßtà
d
¼ó
%Þ /ßtà
Ú
ÿ
Ú
, D 4 ç
ß
©^
ß îà
W
^
Þ
d
Þò•ã
îà /ãß tà
Þò•ã
hãÒÞ
d
Þò•ã
hãÞ
á
×
ê
òñîã
ãEò
Þò“ã
¦à ‰ãEß tà
Wh 9
9
+Iî [
#
HG G
!NÃydS^Y
{
Ð&Ò<*!)8^
^
"
×
Wh?
9
ã
[ » =A
5 E& 3 F & M3 5
ß
áäã ÒãEß
W
B
9
©ndlW
" ^
^Ã
9
/ã
W
B
^
9
@
9
5 ¨ >
7
5
d
^
W
W
^
µ
¸
W
EãEò
N
§
{
+I<¦$
Wh A ûNÄ
9
8^
c c
I
û!NÃ 9
7 {
5 ¨ >
7
5
¾ &Ò
²
<6d
9
9
ê
òñîã
+I
^N
c W
§
î1ˆd
²
²
c
ë
!NÃ@)+I<
ˆW&
9
OIî [
#
W
A îb$ˆd
µ< û!NÃî
^
<t<
9
!NÃ
9
! 9†R !)
y
ˆd
W
ûYd
^
¦¦+î
+¦
F
µ
¸
!NÃ
9
Wh<
©ndlWhËdû^6
^f
9
W
9
9
9
µî û!NÃa
^Ã
²
dW
A
O
ë
<y
9
+I<
?
9
Wh
&©<*!)é
dW
9
W
{
&
W ± ²
show that for §
translated Laguerre polynomials and for
Krawtchouk polynomials are solutions of (6.1).5
5 The
A
§
PN O @
^ÃN
9
B¦
W‰6d
9
W
äd‹ Q
9
ã
N O @
P
!B
Yd
²
??
^Y û!NÃa
Q
!)OI
465
REtodiscreteDE(RE,P(n),x);
, D 4 7
/ã
d€W
"
"
/ò bß
á
^(6dV!ndüN‚dlWh û!N
9
áäã ÒãEß
W
d
ªõ à
æèç
5 E& 3 F & F3 5
HG G
LK
J
Þ
" ^
W
Wh A
9
REtoDE(RE,P(n),x);
ß îà
Ú
û!N
á
/ò bß
ÿûç
ã
Bõ
ñá
õ
Ý
N
§
! " #
$ % ' & ##( *)+) -, . 0/- '. 21 43 5 46./ 55
read "retode.mpl";
+
7 8 " (
$ 9 :& ;)) , . </= >. ?1 43 5 4@A.B/ 55
RE:=P(n+2)-(x-n-1)*P(n+1)+alpha*(n+1)ˆ2*P(n)=0;
C
Ý
c
! 9
45 ¨ >
parameters
R S
and
"Q
§
W ± ²
Meixner and
UTVWR YX-S
that appear in Maple’s output correspond to the translation ’
‰’
.
ETNA
Kent State University
etna@mcs.kent.edu
ORTHOGONAL POLYNOMIALS, RECURRENCE EQUATIONS AND FACTORIZATION
121
Example 2: Let the recurrence equation
!
¨ A
4 5
(6.2)
d‹
9
¨ >
4 5
5 §Y-
5 ¨ >
-
dW‰
!B+I
4 5
be given. The computations
Ú
RE:=P(n+2)-x*P(n+1)+alpha*qˆn*(qˆ(n+1)-1)*P(n)=0;
Z
Ü
ÿûç
Ú
ûN
[
^(d
9
, D 4 9
§
5
-
» 5 ¨ > ¾ dlWh
-
9
\^] @\^] @
§
*
* !
?
A A
-
W
û!NÃît
9
ß
¨
5
7
§
<t
7
§
5 &C 3 F & M3 D5
\^] @
Þ
á~ã ãß
8I<<18î c
ê [
A
Wh
9
!NÄÃ+I
REtoqDE(RE,P(n),q,x);
ß ¦à
"
ûN
kd
-
‰ã
§
9
-
<d
ã
¦
§
òxñîã
ôd¼
#
? ûNÃt
dlW
-
-
îî$f8I
9
-
9
ë
d€W
5 dVWh A
9
-
-
!NÃ
OI<
5
ôWE&
>
5
show that for every §U032 there are - -orthogonal polynomial solutions of (6.2).
7. Associated Orthogonal Polynomials. A monic orthogonal system
)+ 5
45
9
7 ; 5@=?>
5
7 ; ; 5@=A
5
9
9+DEDED
satisfies a recurrence equation of the form (see, e. g., [3])
¨ >
4 5
(7.1)
!)k…düÁ
5
!Ãd
4 5
ý 5
4 5@=?>
!
P
The polynomials defined by
» Ó ¾ !Bk!‚d‹Á
» Ó ¾ d
5 ¨ Ó 4 5
4 5 ¨ >
_
» Ó ¾ !f
¨ Ó
ý 5
4 5@=?>
called the th associated orthogonal polynomials, are also orthogonal by Favards Theorem;
see [3].
It turns out that the associated polynomials can be represented as linear combinations
` a
4 5
`
4 Ó =Ë>
» Ó ¾ !)
» > ¾ !
4 Ó = A
Ó =?>
`
» > ¾
!Ãd
4 5 ¨ Ó =?>
45 ¨ Ó
Ó =?>
5
¢t£ > ý ¢ ; see [4].
where 5
As examples, we consider the classical discrete polynomials. Then it turns out that the
!B
» Ó ¾ !
4 5
associated polynomials
satisfy a fourth order recurrence equation of the form
b
þ
5
» Ó ¾
with polynomials
c
b
!B
¡
c
í
¢
*d b
ÄyNÄ
¢
¢t£?¤
!NÄ
0g2
"
ÄN&
¢
!
, where
¡
c
í
¢
!ÄyNÄ
¢£?¤
d
b
!
9
7<
8I
denotes the shift operator.
8. Factorization of Fourth Order Difference Equations. By linear algebra, one can
» Ó ¾
prove that a certain multiple of the difference operator þ 5 can be factorized as product of
two difference operators of second order [5]
e
Q R 4 Ó =Ë>
þ
5
f hg
» Ó ¾ 5
» Ó ¾
5
» Ó ¾
ETNA
Kent State University
etna@mcs.kent.edu
122
WOLFRAM KOEPF
e
R !Bˆd
5
Q
4 Ó =?> . Using computer algebra, in each specific case this factorfor some function
ization can be computed explicitly.
For example, let’s consider the Charlier polynomials and their associated. The monic
Charlier polynomials are given by
45
#
c » ¾ )ˆd
5
#
dfNÃEdf
5
A ™ ¤ ›
W ž
d
d
œ
œ
_
P
#
œ
œ
œ
The fourth order difference operator of the th associated Charlier polynomials
is given by
þ
5
» Ó ¾ #
!N
9
dû^
9
©^
#
^
g
5
» Ó ¾ f _
9
9
9
d
!
9
9
!
#
W‰
9
^
dS^
i
(?
9
d
C
¼NÄ
#
C)9
#
9†ï
d
d
A
²
° N
düN A
A
ädlWŠI
#
#
9
N
9
N C
9
²
#
adÐ^NÄ
d
ï
A
C
9†ï
N
A
A
d
²
N
°
#
Wht
9
ï
d‹N A
A
düN A
N
9
(
. The factorization yields the second order right factor6
= >
4 Ó ?
9
9
WhË!
²
9
dÐ^
adÐ^
9
#
W‰N
9
^ÃN A
CB9
^
N
²
^
9
d‹nd
N
#
#
N3dS^
#
:
= >
4 Ó ?
d:
í
² 9
adS^
²
N A èd
²
9
i _
*d
i i
i i
i kj i ;d
i i
i
Hi
i Ii
i i
7d
i Hi
i
i i *d
i
(?
äd
#
ädÐð
9
where
i
^
9
!Ë!
WhE!
^(
4 Ó =?>
^( A
9
^(
9
#
d
li
A
! A d <!N
4 Ó =?>
Wh
d
9
4 Ó =Ë>
li
i
9
#
Wh
9
!
^(
9
9
4 Ó =?>
e
!
^(
9
= >
4 Ó ?
9
W A t
h
Wh
4 Ó =?>
still denoting the monic Charlier polynomial. The function as well as the left factor
5
turn out to be rather complicated.
One main advantage of the factorization is the following: In the general case, using the
» Ó ¾
right factor 5 , one can find a solution basis for the fourth order difference equation of
the th associated polynomials (for arbitrary ) consisting of the four linearly independent
functions
465
_
g
Å
_
n
p
¥
» Ó ¾ !BV*!
¨
4 5
¿
4 Ó =Ë>
5 ¨
5
!
» Ó ¾ !BV*! ¿
Ó =?>
45 ¨
5
! ¿
» Ó ¾ !BV*! ¿
Ó =?>
5 ¨
5
5
» Ó ¾ !BV*!
4 Ó =Ë>
!t
Ó
t
Ó
t
Ó
Ó
!
P
In a similar manner, the fourth order difference equations and their factorizations of differently modified polynomials like the generalized co-recursive and the generalized co-dilated
polynomials can be detected [5].
9. Conclusion. The software used was written in connection with my book [8] and is
available from my home page.7
I hope to have shown that new and interesting research results in the classical topic of
orthogonal polynomials can be obtained using computer algebra algorithms.
The most important computer algebra algorithms utilized are the algorithms of linear
algebra, polynomial factorization and the solution of polynomial systems, e. g. by Gröbner
bases.
6 which
doesn’t necessarily look simpler but is of second instead of fourth order
7 http://www.mathematik.uni-kassel.de/˜koepf
md
y
ETNA
Kent State University
etna@mcs.kent.edu
ORTHOGONAL POLYNOMIALS, RECURRENCE EQUATIONS AND FACTORIZATION
123
Software development is a time consuming activity! Software developers love when their
software is used. But they need your support. Hence my suggestion: If you use a computer
algebra package for your research, please cite its use!
REFERENCES
[1] R. A SKEY , An integral of Ramanujan and orthogonal polynomials, J. Indian Math. Soc. (N.S.), 51 (1987),
pp. 27–36.
[2] S. B OCHNER , Über Sturm-Liouvillesche Polynomsysteme, Math. Z., 29 (1929), pp. 730–736.
[3] T. S. C HIHARA , Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.
[4] J. D INI , Sur les formes linéaires et polynômes orthogonaux de Laguerre-Hahn, Thèse de Doctorat, Université
Pierre et Marie Curie, Paris VI, 1988.
[5] M. F OUPOUAGNIGNI , W. K OEPF, AND A. R ONVEAUX , On fourth-order difference equations for orthogonal
polynomials of a discrete variable: derivation, factorization and solutions, J. Difference Equ. Appl., 9
(2003), pp. 777–804.
[6] W. H AHN , Über Orthogonalpolynome, die -Differenzengleichungen genügen, Math. Nachr., 2 (1949), pp. 4–
34.
[7] R. K OEKOEK AND R. F. S WARTTOUW , The Askey-scheme of hypergeometric orthogonal polynomials and
its -analogue, Report 98–17, Delft University of Technology, Faculty of Information Technology and
Systems, Department of Technical Mathematics and Informatics, 1998; electronic version accessible at
http://fa.its.tudelft.nl/˜koekoek/askey.html.
[8] W. K OEPF , Hypergeometric Summation, Vieweg, Braunschweig/Wiesbaden, 1998.
[9] W. K OEPF AND D. S CHMERSAU , Representations of orthogonal polynomials, J. Comput. Appl. Math., 90
(1998), pp. 57–94.
, Recurrence equations and their classical orthogonal polynomial solutions, Appl. Math. Comput.,
[10]
128 (2002), pp. 303–327.
[11] P. L ESKY , Über Polynomsysteme, die Sturm-Liouvilleschen Differenzengleichungen genügen, Math. Z., 78
(1962), pp. 439–445.
[12]
, Orthogonale Polynomsysteme als Lösungen Sturm-Liouvillescher Differenzengleichungen, Monatsh.
Math., 66 (1962), pp. 203–214.
[13]
, Endliche und unendliche Systeme von kontinuierlichen klassischen Orthogonalpolynomen, Z. Angew.
Math. Mech., 76 (1996), pp. 181–184.
[14]
, Orthogonalpolynome in ’ und als Lösungen von reellen -Operatorgleichungen zweiter Ordnung, Monatsh. Math., 132 (2001), pp. 123–140.
[15] M. M ASJED JAMEI , Three finite classes of hypergeometric orthogonal polynomials and their application in
functions approximation, Integral Transforms Spec. Funct., 13 (2002), pp. 169–190.
[16] J. C. M EDEM , R. Á LVAREZ -N ODARSE , AND F. M ARCELL ÁN , On the -polynomials: a distributional study,
J. Comp. Appl. Math., 135 (2001), pp. 157–196.
[17] A. F. N IKIFOROV AND V. B. U VAROV , Special Functions of Mathematical Physics, Birkhäuser, Basel–
Boston, 1988.
[18] A. F. N IKIFOROV, S. K. S USLOV, AND V. B. U VAROV , Classical Orthogonal Polynomials of a Discrete
Variable, Springer, Berlin, 1991.
[19] A. F. N IKIFOROV AND V. B. U VAROV , Polynomial solutions of hypergeometric type difference equations
and their classification, Integral Transforms Spec. Funct., 1 (1993), pp. 223–249.
[20] V. R OMANOVSKI , Sur quelques classes nouvelles de polynomes orthogonaux, Comptes Rendues, 188 (1929),
pp. 1023–1025.
:o;p