ETNA
Electronic Transactions on Numerical Analysis.
Volume 24, pp. 24-44, 2006.
Copyright  2006, Kent State University.
ISSN 1068-9613.
Kent State University
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QUANTUM ALGEBRAS
AND
ASSOCIATED WITH CERTAIN
-HAHN POLYNOMIALS: A REVISITED APPROACH JORGE ARVESÚ
!#"%$&"'
(*)+ (),#"%$'"'
Abstract. This contribution deals with the connection of -Clebsch-Gordan coefficients ( -CGC) of the WignerRacah algebra for the quantum groups
and
with certain -Hahn polynomials. A comparative
analysis of the properties of these polynomials and
and
Clebsch-Gordan coefficients shows that
each relation for -Hahn polynomials has the corresponding partner among the properties of -CGC and vice versa.
Consequently, special emphasis is given to the calculations carried out in the linear space of polynomials, i.e., to the
main characteristics and properties for the new -Hahn polynomials obtained here by using the Nikiforov-Uvarov
approach [29, 30] on the non-uniform lattice
. These characteristics and properties will be important
to extend the -Hahn polynomials to the multiple case [7]. On the other hand, the aforementioned lattice allows
as a limiting case, which doesn’t happen in other investigated cases [14, 16],
to recover the linear one
for example in
. This fact suggests that the -analogues presented here (both from the point of view
of quantum group theory and special function theory) are ‘good’ ones since all characteristics and properties, and
consequently, all matrix element relations will converge to the standard ones when tends to 1.
-./('10 6 2%3354 4
-./('07(
-.8(&09 <: ;
Key words. Clebsch-Gordan coefficients, discrete orthogonal polynomials ( -discrete orthogonal polynomials),
Nikiforov-Uvarov approach, quantum groups and algebras
AMS subject classifications. 81R50, 33D45, 33C80
The author dedicates this contribution to the Russian (Soviet) physicists and mathematicians A. F. Nikiforov and Yu. F. Smirnov for their scientific works in this area. Both are prize
winners for research works. The first one was awarded with the Lenin Prize in 1962 while the
second one was recently awarded (in 2002) with the Lomonosov Prize.
1. Introduction. The theory of quantum groups is a very fascinating subject at the border of many areas, mainly group theory, differential and difference equations and special
function among others. Consequently, many techniques have been developed and are available; therefore there is some productive competition between various approaches to the subject. The crucial role that the group representation theory concepts play as effective tool to
unify independent areas of Physics and Mathematics such as the Quantum Theory and the
Special Function Theory (Orthogonal Polynomials) is magnificently presented in [44, 45].
This role has inspired the main goal of this contribution, i.e., the study of the interrelation
between these two areas by constructing a = -analogue of the Hahn polynomials using the
Nikiforov-Uvarov approach. In such a way, a useful and fruitful parallelism for the study of
= -Clebsch-Gordan coefficients and = -Hahn polynomials is established (see Section 4). Moreover, the construction presented here for the = -Hahn polynomials is a very simple one and
it corresponds to the standard theory of orthogonal polynomials i.e., the orthogonality conditions are considered with respect to a regular linear functional [15]. Indeed, the regularity
condition for certain modifications of the linear functionals [5] is the corner-stone that will
allow the extension of the quantum algebras studied here.
During the last years the study of = -discrete analogues of the classical orthogonal polynomials and the connection with the representation theory of quantum algebras in relation with
several applications has received an increasing interest (see for instance [6, 9, 17, 23, 27] as
well as [39]-[40]). The notion of quantum groups and algebras (Hopf algebras [18]) appears
as a consecuence of the study of solutions of the Yang-Baxter equation [26] and the development of the quantum inverse problem method [19]. Their applications in quantum and
> Received December 1, 2004. Accepted for publication May 13, 2005. Recommended by F. Marcellán.
Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911, Leganés,
Madrid, Spain (jarvesu@math.uc3m.es).
24
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25
statistical physics were practically immediate [43], especially for the study of the = -deformed
oscillator [9, 27], description of the rotational and vibrational spectra of deformed nuclei
[10, 12, 32] and diatomic molecules [3, 11, 13]. As a consequence, a big amount of material can be gathered under the title ‘quantum algebras’, and therefore, any communication
on this topic that serves both as a survey and research paper -like this communication- must
necessarily be summarized. This partially answers to the question about the dedicatory of
this contribution, since here are summarizing some ideas of both Researches. Despite the
contributions of other authors we have focused the attention of the reader in the techniques
and methods developed by A. F. Nikiforov and Yu. F. Smirnov. However, a suitable list of
references is included.
The knowledge of CGC, Racah coefficients (?@ symbols) and A@ symbols is essential for
understanding the corresponding quantum physical problem since all matrix elements of the
physical quantities are proportional to them (see [36, 37]). Based on this well known fact we
investigate the relation between the
G+Clebsch-Gordan
HI coefficients –also known as B@ symbols–
[42] for the quantum algebras CDFE
and CD1E
with = -analogues of the Hahn polynomials
LK E 2 M1N
on the non-uniform lattice J C
E M1N . Although several authors have investigated the
connection between different constructions of the Wigner-Racah algebras for the = -groups
and = -algebras and the polynomials orthogonal with respect to discrete measures (see [4,
16, 22, 25, 28, 33, 34, 45]), the resulting properties for such polynomials could not have in
return classical analogues.
For
O
K instance, the connection of CGC with some = -analogue of the
=,P6Q was studied in [16], however to recover the parameters
Hahn polynomials for J C
involved in different characterizations for these polynomials as well as their connection
with
standard group representation is not possible as a simple limiting case ( =SR
) since many
of these parameters under this limiting operation disappear (go to zero); consequently the
corresponding physical relations lose sense (see Section 5). The same
TK could happen with
some = -analogues of the Kravchuk and Meixner polynomials for GJ + C
=+P6Q in
HIconnection
with the Wigner D-functions and Bargmann D-functions for CD E
and CD E
algebras
that was established in [14]. These questions should be study in forthcoming publications.
Therefore a natural question appears: What = -analogues in connection with physical
problems should beUconstructed?
Recently, in [2] based on the Charlier case the authors show
K
that
the
lattice
is
not
a
proper choice in the sense that to recover the linear ones
J
C
=
Q
VK
J C
C is WnotK possible by taking limits as we have already mentioned. Instead, a good
E 26MFN , such that when = tends to one we recover the classical linear lattice,
choice is J C
E MFN
and the constructed polynomials are ‘good’ = -analogues of the classical ones. Moreover, all
characteristics related to such = -polynomials will tend to the classical ones for the aforementioned limit case ( =SR
). This fact, is crucial when we are dealing with physical concepts
instead of mathematical ones.
The present communication is written in a narrative way without loss of mathematical
rigor. We have stressed simplicity of ideas at the expense of a formal mathematical presentation. Consequently, the customary sequences of definitions, lemmas and theorems will be
omitted; nevertheless a mathematical rigor in all the arguments has been kept. Thus, the presentation of the results is self-contained and mostly elementary. As a consequence the only
prerequisites are a good understanding of the fundamentals of group theory and orthogonal
polynomials. The structure of the paper is as follows.
In Section 2 we summarize some nec
essary formulas and relations concerning CHD E
quantum algebras and = -analogues of the
CGC. In Section 3 we quickly sketch the more important aspects of the Nikiforov-Uvarov
approach. This will facilitate the willpower of the reader for understanding the next calculations carried out in Section 4 in order to obtain the main data of the = -Hahn polynomials.
It is important to remark that the aforementioned approach is connected with the solution of
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the eigenvalue problem for the second order finite difference equation on a quite general lattice. The solutions for such an equation have several properties similar to the solutions of the
Schrödinger equation. Thus, this approach is close to the standard quantum mechanical methods since it allows to use rather fruitfully the physical intuition in the analysis of appeared
problems. The Section 4 contains a lot of calculations so the author kindly ask the reader to be
patient during the reading of this section. In fact, a lack in [29] will be covered since in most
of the papers related to the Nikiforov-Uvarov approach the authors did not study concrete
families (up to the papers [2, 4, 33]). Actually, a non-included in [29] expression involving
the difference derivatives of the = -Hahn polynomials with the polynomials itself (difference
recurrence relation) is obtained. Furthermore, a comparative analysis of the properties of the
= -Hahn polynomials
G+ and = *-CGC
is also given. Hence, the interpretation of the representation
theory for CD1E
and CDXE
in terms of the = -Hahn polynomials is established. The Section
5
is
devoted
to
a
brief
presentation
of the = -Hahn polynomials in the exponential lattice
WK
J C
=,Q in order to emphasize the ‘improper’ choice of such a lattice for the construction
of physical = -analogues. Finally, the Section 6 includes some conclusions and remarks.
2. = -Clebsch-Gordan coefficients for the CDFE
quantum algebra. The aim of this
section is to prepare all the necessary results concerning the = -Clebsch-Gordan coefficients
[41] for the comparison with the properties of = -Hahn polynomials given in the Section 4
below.
G+
In [35, 36] the authors define the quantum algebra CDYE
in the standard way using the
three generators Z[ , Z\ and Z M with the usual properties
] K b
c
] K ]
Z^[ K Z _a` Z_ K Z \ Z M `
^Z [ `dE
ZS[ e Z [ ZO_ e Z f
where
]
K
lk!mVKonqpr %br6sr&tvu+
Zg Z h,` Zg
ZhUijZh.Zg
i
and
(2.1)
]w
`E
K =1x yzi{= M x y | E
=
K~}1y | E K Y= y
€ i{= M y€ ‚
]
The above expression Z[H`/E means the corresponding infinite formal series.
K…For
t! N C
D E quantum algebra the irreducible representations ƒO„ with the highest weight
@
‚‚H‚ are determined by the highest weight vector † @@ˆ‡ E , such that
P
Z\L† @I@ˆ‡ E
K…t!
Z^[‰† @I@Š‡ E
K
@‹† @I@Œ‡ E
Ž
@I@5† @I@.‡ E
Ko+
and
]
Ko‘ ] @ p ] V` EI’ „ MX”
† @+‡ E
† @I@ˆ‡ E with i•@z–—˜–j@
‚
@
`/E ’ U
@ iŒV`/E ’ Z“M
]w
Here we have used the symmetric = -factorial symbol `GE ’ which can be expressed through the
š
symmetric = -Gamma function
™ E C defined in [29] and that is connected with the classical
š = -Gamma function E C [20, 29]. Indeed, by [29]
š ™ E C K = M› 2*œ €# ž › 2*œ y  š E C %
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and
27
¨ iŒ= \ N
t«ª ªc
¤^ K¦
N6M Qa¨§ © [
=
+
C
&
¥
=
Œ
i
=
¨
¢
\
š E C K Ÿ¡¡¡
iŒ= Q
(2.2)
¡¡¡
¨© [
¡
§
¡¡¡ = › 2*œ €G y › 2*œ y  ¤^ C
¥&= M1N %
=r¬ ‚
¡¡¡
] w K š w p7I
w
š p7I¡£K ] CH`/E!š ™ E C ™E
Notice that ™ E C
and `dE ’
for nonnegative integer .
On the other
hand,
if ­ „ € and ­ „ y denote the two irreducible representation of the quantum algebra CD1E
, then the tensor product can be decomposed into the direct sum
„ ¯€ \ „ y
K
y
­ „ €® ­ „
ƒ „
„'°± „ € M „ y ±³²
*
usually known as Clebsch-Gordan series. Its generators (co-products) are given by the expressions
Z_
*
6
Z K [
*+&
K
Yp
G
Z [ Fp Z [
G = yI€ ´.µ ¶·P&¸Z_
= M Iy€ ´.µ ¶ N *¸ Z_ ‚
Taking into account the explicit form of the irreducible representation

(2.3)
K7¼
@+º¹<† Z [ † @+»‡ E
”¾½ ”“¿
] b

K¦À ] s
p ¼
~
@+ ¹ † Z_† @+»‡ E
@ S`dE @ 
`dE “
” ¿d½ ” \ N
as well as the Casimir operator
(2.4)
K
p ] p
Z‰MaK Z \ ] p Z [ y€ ` PE PÁ
Á P † @ Â‡ E @ y€ 8` PE † @+Ã‡ E
one can define the = -CGC in a similar way to the classical case. Thus, for the basis vectors of
the irreducible representations ­ „ we have
K ¯

† @ N @ P +@ Ã‡ E
@ N  N @ P  P @+»‡ E † @ N  N ‡ E † @ P  P ‡ E
” ½” y
€ *I
K ] p
† @ N @ P @
‡ E @ y€ ` PE † @ N @ P @+‡ E
(2.5)
ÁP

where @qN%•N@ 
P P @
•‡ E denotes the = -Clebsch-Gordan coefficients. Notice that for these
coefficients the following orthogonality conditions
¯


K7¼ ¼
@qN%LN*@ P  P @+•‡ E @qN •N@ P  P @ŏ»¹‡ E
½ „ ¿ ”¾½ ”‰¿
„
” ½” y
€
¯ 

K…¼
¼
@ N  N @ P  P @+»‡ E @ N  ¹N @ P  ¹P @+•‡ E
” ½ ” ¿ ” y ½ ” ¿y ‚
€ €
„ ½”
¼¨
hold. Here ½ Å represents the Kronecker delta symbol.
 Using the *Casimir
6
operator (2.4) as well as the expression (2.5) the matrix elements
@qN •N@ P  P † Á P
† @qN*@ P @+»‡ E can be computed. From these matrix elements one deduces
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the following three-term recurrence relation in 
[33, 35]
Æ 34HÇ È É
(2.6)
:ÊÌË :ʊÍ&Î Ç ÈÉ
È
:aÊÌË :'Î Ë :YÏ
Ï
Æ 3.Õ È
Ï{Ô € Ë : Ï
È
Ï ËÏ y€ Î :
È
N  P
É ÈÉ
È
Ë Ð: Ï :Î 4 Ê«Ë 4 Î Ë 4
É
È
É ÈÉ
:YÏ{Í&Î Ë 4 Ï 4 Î 4
É
È
É
: Ï{Í&Î Ë : Ê : Î Ï
ÈÉ
Æ 3 Õ 3vÕ
Ê
Ï y€ Î : Ö y
€ × y
for the = -Clebsch-Gordan coefficients
É 4
É
É
ÉÌÓ
Ï Í&Î ÐÑ Ë 4 4 ÏjͳËH: :^ʊÍqÒ Ë Œ
É
É
ÉÌÓ
ÊÌË 4 ÊLÍ&Î ÐÑ Ë 4 4 ʈͳËH: :YÏ{ÍqÒ Ë È
É
ÆÕ yÈ 4 É 4
Ë Ï
Ï{Í&Î Ë 4 Ê 4 Î 4GÙ É É ÉzÓ
€Ø Ñ Ë 4 4 Ë : : Ò Ë ÚjÛ
Ü
Ï
Based on the symmetry property for the = -Clebsch-Gordan coefficients

KÝ I „ \ „ y M „ 
i €
@ P  P @ N  N † @+»‡ E œ
€
the expression (2.6) is invariant with respect to the change @ N by @ whereas = is replaced by
P
= M1N .
Following
the same ideas

used
for the computation of (2.6), but in this case for the matrix
element @qN •N@  † Z [
†
q
@
N
@
P P
P @+»‡ E one gets a similar three-term recurrence relation with
respect to the variable @ [38]
(2.7)
@ N  N @ P  P † @
•‡ E
À Þ „ MX”ßáà Þ „ \ ”ßâà Þ „ \ „ y \ „ \ N<ßâà Þ „ y M „ \ „ ßâà Þ „ M „ y \ „ ßâà Þ „ \ „ y M „ \ N<ßâà  @qN •N@  † @ãi •‡
E
€
Þ P „ \ N<ßâà Þ P „ MF€ N<ßâà Þ P „ ß yà
€ €
P P
p«ä Þ „ MX” \ N<ß à Þ „ \ ” \ N<ß à Þ „ \ „ y \ „ \ P ß à Þ „ y M „ \ „ \ N<ß àHÞ „ M „ y \ „ \ N<ß à 
p~
@ N  N @ P  P †@
•‡ E
€
Þ P „ \Yå ß àH€ Þ P „ ß à Þ P „ \ P ß yà Þ „ \ „ € y M „ ß àœ €
€
]
pçæ Þ P „ ß àHÞ P „ \ P ß à M Þ P ß à%Þ „ \ „ y M „ \ N<ß àHÞ „ M „ \ „ y ß à æ M \ N ] p
= y,€ ¶ „ ¸ @ V`/E‰i{= y
€ ¶ „ \ N ¸ @ãiŒS`dE%è
Þ P ß à%Þ P „ € ß àHÞ P „ \ P ß à
€
€
y æ
] p
]

K…t
i E œYÞ éß ày = M y
€ ¶ „ € \ N ¸ @ N  N `/E‰i{= y+€ ¶ „ € \ N ¸ @ N iŒ N `dE%èXê @ N  N @ P  P † @+•‡ E
‚
P
Finally, from (2.3) a straightforward calculation leads to
] b
K À ] s
p7 
@ N  N @ P  P † Z_† @ N @ P +@ »‡ E
@ V`dE @ 
`/E @ N  N @ P  P † @+•‡ E ‚
nIëì1 %uì ©
3. Nikiforov-Uvarov approach. A polynomial sequence
J
w [ orthogonal with
ëì
respect to a positive measure í on the real line is such that
has degree and satisfies the
conditions
î.ï
ë ì ¨ð K~t!lñÌK~tH
w + ò~ó7ô
J J í J
‚H‚‚ i
‚
(2.8)

This defines the polynomial up to a multiplicative factor. In the case of discrete orthogonal
polynomials, we have a discrete measure í (with finite moments)
í
Kö¯ õ ¨ ¼HøIùÄ
¨ [ ÷
°
÷
¨ ¬ !t J ¨rú ô
and û
ú nÄ
&! ‚‚H‚ uüŠnqpTý7u+
pþ
which is a linear combination of Dirac measures on the û ëì
points J [
orthogonality conditions of a discrete orthogonal polynomial
on the lattice
n ‚H‚‚
J C
Jõ ÿ . ô The
\
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K~t!H+
u
‚‚H‚ û
(), AND
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29
ASSOCIATED WITH -HAHN POLYNOMIALS
are usually written as
¯ õ aë ì1 ¨ K…t! ñzK7t!
w J C J C C
‚H‚H‚ i ‚
[
÷
Q°
There exist several mathematical methods and approaches to the study of classical polynomials orthogonal with respect to a discrete measure, usually known as discrete orthogonal
polynomials [29]. Nevertheless, for physicists perhaps the more easy method to get in touch
with these kind of mathematical objects (discrete
orthogonal
polynomials) is via the discretization of the first and second derivatives ¹ J and ¹ ¹ J involved in the hypergeometric
differential equation [29]
K
™ J v¹ ¹ J Yp ™ J Yp J K~t
K w C . This method is usually known as Nikiforovand
where
™ – ,
™
Uvarov approach.
Thus, the corresponding hypergeometric-type difference equation is [29]
p p K~t
C C C C
J
C
i
J
C
J C
y€
(3.1)
C K ™ J C i y€ ™ J C J Ci y€ % C K ™ J C '%
TK C i Ci and C K C pç i C denote the backward and
where C
forward finite difference, respectively.
Of course, any partition J C can
not guarantee the existence of polynomial solutions of
(3.1). In [8, 29] it is shown that J C should be of the form
K Q p M Q p U ú ô \ n+,u
K P p p q
å =
or J C
J C
N%=
C
C
P=
«K E2 M1N
where N , , å , , and are constants. In fact, the lattice J C
E M1N belongs to this
P
C
class.
The polynomial solutions of (3.1) can be orthogonalized constructing a Sturm-Liouville
problem. First, the equation (3.1) is written in the self-adjoint form
(3.2)
J C“i y€
C C
÷
C J C p w
÷
K~t
C C
for two different polynomial solution of degree and  , respectively. The function C
÷
(the so-called symmetrization factor of (3.1)) is the solution of the Pearson-type difference
equation [29]
(3.3)
] K
C C `
J Ci y€
÷
C C
÷
where
÷
C
t
J ¾
C i y€ ¬
–7CT–
w
i
‚
Second, the equation (3.2) for polynomial solution of degree is multiplied by the other
polynomial solution of degree  of the same equation and vice versa, and then one
subtracts
the resulting equations (one from the other). Finally, we sum over – C̖
i for which
we obtain the orthogonality property (see [29, pages 70-72])
(3.4)
¯ M1N ë ì ë ' J C ” J C
C
÷
Q° K…¼ ì ë ì P J W
C i y€
½ ” †† † †
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under additional (boundary) conditions
¨ "!
K7tlñzK7t!
C C J C i y€ ! ½
‚H‚H‚
Q° ÷
Since
ñ we are interested in polynomial solutions the same procedure could be carried out
for the -order finite difference of a solution of (3.1), defined by
$### J C '% ¶ ¨ ¸ C %
' K
C
¨
J MFN C J ¨ M P C J C
‰K p &
ñ
where J ¨ C
J C : , because this -th finite difference verifies a difference equation of
¨ J
the same type (hypergeometric type) [29]
(3.5)
being
¨ C K C
¨ C E p ¨ C E
J ¨ C ' ¨ C J ¨ C J ¨ Ci N
P
p !ñ C
i
Y p
C
p !ñ C
¨J MFN C p ñ N
J C
i
P
For (3.5) the symmetrization factor is
(3.6)
÷
¨ C K
÷
p ñ!
C
¨
(°
§
N
p
K~t
í ¨¨ C E
¨
K ì p ¯ FM N
¨
and í
” ° [
”
C 5J ” C ‚
C p)* ‚
As a consequence of (3.4) the polynomial solutions of (3.1) and (3.5) verify several
crucial relations that will help us to find the connection between the = -Clebsch-Gordan coefficients for the quantum algebras and the = -Hahn polynomials. Now we will write some of
them (see [29] for more details).
ëìF It is well known that
ì the polynomial solutions J C of (3.1) are determined up to
a normalizing factor
, by means of the discrete analog of the Rodrigues formula [29, Eq.
3.2.19, page 66]
*
ëaìF K * ì ¶ ì ¸ ] ì1 ¨¶ ì ¸ %
+
#
#
#
C
`
J C
[
J ¨\ N C J ¨\ C J ì C C
÷
P
÷
ì C isì given in (3.6). Notice that these polynomial solutions correspond
where
ë ì to certain
÷
J C into
eigenvalues
(see (3.1)) which can be computed by simple substitution
of
the equation (3.1) and comparing the coefficients for the powers of J C , i.e.,
ìzK i ] w ` E-, / . = ì M1N p = M ì \ N 0 ™ ¹ p ] w i ` E ™ ¹ ¹ ê (3.8)
2
K 3 1 ¿ ¿ p #tÄ Yp #tÄ
K Yp Gt+
where (see (3.1)) ™ C ñ
and ™ C
.
J C P ™¹ J C
¹J C
™
P
Similarly, for the -th finite
difference the following Rodrigues-type formula is also valid
ì ëaì
ì ¨ ì
ì ] ì ¨
K
ë
'
5
K
4
K
ì
ì
ì
½
*
¶
¸ì ¨¶ ¸ C ` where *
(3.9)
ì
¨½ J C ¶ ¸ J C
¨ C 4 ½
÷
÷
and
ì
¨ M1N
]w
p M ì \ ”¾M1N p ] w p
\
¾
”
1
M
N
ì
K
™ /
q
E
’
=Yy
€ ¶
¸ = y
€ ¶
¸¹
4 ½ ¨ ] w i ` ñ `dE ’
(3.10)

i
` E ¹¹ 8 ‚
™
[ 6
” ° 7
§
(3.7)
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From (3.7) follows an explicit expression for the polynomials
ë ì
31
[2]
ì
ì
ì
ì
ž ¶ ì \ N ¸ ¯ ] w `/E ’ = M é ya¶ MFN ¸ i ” \
x
]
]w
9
(3.11)
S` Eq’ iŒS` Eq’
” ° [
ì
M5”¾M1N ] CWi;: ` ”¾M1N ] C p : Fp C p : J C p :Xi y€ ` Å° [
Å° [
§ N ¤^ =% §
1
M
:
with the assumption < Å [
. Notice that this expression depends only on the coeffi°
cients of and given in (3.1).
9K E 2 M1N
4. = -Hahn polynomials in the non-uniform lattice J C
E MFN and = -ClebschGordan coefficients. It is well known [29] the relation
between
Clebsch-Gordan
coefficient
‹K
and Hahn polynomials in the uniform lattice J C
in
[33]
is
used
a relaC . Analogously,
K
tion (see (4.24) below) for the = -case in the non-uniform lattice J C
. In fact, we will
=
6
P
Q
‰K E 2 M1N
deduce a similar relation but in the lattice J C
E MFN . In Gsuch
+ a way an useful and fruitful
parallelism between = -Clebsch-Gordan coefficients for CDYE
and = -Hahn polynomials will
ë ì J C
ì
* ì = ì M Q \
N =‹i
K
be constructed. Therefore, studying in details the relations and properties satisfied by the
= -Hahn
polynomials we have the corresponding partner among the relations and properties of
CDXE Clebsch-Gordan coefficients and vice versa. For such a purpose let first to determine
the = -Hahn polynomials as well as their main characteristics based on the aforementioned
w
Nikiforov-Uvarov approach. Second, selecting a special choice of C , û , , , and we will
establish the connection between both mathematical objects.
In [30] has
that the most general orthogonal polynomial solution of (3.1) in
K been proved
p
the lattice J C
NH=,Q å corresponds to the choice
> ?
(4.1)
KA4 @ MCB H MDB y I%
C
=,Q € i
=,Q i
K 4 @ FM B E H MGB E y =,Q € i
,= Q i ‚
C
J CWi y€
Y p
C
From (3.8) considering (4.1) the following expression for the eigenvalues of (3.1)
ì
(4.2)
K
i
4 @
PN
]w ]H p H H H p w = M y
€ ¶ B € \ B y \ B E € \ B E y ¸ d` E N P i @ N i @ P
i `/E
š H@ š
K
C
š E CWi E H CN i p~N š
÷
H@ E C i H P p~ ‚
E Ci P
yields. The corresponding weight function is, in this case,
(4.3)
š Here E C functions are those defined in (2.2).
In this case the corresponding polynomial solution of (3.1) can be represented in terms
of the basic hypergeometric series [2, 31]
ì
(4.4)
(4.5)
ëaìF öK 9
J C
4@
ì œ
M
›
M
B € = B € MFB E € ¥&= *ìF = B € MGB E y ¥&= *ì
ž
x
x
#
€

* , N = ì y€ i{= M y€ ê = ì
M = B \ B y MFB E MFB E y \ M1N = B M Q
€ B MGB E € B FM B E y
€
åJI P , =
¥= = Q MCB y \ N ê
= € € = €
ì
@
ì
9
ë ì 'öK ì
J C
* , i N = y€ 4 iŒ= M y€ ì ê = +M K x › x ž œ €# M ¶ì B € \ B y MFB E € ¸ = B € MFB E € ¥'= ì
M = B \ B y FM B E FM B E y \ M1N =,Q MFB E
B y MFB E ì
€ B GM B E € B y MFB E
€ ¥= =Iê =
¥€ '= å I P , =
= € € =
€
ì
ETNA
Kent State University
etna@mcs.kent.edu
32
L
where
I=M-NPO S 4R4 QQ OS
and
i
JORGEARVESÚ
L
L
ù
VÆ Æb& a &ed ÊÍ a & Æ yÄ× & 34GÙgf M 3
_ T _
Æba
ÆJa
: Q ÜáÜáÜ Q S O MUT Æ WQ VJX ÚUZ Y O S 4 T ÆJa &G&Fc\cRcRc\cc SO M T ÆJa & &
&\^[ ]`_ 4 T
_
: Q ÜáÜáÜ Q
_
_ T
h
¥'=
¨ FM N
* ” i = ‚
” ° [
§
9K t
¨ K
points
interval
å Let
K the
K ofE 26the
MFN , orthogonality
N , end
J
C
i.e.,
and
E M1N
E M1N
and
LK
û
L
Ø
4
. Moreover,
Q
N
K
M ì H K…t! H K
K
p H@ K
H @ K
ì
i ]w
*
û
>
=
i
•
?
i
û
i
(4.6) 4 N
N
N
P
P
` EI’
w ª
with the restrictions > ?~¬þi and
û . Taking into account this choice of parameters
(4.6) the functions C and C take the form
C K i=,Q M1N ] CH` E ] û p >LijCH` E C Yp C J Ci y€ K i= Q \ m
koy n ] C p ? p~ `dE ] Ci{û p7 `/EI‚
K
i =jilykWm =‹i
Consequently, from (3.8) or (4.2) the eigenvalues of (3.1) are given by the expression
The solution
(4.7)
÷
C
ìzK = n k y y œ i
]w
`E
]w p
> p ?
p7
`E ‚
of the Pearson-type equation (3.3) is
p š
K ¶ m
koy n ¸#Q š ™ E C p ? ~
™
C
=
p š
š™ E C ~
™
÷
p
E û
> i{C ‚
Š
E û i{C
š K
2*œ *2 œ y  š E C .
Notice that the same result can be found if we use (4.3) and the relation ™ E C
= M› €# ž ›
Considering (4.5) the hypergeometric representation for the = -Hahn polynomials
ì ì
(4.8)
ì ì
pCq
=r\
q
N ¥'= q = õ \ \ r \ N ¥'=
ì
ì
= x y!¶ \ \ P r \ õ ¸ =.¥'=
q
= \ \ r \ q N ,= Q \ r \ N ¥!= = ê =r\ N =õ \ \ r\ N
yields. From here follows other useful characteristic of the = -Hahn polynomials,
] t
i.e., the
value of this polynomials at the end points of the orthogonality interval û i ` . Indeed,
pDq
q
]
š ] p w p7
` = y€ ì ¶·P \ r \ \ y,€ ¶ ì M1N ¸d¸ ì r Gt! û ¥'= Ko i ì ] w û i ] ` E,’ ™ E ] ? w
p~
õ
` Eq’š ™ E ?
` û i i ` EI’
q
pq
]
š ] p w p~
` = y€ ì ¶ \ M y
€ ¶ ì M1N ¸d¸ ‚
ì r û i û ¥'= K ] w û ] i `/E w ’ ™ E >
] p
õ
`dE ’ û i i `dE ’ š ™ E >
`
ë ì
ë ì
By †† †† P we will denote the square norm of . In [29, Chapter 3, Section 3.7.2, page 104]
ì ½ r C û ¥'= K
åbI P
,
=M
the authors show a convenient way to compute it. In fact,
(4.9)
ì
ë ì P KÝ ì 4 ì ì ì P M ¯ FM N ì †† †·†
i
C
½ *
÷
Q° J
ì Ci y€ ‚
ETNA
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(), AND
(*)
#"%$"'
TABLE 4.1
Main data for the -Hahn polynomials in the lattice
Interval
C
C
C ìF C ì
ì
*
††
÷
ëÐì
pq
-.8(&0 6 2635354 4
.
ì ½ r C ûj¥&= ] t
û i `
š p p7I š p
>Š i{C = ¶ mJky n #¸ Q ™ E C š ? p7I š ™ E û
™E C
™ E û ijC
] p
p
i= M ilksy m J C P = M y€ û
Y`/EHJ C
>
q
]
] p7 ]
p
p
a
p
y
œ
i= nk y i > ?
`dE6J C = \ r \ N ?
`/E û i `/E
] w p p p ìF > ? `E
i= nk y y œ i J C i{= nk y y œ i
] w ] w p p p~
= nk y y œ i ` E M > ì ?
`E
i ]w
` Eq’
w p p7I š w p p~ š w p p p
p I
7
™E
™E
= õ ì ¶ mJ koy n \ õ ¸ M m › m œ ž K  k y š ™ E
>
?
>
?
û
]w ]
w w p p p~ š G w p p p = m œy n \ õ \ N `dE ’ û i i `dE ’ š ™ E
> ? ™E
> ?
q ì
š p w p p7I š p
ì
\
\
F
M
N
¸ x yv¶ \ r \ M1N ¸ \ N ™ E C š p7? I š ™ E û w >L ijC
= Q6¶ m
koy n
™E C
™ E û ijCi
ë ì '
J C
÷
33
ASSOCIATED WITH -HAHN POLYNOMIALS
†† P
ì1 C
Based on (4.9) after some cumbersome calculation we arrive to the expression,
pq
tt
(4.10)
K9
ì ½ r C ûj¥'= P
tt
pq
š ™=
š™ E
ì =õ
œ \
m yp n w
E ?
p
> ?
š õ M y€ ™ E >
\ \ › œ K ]w
õ p~N š m m ž p  `dp E
™E > ?
p w p~ š p
™E >
mJky n \
p w ~
p ]
w ’ û p i w p~
i `dE ’
û
p w p ‚
?
Finally, after the calculations of the very basic characteristics of the = -Hahn polynomials
ì
ì ½ C û ¥'= P we summarize in Table 4.1 the results of the
like C , C , C ,
and
÷
remaining
calculations by using the Nikiforov-Uvarov approach.
tt
r
tt
4.1. Three-term recurrence relation. A simple consequence of (3.4) is [29]
ë ì ‰K ì ë ì 'Yp
J C
J C
> \N J C
ë K~t
ë ‰Ko
with 1
.
M N J C ì and ì [ J C
(4.11)
##
#
?ìë
ì ap;u ì ë ì %
J C
1M N J C
wwv t!
ë ì FKx ì ì 'p- ì ì MFN &p
C
J C
J C
J
Denoting by and the coefficients in the expansion
we have the following explicit expressions for the recurrence coefficients [2, 29]
(4.12)
>
ì K .ì ì \ N
?ì
K Ä ìì Ä ìì \ N i
\ N
uì
K
ì M1N
Äì
ë ì
a놆 ì † † P ‚
†·† M1N+†·† P
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34
JORGEARVESÚ
TABLE 4.2
Three-term recurrence coefficients and leading coefficient for -Hahn polynomials in the lattice
Coefficient
Explicit expression
ì
>
=M
q
ì
?
-./('0 6 2 3534 4
= \ õ \
p ] = w M p "y€ y'P õ p \
>
u.ì
= M ilykWm M P
ì
] w p~ ] w p p p7
nk i ] w p > p ? `dE p ` E ] > w p ? > p `/? E p~ ` E
] w p p p7 ] w p p~ ]
w Mxy 1N ] w> p ? p `/Ep ] ? w p `dE p û i p~ i `/E
ì q
r \ \Få ¶ \ N {¸ z ] w > p >Y? `/E ] w p ` E > p ? > p û»? `dE ] û `i E w `dE ] w `/E
? p~ ` E ] w p > p ?F` PE ] w p > p ?•i ` E ] û i w i ` E
]w p
]w p
]w p p p
]
w
E
E
E
Y
>
`
1
?
`
>
?
»
û
`
û
i
`E
] w p p p~ ] w p p
> ? `dE
> ?1` PE ] w p > p ?»i `/E
š G w p p p~
ì
= nk y y œ i ] w ™ E š w p > p ? p~
` Ev™ E
> ?
yy œ
To find the leading coefficient
ì
it is enough to note that
ì
ì ì
J C K ] w /` E J MFN C p Yp ##
# so , ê J ì C K ] w `dE ’ ‚
y€
J C
J C
ì
] w .ìOK ì ì ì
æ |
}
ëÐì1 '‰K ] w .ì
è
On the other hand, | ø
J C
` Eq’ , then using (3.9), ` Eq’
* 4 ½ , and
¶Q³¸
taking into account (3.10)
ì
ì
ì
.ìÌK * ì M1N = y
€ ¶ \ ¨ M1N ¸ p = M y
€ ¶ \ ¨ MFN ¸ ™ ¹ p ] w p ñ i ` E ™ ~¹ ¹ 8 ‚
¨ [ 6
§°
ì
ì
ì
Frequently,
the
computation
of as well as ? is very tedious, then if we know > and
u ì , and ë ì h v€ K… t for all w , we can
use (4.11). Indeed,
{ v*ëaìFh v ;i > ìvëaì \ N hv i uvìvëÐì MFN {v
ë ì { v
? ìÌK J
‚
See Table 4.2 to find the result of computations for the coefficients (4.12)
4.2. Other recurrence relations. Using the Rodrigues-type formula (3.9) we find [29]
(4.13)
C
ëÐì1 ' K ì
J J C C ] w d` E ì ¹
ì1 C ëÐìF J
' C i
*
*ì ì
\ N
ëaì
\ N J C ‚
Despite the fact the results shown here are given in [2] we sketch below most of the steps
to obtain the so-called structure
Srelations
K
pfor the corresponding polynomial solutions of (3.1)
å . Thus,ì1 the
in the non-uniform lattice J C
N%=,Q
reader could follow easily all the
calculations. For such a purpose we will substitute
C as follows

‚
ìY C K ì ¹ J ìF C Ðp ì1Gt+K ì ¹ =
x y J C Ðp
ì1#tÄ i ì ¹ å =
x y i I%
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(), AND
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35
ASSOCIATED WITH -HAHN POLYNOMIALS
in (4.13). Then, based on the three-term recurrence relation (4.11) we obtain the first structure
relation
(4.14)
where
(4.15)
C
ì K ] w ì
™
ëaì1 K 1ìvëaì ap ƒ ìvëaì1 Ðp „ ìvëÐì '%
J J C C ™ \ N J C ™ J C ™ MFN J C
K ] w ì
xy ? ì p
ì =
\ N
`E
„ ™ ìOK ì ] =w xy u ì ‚
`E
ì
ì ¹ **
xy > ì i
=
`E
ƒ™ ì
ì1Gt+ ì¹ i å =
x y i I To deduce the second structure relation we will transform (4.14) with the help of the
identity
ë ì ' K ë ì ë ì J C i
J C ‚
J J C C
J C
J C
K
| E J C i å | E , as well as (3.1) and (4.11), one gets
Thus, using J Ci y€
] Yp ëaì1 K ì ë ì Ðp ƒ ì ë ì 'Ðp „ ì ë ì '%
C
C J CWi y€ ` J J C C
\ N J C
J C
FM N J C
(4.16)
where
FìzK 1ì
ìì | E
™ ix>
ƒ
or equivalently, using (4.15),
ì K ] w ì
M xy ì
`E = > i
ì ¹ **
ƒ ™ ì iw? ìì | E p å ì | E „
ìzK
ì
ì
\
N
„
ƒ
ì K ] w ì
ì p
= M xy ?
E
`
ìÌK ì = ] w M x y u ì
`E ‚
ìOK
„ ™ ì i uvìì | E ì1#tÄ M
ì¹ i å =
x y i Now we will find a difference-recurrence relation of the form [2]
ëaì1 ‰K†…ì ë ì \ N J C p‡jì ë ì J C p ì ë ì MFN J C J C
û
J C
J C
J C
ì ‡ ì y û ì constants. Notice that this relation is not include in [29].
being ,
| on both sides of
Let describe briefly how to prove (4.17). Applying the operator | ø
¶Q³¸
K M
(4.14), and using (3.1) as well as = y€ J C
J Ci y€ one obtains
C ëaì1 J C ì ë ì 'K
= y€ J C i C J C i
J C
ë
ë
ë ì ì
ì
(4.18)
K y€ 1ì \ N J C p y€ ƒ ì J C p = y€ „ ™ ì M1N J C ‚
™
= ™
=
J C
J C
J C
(4.17)
Since
K
C
¹ ¹ J P C Ðp
¹
#tÄ Yp
J C
#t+ and
K
C
¹J
Ð p
C
#tÄ%
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36
I
=YJ C i å =i
pjIK
and J C
Hence,
where
4
K
¹ ¹ p = = y€ i
JORGEARVESÚ
the expression
|| 3ø ¶Q³¸
= y€ J
C i
C
C K 4
¹
and
*
K
¶8Q³¸
is a polynomials J
Yp
J C
#tÄ
= y€ ¹ i
*
C
of degree one.
¹¹ å=
y€ =Wi i
#tÄ
‚
Thus, (4.18) becomes
aë ìF ìvëÐìY K
= M y€ , 4 J C J J C C i
J C ê
ë ì ë
'
ë
ë ì '
ì
ì
(4.19)
ì \ NJ C p ƒ ì J C p „ ™ ì M1N J C i * J C ‚
™
™
J C
J C
J C
= y€ J C
ø| on both sides of (4.11) we can eliminate the term
Now, ø if we apply the operator |
¶Q³¸
|‰ˆ
p~K
I
J C | x ø ¶ ¶8Q³¸d¸ in (4.19) by using the identity J C
=aJ C i å =‹i , i.e.,
¶Q*¸
s ëaì1 J C K ìŠ ëÐì \ N J C p ] ìUp I ëaì1 J C =IJ C J C
> ëaì J C ? å =‹i ` J C
‹p u.`ì M1N J C i ëaì1 J C ‚
J C
Finally, multiplying (4.19) by = and using the above equation one gets
æ ì4
> {i = Ky™ ì è ë ì \ N J C ' p æu ì 4 iŒ= yK „ ™ ì è ë ì
4 p ì
ƒ ì 4 p = ì
æ ì 4 p = 4 Fp J C
K
ë
ì
å =‹i
="*oiŒ= y¾™ è p ?
J C 4 p ì
J C =
p ì K
 t . In particular, we have
=
which is of the form (4.17) if 4
q
ž ] p
]
K
K
p
p œ
M
œ
4 = n ily k > ?1` E * û >Y` E i{= mJksi y € i{= \ r \ N ] ? ~
`E
ëaì1 J C
M1N J C
J C
K
consequently,
4
p
=
ìOK = n œ ily k ž ] > p F? ` E
p ]w
`E
]w p
> p ?
]
û i
`E
p~ `E ‚
See Tables 4.3 and 4.4 to find all the recurrence coefficients described in this subsection.
4.3. = -Clebsch-Gordan coefficients. It is relatively easy to verify that the hypergeometric-type difference equation (3.1) for the = -Hahn polynomials is equivalent to the recurrence
relation (2.6) for the = -Clebsch-Gordan coefficients with respect to the projection ŒN or 
P
of the angular momentum @
N or @ , respectively. For such a purpose let to rewrite (3.1) as a
P
recurrence relation in C , i.e.,
(4.20)
ΠC ~
p Fp1 apŽ K7t
C
C C
C W
C i
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(), AND
(*)
#"%$"'
ASSOCIATED WITH -HAHN POLYNOMIALS
TABLE 4.3
Recurrence coefficients for -Hahn polynomials in the lattice
Coefficient
1
x
‘1
x
™1
x

x
‘
x
™
x
ì q ì ì q q
Þ
M E œ › x ksmJky nk €G Þ ì \ \ \ N³ß àr Þ \ ß à%Þ ß à Þ\ ì \ \ r \ \ r N<ß\ à N<ß à
P
P P
q
ì q
q ì
qì
ì
E nk y y œ i Þ \ \ r \ N³ß à
 E œ Þ œ\ œ \ Þ r ì \ \ N³ß à%Þ \ \r \ r \ ßáà N³Þ ß à ì Þ õ \ M \ M r P \ ß à N³ßâà
Pì P ì
ì q q € xì m q i q P
q
y
œ
E
Þ
³
K
›
\ Þ ì \ yb€ ’ \ ilkor n\ k N³ßámJà k Þ €Gì ”“ \ \ \ r,ßâà ß Þ yà Þ \ ì \ \ r \ \ rõ MTßáN³à ßâÞ à õ Þ M M ßâì à Þ MTßáN³à ßáà \ãE x œy y Þ x y ß à
q
P
P
P
õ
E
\
\
\
\
\
r
³
N
ß
M
<
N
ß
M
ß
ß à—–
Þ
%
à
Þ
à
Þ
à
Þ
y
m
kon xy
xy xy õ
ì xy q õ
\YE i œ x y œ n œ K • x k € k
Þ \ \ r \ P ßà q
˜
ì q ì q ì P q
ì
ì
q
q
\
\
\
\
\
\
\
\
ß
,
r
ß
r
³
N
ß
r
ß
M
ß
E › x kon œ m y œ y i œ y  Þ Þ ì à Þ \ \ à6r Þ \ N<ßâà Þ ì \ \ à%Þr,ß yà Þ ì \ \ õ rWMTà6Þ õ N³ßáà à
P
ì q ìP ì q q P
Þ
M E x kWm
koy nk € Þ ì \ \ \ N³ß àr Þ \ ß à%Þ ßâà Þ \ ì \ \ r \ \ r N<ß\ à N<ßâà
P ì q P P ì
q
q
q ì
ì
E nk y y œ i Þ \ \ r \ N<ß à  E Þ œ \ œ \ Þ ìr \ \ N³ß à%\ Þ r \ \ r ß \ à%Þ N³ìß à Þ\ õ M \ r M \ P ßN³à ß à
P Pì ì
ì €q m ì i P qq
q
q
y
y
6
E
œ
\
\
\
\
ß
r
ß
Þ
à
Þ
%
à
Þ
³
K
›
€
y
"
’
\ Þ ì \ il\knr k \ x kN³ß à%ÞmJk ì €#\ š “ \ r,ß yà Þ ì \ \ rMUõ N<ß à Þ õ M M ì ß MTà Þ N³ßß àà M E œ x kž y
q
P
P
P
õ
\
\
\
\
\
E
r
³
N
â
ß
à
M
<
N
â
ß
F
à
M
â
ß
à
ßâà –
Þ
Þ
Þ
Þ
y
m
o
k
n
q
xy xy õ
xy
ì xy õ
\YE i œ x y œ n œ K • x k € k
\
\
\
r
â
ß
à
Þ
˜
P q
ì q ì q ì Pq
ì
ì
q
q ßâà Þ õ M ßáà
E n œ m y œ x œ i œ € Þ \ Þ ßâà ì Þ \ \ r,\ ßâà r Þ \ \ N<ß à \Þ ì r \ \ N<\ ßâà r,Þ ß yà \ Þ ì \ \ r \ \ rõ MU
N<ß à
P
P
Explicit expression
TABLE 4.4
Recurrence coefficients for -Hahn polynomials in the lattice
Coefficient
›
x
¢
x
õ x
-.8(&0 6 2%3534 4
-.8(&0 6 2%3534 4
ìq
ì q q
ì q q
y
œ
œ
œ
q
E
\
\
\
\
³
N
ß
r
<
N
ß
Þ
%
à
Þ
à
Þ
i xy n ì
ì
ì \ ì \ r,ß à
Þ P \ \ r \ P ß à%Þ P \ \ r \ N³ß à ¶ Þ \ r
ß à \ Þ ß à%Þ \ \ q r \ N<ß à ¸
ì
ì
q
Þ \ \ r \ N<ß à
ì Nì q
ì M Þ q \ N<ß à
Þ \ r,ßâà \ Þ ßâà Þ \ \ r \ N<ßâà s¸ œ Þ P \ q \ r \ N<q ßâàžŸ E € œ m y œ n œ œ œ
m ì i xì
ì¶
ì
ì
ì
Þ \ Þq r ì \ \ N³q ß à%\ Þ õ r M \ ßâM à P ß à \ E œ Þ i ì œ \m œ q € Þ\ r,\ ß yà q Þ ß à ì Þ \ \ q \ \ rr MT\ N³ßâõ à Þ ß à%q Þ õ M ìM MTß à%N³ßáÞ àß à ¡
P
P
P
P
ì
N M Þ r \ N<ß à Þ õ MUN<ß à M Þ \ r \ P ß à M Þ \ õ \ r \ N³ßáà Þ xy ßâà
\ Þõ \ y œ ß à \
E
E œ xž q
E nk y q i E mJkynk K
E œ m œ ilykn
ì
Þ \ \ r \ N<ß à E x k € k mJkoy n Þ x y \ r \ N³ß à q Þ x y M õ \ N³ß à M Þ xy ß à%Þ x y \ õ \ ß à—–
M
• E x œ ily konk y Þ ì \ \ r \ ßâà
£
q P ì ì Pq
ì
ì
ì
œ
œ
K
ì M E q x ily kn Þ ì \ N<ßâq à Þ \ ßáà Þ \ q r,ßâà Þ \ ì \ r,ì ß8àœ Þ q M ßâà
Þ P \ \ r \ N³ß à%Þ P \ \ rMTN³ß à ¶ Þ \ r
ß à \ Þ ß à6Þ \ € õ \ r \ N³ß à ¸
Explicit expression
37
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38
JORGEARVESÚ
where
(4.21)
ΠC K
Let
K
(4.22)
C
Ž! K
Y p C C J C i y€ ¤
C
J C i 1y€ KJ C ì Ž J Ci
C
i C ixŒ C ‚
@qN‰iˆLN
?
Kû
C
J
C
y€
K
p
p ~
I@ N p @ P iŒ  w K
> K  p @IN“i»@ P  i•@qN @ P
@ iŒŒ‚
T
Therefore, substituting in the above equations (4.21) the choice (4.22) we have
] p
p~ ]
i=qP ” € \ ” y M „ € \ ] y€ @ P  ] P p ` E p7 P i•p @ P ] ` E
] p
p7
\
”
M
”
@IN“iˆLN p = ” € \ ” „ € ] y€ = p y @ãp7
iˆ V` ] E @  1 ` E @qN‰iŒLN&` E @IN LN
`E
y
E
E
=

@
`

»
i
@
`
P Ž! @ N iˆ N K = ” € M „ \ y€ ] P @ N iˆP  N `dE ]  N pP @ N p~
`dEI‚
€ €
Consequently, the equation (4.20)
p for the = -Hahn polynomials becomes
” p\ „ M „ y ½ ”¾M „ \ „ y Œ p€@ 1N iŒ N p „ M5” ” € \ M ½ ”¾€ M \ @ N iˆ N p~
@ N p p @ P iˆ p~p7 ¥'= (4.23)
-p Ž @ N iŒ@q N‰N iŒ L” M5N \ ” „ € „ M M5„ ” y „ ½ €”¾M „ y„ € \ „ y „ € @ N „ yiˆ@q N‰N iˆi L
N @ @qN N p @ @ P iŒiŒ  p7 ¥'¥&= = K…t ‚
P
„
Comparing the recurrence relations (2.6) and (4.23) one deduces pDq

KÝ ì \ bQ ¥ b¦ %
(4.24) p q
C
J CWi y€ ì ½ r C ûj¥&=
@IN%•N@ P  P † @+»‡ E œ
i
€
÷
w
¦ì where ½ r C ûj¥&= denotes the orthonormal = -Hahn polynomials and C , û , > , ? , coincide
K
1Œ @IN“iˆLN K
with (4.22). To efficiently carry out a comparison between these two recurrence relations we
should consider (4.10) as well as the following useful expressions
÷
] p
]
p~
N K = M5” ] @IN LN&` E ] @qN‰piˆLN p~ ` E ÷ q@ N‰iŒLp~
@ N iŒ N
@ iˆ `dE @
 P `dE
÷ @ N iŒ N K ” ] @ P p P  P `dE P ] @ P iˆP  P p~
] p
÷
= ]
p~ `dE @IN“iˆLN‰i
@I“
N iˆLN&` E @qN LN
`E
J C i
J Ci
J Cpi y€
J C y€
yå € K =
PK
= M1N ‚
For the ratio of weights we have used (4.7).
Notice that (4.24) constitutes an extension –in this case a = -analog– of the well known
relation between the classical Hahn polynomials and Clebsch-Gordan coefficients. Furthermore, this expression is in accordance with the results obtained in [33] (see also [1] for more
details).
From (3.4) as well as from (4.8) the symmetry property for the = -Hahn polynomials
pCq
q
p q
ì ½ r C û ¥'= KÝ i ì = ì ¶ \ r \ ¸ r!ì ½ û ijCWi + j
û ¥&= M1N
õ
holds. It leads as a consequence the corresponding symmetry property for the = -ClebschGordan coefficients given in (2.7) by a simple substitution of the parameters (4.22). Now, the
substitution of the parameters (4.22) into the
expression (3.11) with the help of (4.24) gives
the = -analog of the Racah formula for CDFE
Clebsch-Gordan coefficients

K¦ I „ M5” ª
i € €¨§©
©
¨
\
\
”
¯ I ¨ = y€ ¶ „
]ñ ]
i
` Eq’ @Uiˆ i
¨
@qN%•N@ P  9 P † @
•‡ E
(4.25)
Þ \ N³ß à Þ MX”ß àR«áÞ \ ”ß àR«áÞ MX” ß à\«áÞ M5” ß àR«áÞ \ M ß à\«
Þ „ P \ „ „ y \ „ \ „ N³ß àR«áÞ „ M„ „ y \ „ ß à¬„ «á€Þ „ y M „ € \ „ „ ß à¬y « Þ „ \ y ” „ ß àR€ «áÞ „ „ y y \ ” „ y ß à\«
€ = 5M ” ” \ € N \
€
€ €
¶ ¸ y,€ ¶ „ ¶ „ \ N ¸ \ „ € ¶ „ € \ N ¸ M „ y ¶ „ y \ N ¸d¸
€
N ¸ ] @qN‰iŒLN p ñ ` Eq’ ] @ p @ iˆLN‰i ñ ` E,’
ñ ]
ñ ] P p
ñ ‚
` Eq’ @IN“iˆLN‰i ` Eq’ @ P i»@ •Ni ` Eq’
ETNA
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(), AND
(*)
#"%$"'
ASSOCIATED WITH -HAHN POLYNOMIALS
39
Two particular casesK are immediately derived from (4.25):
 N
i) First, when @ N
Þ \ <N ß àHÞ \ ”ß à\«áÞ ß à\« Þ MX” ß à\«áÞ M \ ß àR«
Þ „ \ „ y \ P „ „ \ <N ß à¬«á„ Þ „ \ „ y M P „ „ ß € àR«áÞ „ „ My „ y \ y „ ß àR«á„Þ „ y \ „ € ” y „ ß y à\«áÞ „ M5”ß à @ N @ N @ P  P † @
•‡ E
€
= „ € ¶ „ 5M ” € ¸ M y,€ ¶ „ € \ „ €y M „ ¸<¶ „ M „ € \ „ y MFN ¸
K~t
which corresponds to the choice C
for the = -Hahn polynomials.

K
ª ©§
©
ii) Second,
Þ P „ \ N<ßâà «áÞ „ \ ” ßáà «áÞ „ y \ ” y ßâà «áÞ „ \ „ y M „ ßâà «
„ \ \ N³N ßáà «áÞ „ € M5M ” €„ y \ „ M €ßâà « Þ „ y M \ „ € \ M „ ßâà «áÞ „ € M € M5” \ € ßâà \ «áÞ N„ y MX” y ßâà « ¶ „ ¸<¶ „ € € ¸d¸ y
€ ¶ „ € „ y „ ¸³¶ „ „ € „ y ¸#¸
K
@
which coincides with the expression (4.24) for 

p
¦ [” \ „ € M „ y ½ ”¾M „ € \ „ @qy N6 L@qN‰N*@ iˆP  LP N † @I@q@.N ‡ Ep @ iŒ p~ ¥&= K¦ i Qb¥ ÷ C J CWi y€ ‚
P

@qN •
 N@  † @@.‡
I P M5P ” E K
i „€ €
pq
ª ©§
©
ބ \ „ y \
€ =
pq
pq
From the Rodrigues-type formula (3.7) and the identity (4.16) it is easy to obtain
ì FM N6½ r!MFN C p~
û ¥'= i
\ N
m
kon y œ x ì \ ½ r N C ûj¥&= ‚
ì 1M N%½ r!M1N C ûj¥&= K = Q \ N \
\ N
Now, if we put in this expression the above selection of parameters (4.22) and consider (4.23)
we deduce an interesting relation for the = -Clebsch-Gordan coefficients. Indeed,
p
¥
¥
b
@Nb
@P ¥
H s
p7I é y 
 N H @ N s  N ~
p = M y é @  N 
 s P @ HP b  P ~
p F=  y € @ N 
@  @ 
@qN%LN*@ P 
N
s7
@ P  s P † @
•‡ E K
N @ P  P p~ † +@ •‡ E
‡E
P † @+
which can be also obtained from (2.8).
Relation
the Clebsch-Gordan coefficients for the quantum
Hbetween
I
*+HI algebras
and CD E
. It is well known [21] that the quantum algebra CD E
can be generated from the operators
CD E
4.4.
]
#
]®­ ­ Kcb ­ [ a_ `
_
­ [e K ­ [ ­ e_ K
#
­
]š­
f
\
­
M `
K
i
]
­
[ `/E
where
` represents as in the Section 2 the commutator product.
The description of this subsection is quite similar to the already presented in the Section
2 as well as [1]. Here, to avoid confusion
K
po+with
p the
v Section 2 we use the notation ‘prime’ over
@¹
@¹
‚‚H‚ These basic vectors of the irreducible
the basic vectors † @ ¹  ¹ ‡ E ,  ¹
p~
¿
\
representations of the discrete positive series ƒ „
are determined by † @ ¹ @ ¹
‡ E , such that
­
M † @Ĺ @+¹
p7
‡E
K~t
and
†@ ¹  ¹ ‡ E
­
[‰† @Ĺ @+¹
p7
‡E
K¦ ~
p p~ Ð  p7
p7 K¦
@Ĺ
† @+¹ @Ĺ
‡ E Ä@ ¹ @+¹
† @Ĺ @+¹ ‡ E
] p7
Kݑ ] p
@ ¹ ] ` Eq’ ­ ” ¿ M „ ¿ M1N † @ ¹ @ ¹ p7 ‡ ‚
E
@ ¹  ¹ ` EI’  ¹ i•@ ¹ i ` EI’ \
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40
JORGEARVESÚ
Analogously, the*tensor
+HI product of the irreducible representation ­
quantum algebra CDFE
can be decomposed into the direct sum
K
­ „ €¿ \ ® ­ „ y ¿ \
„ €¿ \
and ­
„ y¿ \
of the
¯¯
ƒ „ ¿\ ‚
„ ¿ °5„ €¿ \ „ y¿ \ N ²
Its generators (co-products) are given by the expressions
­
_
­
*+&+^K [
*+&
K ­ *IYp ­ [ *Fp [
­
­ G +
= Wy€ ° µ ¶·P&¸ _
= M Wy€ ° µ ¶ N ¸ _ ‚
±
The explicit form of the irreducible representation
±
­
K…¼ 1 @Ĺ  ™ ¹ † [ † @Ĺ⏻¹h² E
”“¿d½ ”‰¿
­
] b b7 ¼ 1
K À ] s
Ä@ ¹  ™ ¹ † _ † @ŏ»¹ ² E
 ¹ @ ¹` E  ¹ @ ¹ ` E ” ¿ ½” ¿ _ N
as well as the Casimir operator
K ­ ­ p ]š­ ]®­
[ ` E [ i ` E Á P† @ ¹  i¹ ‡ \ K ] M@ ¹ ` E ] @ ¹ p~
` E †@ ¹  ¹ ‡ E
E
ÁP
(4.26)
are given in a quite similar form to those presented in Section 2. Analogously, the definition
of the = -Clebsch-Gordan coefficients is
K
† @ÄN¹ @ÄP ¹ Ä@ ¹·º¹‰‡ E
¯

@ÄN¹ »¹N @ÄP ¹ ºP¹ Ä@ ¹·º¹8‡ E † @ÄN¹ »¹N ‡ E † @ÄP ¹ º¹P ‡ E
” ¿ ½ ” ¿y
+ € K ] ] p~
¹
¹
¹
¹
¹
¹
¹
¹
†
@
@
@

‡
@
/
`
E
@
d
`
E
†
@
@
(4.27)
N
N
E
P
P @ ¹ ¹ ‡ E ‚
ÁP
Now,
(4.26) as well as the expression (4.27) the matrix ele using the Casimir
&+ operator
†
@
@
@

‡
ments @ N ¹  ¹N @ ¹  ¹ †
¹
¹
¹
¹
N
E can be found.
From these matrix elements (see [16])
P P ÁP
P
the following three-term recurrence relation in  ¹N  ¹ for the = -Clebsch-Gordan coefficients
P
³ É ³ ³ É ³ ³É ³Ó
Ç È É´³ ³
È ³ µ
É ³ È´
É ³ 4 ³4 È ³4 ´
É ³4
ÊÌË Î Ë Ï
ÏjÍ'Î ÐÑ Ë 4 4 ÏjÍ³Ë : : ʊÍIÒ Ë
: Ê«Ë : ÊLÍ'Î Ë : Ï : Î
³ É ³ ³ É ³ ³É ³Ó
É ³ ³ È ³ µ
É ³
È ³ µ
É ³ ȵ
É ³ ³
Æ Ç Èµ
Ï
: ÊzË : Î Ë : Ï : Ï{Í&Î Ë 4 Ï 4 Î 4 Ê«Ë 4 ÊLÍ&Î aÑ Ë 4 4 ʊÍ*Ë : : Ï{ÍqÒ Ë
(4.28)
ÈÉ ³ ³
ÈÉ ³ ³
Æ 3vÕ ¿ È ³ É ³
ÆÕ ¿È ³ É ³
Ï ¶
€ Ë : Ï : Ï{Í&Î : Ê«Ë : Î Ï y Ë 4 Ï 4 ÏjÍ'Î 4 ÊÌË 4 Î È ³
ÈÉ ³
Æ 3 Õ ¿ 3vÕ ¿ 354GÙ ³ É ³ ³ É ³ ³ É ³ Ó ÚjÛ Q
Ï Ë Ï y€ Î : Ê
Ï y€ Î : Ö y
€ × y € Ñ Ë 4 4 Ë : : Ò Ë
yields.
Finally, to conclude this section
G+ let establish
*+HI a useful relation between the = -ClebschGordan coefficients for the CD E
and CD E
algebras, i.e.,
(4.29)
where

@qN6LN'@ P  P † @+•‡ Q¸· à ¶·P&¸
Kݍ
@ N¹  ¹N @ P ¹  ¹P † @ ¹  ¹ ‡ ¸Q · à ¶ N6½ N ¸
K ¿ ¿ ¿ K ¿
@ N ¿ ¿ ” \¿ „ € M „ y MFN  N ¿ ” ¿ € M5¿ ”
P
K
K
@ P ” M „ € \ „ y MFN  P ” y M5” € \ „ € \
P
P
¿y \ „ ¿ \ „ y¿ \ N K
@
@¹
€
„ y¿ P \ N  K @ N¹ p @ ¹ p7 ‚
P
The relation (4.29) can be deduced by a mere comparison between the relations (2.6) and
(4.28). This result is in accordance with [4, 16].
ETNA
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etna@mcs.kent.edu
(), (*)
#"%$"'
AND
TABLE 5.1
Three-term recurrence coefficients for -Hahn polynomials in the lattice
-.8(&0º ;
Coefficient
q
>
] p
ì
?
p
> ?
= n œy m ,
[1]
Explicit expression
p ] p p w p7
| E = M y€ ¶ \ r \ N ¸ ] w ~
] p p w p ] ` E p > p ? w p7 ` E
> ?
`E > ?
`E
œ
œ€
w ]w ] p p
]
p w = m ] nksy p i p w p `/E > ] ? w ] w `/FE p7¹ û ] i p ` E p ` E w > ?
] p i p û i p w i p7 ` E ] w p7 ` E > ] ? p p ` E p w
> ? ] û p p w M1` E N ` E i > ] p ? p û w p
> ? `E
> ?
ì
41
ASSOCIATED WITH -HAHN POLYNOMIALS
q
w p `
]w
`E `E P
EM1N ê º
`
| E = \ õ M y€ ] > p w ` E ] ? p w ` E ] > p ? p û p w ` E ] û i w ` E
] p p w ] p p w 7
p > ? `/E > ?
`/E
u.ì
ãK
5. Remark on the = -Hahn polynomials for J C
= Q . Here we will present a brief
description on some basic characteristics of these polynomials from the point of view of
the Nikiforov-Uvarov approach [29], specially, the three-term recurrence relation. The importance of this relation in connection with the = -CGC was already
discussed in the above
ì
section.
We
focus
our
attention
in
this
relation
since
when
the parameters
and
{
=
R
.ì
|
clearly go to zero as a consequence of the multiplicative factor E (see the equation (2.1)
as well as Table 5.1). Thus, this relation loses sense in this limiting case. Therefore, this
comparative analysis reveals that the three-term recurrence relation –for instance– is not a = analogue relation because it has not the corresponding partner property
classical Hahn
K E 2%for
M1N the
polynomials, which does not happen in the investigated case J C
(see
Section 4).
E M1N
K
=+Q have been studied in [1, 2, 33]. Despite
The = -Hahn polynomials in the lattice J C
the fact that we can use the same scheme presented in Section
¾K 4 to obtainK them,
we will
K¦t omit
it. Below we will summarize few results. Let chose J C
= Q , i.e., N
and å
. In
this case, with the parameters
>
u
HN
K…t!
H
ì * = xy |i ìE ] w ` Eq’
H @ N K i=?ºi H @ K û i +
P
P
ìÌK
K
û
p
>
q
4
K
q
i= y
€ ¶ õ \ ¸
we obtain the = -Hahn polynomials [24, 33]. Indeed, from (4.4)-(4.5) we have
ì ì q
1
ì
*
ì
\
\
r
N
%
N
M
M
= M Q = \ \ r \ N &¥ = =Iê
ì ½ r C û E K = x y!¶ õ ¸ = ì '¥ = ì = õ ¥'= å I , =
| E =.q¥'=
P ì = r \ N = N%ì M õ q
K = r \ N ¥'= q ì1 = õ ì \ \ r \ì N ¥'= ì
= M =,Q \ r \ N = q \ \ r \ N ¥&= I= êj‚
J
å
I
,
*
ì
=r\ N =õ \ \ r\ N
P
= M x y!¶ õ \ \ P r \ \ N ¸ | E =.¥&=
5ì
pCq
In addition, for
we find
ìOK
q
]w ]w p
= y,€ ¶ \ r \ 'P ¸ ` E
> p ?
p~
`E ‚
ETNA
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42
JORGEARVESÚ
6. Conclusions and remarks. The method presented in this paper connects in a precise
way (see equation (4.24)) the = -Hahn polynomials with = -CGC, i.e., the Nikiforov-Uvarov
approach and the quantum group representation one. The Pearson
type approach to = -Hahn
polynomials used here is based
on
a
specific
kind
of
lattices
J
C . In fact, this approach is very
effective since the lattice J C is always the same for each relation used through the paper, i.e.,
the lattice is the same for Pearson-type difference equation, hypergeometric-type difference
equation, structure relation, difference-recurrence relation, and orthogonality relation which
could not occur in other approaches.
For example, it could happen that the
equation
ãK
rdifference
K M
is given on the lattice J C
=
Q and the orthogonality relation on J C
= Q . Thus, our
approach avoids this and unifies the election of the
lattice.
L
=
R
As
the
reader
has
observed
the
limit
do not transform directly the polynomials
VK
=,Q into the classical discrete Hahn polynomials since a rescaling factor before
on J C
taking limits is needed. However, one of the contributions contained in the paper is precisely
to avoid this. In such a way we avoid to answer the following questions: Is the rescaling
factor the same for all the aforementioned relations/characterizations? or different rescaling
factor should be introduced depending onthe
relation for which one is looking for the
K specific
E 2 M1N proposed
classical analogue? With the lattice J C
in the paper one can avoid this;
E
1
M
N
therefore the = -Hahn polynomials on J C just becomes directly into the classical Hahn polynomials. Moreover, there is also another
reason. When one computes all the characteristics
K
of the = -Hahn polynomials on J C
=+Q such as the three-term recurrence relation, structure
relations, etc, and takes the limit =•R
one realizes that all these formulas becomes zero.
For getting a non trivial result one should use not only a rescaling factor and some times the
higher (usually 2) order Taylor expansion. One can avoid this by using an appropriate lattice.
On the other hand, notice that the = -Hahn polynomials obtained in this paper are new
ones despite the fact they are included in the Nikiforov-Uvarov approach. Regarding this to
avoid confusions let us comment a couple of things. First, any change of variable carried out
to transform the weight (measure) which leads to a non-linear change of the support of the
measure never will produce up to a constant depending on the degree the same polynomials.
ë ì
For instance, ifn theK orthogonality
condition
of
a discrete
orthogonal
polynomial
are given
!
ñ
ÿ
ô
º
ñ
ç
K
t
H
+
u
R \
‚H‚‚ û i , being the support of the measure
on the lattice n J ¨ u J
the closure of J ¨ õ¨ [ . Then, for such a support the orthogonality condition
°
¯õ 1M N Ðë ìF ' ¨ K7tlñÌK~t!
w J C J C C J Ci y€
‚‚H‚ i
[
÷
Q°
by replacing C by û i
ie
nt leads us to,au completely different polynomial family. Notice that
the support is not the set
to which belongs the variable C .
‚‚H‚ û i
Second, the = -Hahn polynomials studied in this paper are the solution of the correspond ing second order difference equation of hypergeometric type on the non-uniform lattice J C .
Since
¾»
K this equation
p is linear
P
‹one
K can always make a change of variable to convert the lattice
J C
=
Q , in fact if one writes the solution of the equation in terms
into J C
N =,Q
P
of the basic series it looks very similar (see [29], or the paper by Nikiforov and Uvarov [31]).
Actually, this representation does not depend on . Nevertheless, the polynomials are quite
P
different since ëa
they
ì1 are
polynomials on different
lattice. This means that when one writes the
expansions of
C as polynomials in J C the coefficients are completely different, that is
not evident when one compares the basic hypergeometric
function —the reason is that in both
cases the polynomial can be expanded in the =ÄQI¥'= ¨ basis just in the same manner–.
Finally, we would like to point out two remarkable benefits (among others) obtained from
the approach presented in this paper:
1. A unique lattice is involved in all relations.
ETNA
Kent State University
etna@mcs.kent.edu
(), AND
(*)
#"%$"'
43
ASSOCIATED WITH -HAHN POLYNOMIALS
without a rescaling factor is computed wherever is needed to
2. A simple limit =«R
obtain any classical property.
Acknowledgements. The author was supported by the ’Ramón y Cajal’ grant of the
Ministerio de Ciencia y Tecnologı́a of Spain as well as for the grant BFM2003-06335-C0302. The author is grateful to Professor R. Álvarez-Nodarse from the Departamento de Análisis
Matemático de la Universidad de Sevilla (Spain) for several helpful and insightful comments
on an earlier draft of this contribution.
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-.8(&0½¼ 4 ;l¾
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(),8 5 8 #"%$'"'
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