QUANTUM ALGEBRAS AND ASSOCIATED WITH CERTAIN -HAHN ...etna.mcs.kent.edu/vol.24.2006/pp24-44.dir/pp24-44.pdf · prerequisites are a good understanding of the fundamentals of group

Electronic Transactions on Numerical Analysis.Volume 24, pp. 24-44, 2006.Copyright 2006, Kent State University.ISSN 1068-9613.

ETNAKent State University [email protected]

QUANTUM ALGEBRAS��

AND��

ASSOCIATED WITH CERTAIN�-HAHN POLYNOMIALS: A REVISITED APPROACH �

JORGE ARVESU �Abstract. This contribution deals with the connection of � -Clebsch-Gordan coefficients ( � -CGC) of the Wigner-

Racah algebra for the quantum groups �� and ��!��#"%$&"'� with certain � -Hahn polynomials. A comparativeanalysis of the properties of these polynomials and (*)+�� and (�),��#"%$'"'� Clebsch-Gordan coefficients shows thateach 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 themain characteristics and properties for the new � -Hahn polynomials obtained here by using the Nikiforov-Uvarovapproach [29, 30] on the non-uniform lattice -.�/('�10 ��2%354�63�4 . These characteristics and properties will be importantto extend the � -Hahn polynomials to the multiple case [7]. On the other hand, the aforementioned lattice allowsto recover the linear one -.�/('��07( as a limiting case, which doesn’t happen in other investigated cases [14, 16],for example in -.�8(&��09� :<; . This fact suggests that the � -analogues presented here (both from the point of viewof quantum group theory and special function theory) are ‘good’ ones since all characteristics and properties, andconsequently, all matrix element relations will converge to the standard ones when � tends to 1.

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 mathemati-cians A. F. Nikiforov and Yu. F. Smirnov for their scientific works in this area. Both are prizewinners for research works. The first one was awarded with the Lenin Prize in 1962 while thesecond one was recently awarded (in 2002) with the Lomonosov Prize.

1. Introduction. The theory of quantum groups is a very fascinating subject at the bor-der of many areas, mainly group theory, differential and difference equations and specialfunction among others. Consequently, many techniques have been developed and are avail-able; therefore there is some productive competition between various approaches to the sub-ject. The crucial role that the group representation theory concepts play as effective tool tounify independent areas of Physics and Mathematics such as the Quantum Theory and theSpecial 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 interrelationbetween these two areas by constructing a = -analogue of the Hahn polynomials using theNikiforov-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). More-over, the construction presented here for the = -Hahn polynomials is a very simple one andit corresponds to the standard theory of orthogonal polynomials i.e., the orthogonality con-ditions are considered with respect to a regular linear functional [15]. Indeed, the regularitycondition for certain modifications of the linear functionals [5] is the corner-stone that willallow the extension of the quantum algebras studied here.

During the last years the study of = -discrete analogues of the classical orthogonal polyno-mials and the connection with the representation theory of quantum algebras in relation withseveral applications has received an increasing interest (see for instance [6, 9, 17, 23, 27] aswell as [39]-[40]). The notion of quantum groups and algebras (Hopf algebras [18]) appearsas a consecuence of the study of solutions of the Yang-Baxter equation [26] and the devel-opment 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. Marcellan.� Departamento de Matematicas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911, Leganes,Madrid, Spain ([email protected]).

24

ETNAKent State University [email protected](�),�� AND (*)��#"%$�"'� ASSOCIATED WITH � -HAHN POLYNOMIALS 25

statistical physics were practically immediate [43], especially for the study of the = -deformedoscillator [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 mate-rial can be gathered under the title ‘quantum algebras’, and therefore, any communicationon this topic that serves both as a survey and research paper -like this communication- mustnecessarily be summarized. This partially answers to the question about the dedicatory ofthis contribution, since here are summarizing some ideas of both Researches. Despite thecontributions of other authors we have focused the attention of the reader in the techniquesand methods developed by A. F. Nikiforov and Yu. F. Smirnov. However, a suitable list ofreferences is included.

The knowledge of CGC, Racah coefficients ( ?�@ symbols) and A�@ symbols is essential forunderstanding the corresponding quantum physical problem since all matrix elements of thephysical quantities are proportional to them (see [36, 37]). Based on this well known fact weinvestigate the relation between the Clebsch-Gordan coefficients –also known as B�@ symbols–[42] for the quantum algebras C�DFE �G�+ and C�D1E ��H�I with = -analogues of the Hahn polynomialson the non-uniform lattice J � C LK E 2 M1NE M1N . Although several authors have investigated theconnection between different constructions of the Wigner-Racah algebras for the = -groupsand = -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 inreturn classical analogues. For instance, the connection of CGC with some = -analogue of theHahn polynomials for J � C OK =,P6Q was studied in [16], however to recover the parametersinvolved in different characterizations for these polynomials as well as their connection withstandard group representation is not possible as a simple limiting case ( =SR �

) since manyof these parameters under this limiting operation disappear (go to zero); consequently thecorresponding physical relations lose sense (see Section 5). The same could happen withsome = -analogues of the Kravchuk and Meixner polynomials for J � C TK =+P6Q in connectionwith the Wigner D-functions and Bargmann D-functions for C�D E �G�+ and C�D E ��H�I algebrasthat was established in [14]. These questions should be study in forthcoming publications.

Therefore a natural question appears: What = -analogues in connection with physicalproblems should be constructed? Recently, in [2] based on the Charlier case the authors showthat the lattice J � C UK =Q is not a proper choice in the sense that to recover the linear onesJ � C VK C is not possible by taking limits as we have already mentioned. Instead, a goodchoice is J � C WK E 26MFNE MFN , such that when = tends to one we recover the classical linear lattice,and the constructed polynomials are ‘good’ = -analogues of the classical ones. Moreover, allcharacteristics related to such = -polynomials will tend to the classical ones for the aforemen-tioned limit case ( =SR �

). This fact, is crucial when we are dealing with physical conceptsinstead of mathematical ones.

The present communication is written in a narrative way without loss of mathematicalrigor. We have stressed simplicity of ideas at the expense of a formal mathematical presen-tation. Consequently, the customary sequences of definitions, lemmas and theorems will beomitted; nevertheless a mathematical rigor in all the arguments has been kept. Thus, the pre-sentation of the results is self-contained and mostly elementary. As a consequence the onlyprerequisites are a good understanding of the fundamentals of group theory and orthogonalpolynomials. 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 theCGC. In Section 3 we quickly sketch the more important aspects of the Nikiforov-Uvarovapproach. This will facilitate the willpower of the reader for understanding the next calcu-lations 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


26 JORGEARVESU

the eigenvalue problem for the second order finite difference equation on a quite general lat-tice. The solutions for such an equation have several properties similar to the solutions of theSchrodinger equation. Thus, this approach is close to the standard quantum mechanical meth-ods since it allows to use rather fruitfully the physical intuition in the analysis of appearedproblems. The Section 4 contains a lot of calculations so the author kindly ask the reader to bepatient during the reading of this section. In fact, a lack in [29] will be covered since in mostof the papers related to the Nikiforov-Uvarov approach the authors did not study concretefamilies (up to the papers [2, 4, 33]). Actually, a non-included in [29] expression involvingthe difference derivatives of the = -Hahn polynomials with the polynomials itself (differencerecurrence relation) is obtained. Furthermore, a comparative analysis of the properties of the= -Hahn polynomials and = -CGC is also given. Hence, the interpretation of the representationtheory for C�D1E �G�+ and C�DXE �*�� in terms of the = -Hahn polynomials is established. The Sec-tion 5 is devoted to a brief presentation of the = -Hahn polynomials in the exponential latticeJ � C WK =,Q in order to emphasize the ‘improper’ choice of such a lattice for the constructionof physical = -analogues. Finally, the Section 6 includes some conclusions and remarks.

2. = -Clebsch-Gordan coefficients for the C�DFE �� quantum algebra. The aim of thissection 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 4below.

In [35, 36] the authors define the quantum algebra C�DYE �G�+ in the standard way using thethree generators Z�[ , Z�\ and Z M with the usual properties] Z^[ � Z�_a` Kcb Z�_ � ] Z�\ � Z M ` K ] � Z^[ `dE �ZSe[ K Z�[ � ZOe_ K Z�f �where] Z�g � Z�h,` K Z�gZ�hUijZ�h.Z�g �lk!��mVKonqpr� i �%br�6sr�&tvu+� ] w ` E K =1x yzi{= M x y| E � = K~}1�y �and

(2.1) | E K =Y�y�i{= M �y��The above expression

] � Z�[H`/E means the corresponding infinite formal series.For C�D E �� quantum algebra the irreducible representations �O� with the highest weight@ K�t!� NP �� H� are determined by the highest weight vector � @�@�� E , such thatZ�\L� @I@�� E K�t!� Z^[�� @I@�� E K @�� @I@�� E �� @I@5� @I@.� E Ko�+�

and � @+�� E Ko� ] @ p �V` EI�] � @`/E � ] @Ui��V`/E � � Z�M � MX� � @I@�� E � with i�@z��j@�Here we have used the symmetric = -factorial symbol

] w `GE � which can be expressed through thesymmetric = -Gamma function �� E � C defined in [29] and that is connected with the classical= -Gamma function

� E � C [20, 29]. Indeed, by [29]�� E � C �K = M�� 2*� �#� � 2*� y �� E � C %�


and

(2.2)� E � C �K �¡¡¡¡¡¡¡¢ ¡¡¡¡¡¡¡£

¤^� C+¥&= �K¦�� i�= N6M Qa§¨�© [ �*� i�= ¨ \ N §¨�© [ �� i�= Q \ ¨ � t«ª = ªc��

= � 2*� �G� � 2*� y �y ¤^� C¥&= M1N %� =r¬ � �Notice that �� E � C p7�I�K ] CH`/E!�� E � C and

] w `dE � K �� E � w p7�Ifor nonnegative integer

w.

On the other hand, if � � and � y denote the two irreducible representation of the quan-tum algebra C�D1E �� , then the tensor product can be decomposed into the direct sum � ��® � y K � � \ � y¯�'°�± � � M � y ±³² � � �usually known as Clebsch-Gordan series. Its generators (co-products) are given by the ex-pressions Z [ �*�+�&��K Z [ ��Yp Z [ �G� �Z�_ �*��6��K = �yI´.µ ¶·P&¸�Z�_ ��Fp = M �yI´.µ ¶ N ¸*Z�_ �G� �

Taking into account the explicit form of the irreducible representation� @+�º¹<� Z [ � @+�»� E K7¼ �¾½ ��¿ �� @+� ¹ � Z�_�� @+�»� E K¦À ] @ s �S`dE ] @ b � p~� `dE ¼ ��¿d½ � \ N �(2.3)

as well as the Casimir operator

(2.4) Á P K Z�MaZ \ p ] Z [ p �y ` P E �Á P � @�Â� E K ] @ p �y `8PE � @+�Ã� E �one can define the = -CGC in a similar way to the classical case. Thus, for the basis vectors ofthe irreducible representations � we have� @ N @ P � @+�Ã� E K ¯� � ½ � y � @ N � N @ P � P � @+�»� E � @ N � N � E � @ P � P � E �

Á P �*�I� � @ N @ P � @�� E K ] @ p �y ` P E � @ N @ P � @+�� E �(2.5)

where� @qN%��N�@ P � P � @�� E denotes the = -Clebsch-Gordan coefficients. Notice that for these

coefficients the following orthogonality conditions¯� � ½ � y � @qN%�LN*@ P � P � @+�� E � @qN ��N�@ P � P � @Ä¹��»¹�� E K7¼ � ½ � ¿ ¼ �¾½ ��¿ �¯ � ½ � � @ N � N @ P � P � @+�»� E � @ N � ¹ N @ P � ¹P � @+�� E K�¼ � � ½ � ¿ � ¼ � y ½ � ¿y �hold. Here

¼ ¨ ½ Å represents the Kronecker delta symbol.Using the Casimir operator (2.4) as well as the expression (2.5) the matrix elements� @qN ��N�@ P � P � Á P �*��6� � @qN*@ P @+�»� E can be computed. From these matrix elements one deduces


28 JORGEARVESU

the following three-term recurrence relation in � N � � P for the = -Clebsch-Gordan coefficients[33, 35]

(2.6)

Æ 3�4HÇ È É :�ÊÌË :�Ê�Í&Î � È Ë :ÐÏ É :�Î � È É 4 Ê«Ë 4 Î � È Ë 4 Ï É 4 Ï�Í&Î �ÐÑ Ë 4 É 4 ÏjÍ³ËH: É :^Ê�ÍqÒ Ë ÉÌÓ �Ï Ç È É :aÊÌË :'Î � È Ë :YÏ É :YÏ{Í&Î � È Ë 4 Ï É 4 Î � È É 4 ÊÌË 4 ÊLÍ&Î �ÐÑ Ë 4 É 4 Ê�Í³ËH: É :YÏ{ÍqÒ Ë ÉÌÓ �Ï{Ô Æ 3.Õ � È Ë : Ï É : Ï{Í&Î � È Ë : Ê É : Î � Ï Æ Õ y È Ë 4 Ï É 4 Ï{Í&Î � È Ë 4 Ê É 4 Î �Ï È Ë�Ï �y Î :� Ê È É Ï �y Î :� Ö Æ 3 �y× Õ y 3vÕ ��Ø 4GÙ Ñ Ë 4 É 4 Ë : É : Ò Ë ÉzÓ ��ÚjÛÜBased on the symmetry property for the = -Clebsch-Gordan coefficients

(2.7)� @ N � N @ P � P � @�� E KÝ� i �I � � \ � y M � � @ P � P @ N � N � @+�»� E � � �

the expression (2.6) is invariant with respect to the change @ N by @ P whereas = is replaced by= M1N .Following the same ideas used for the computation of (2.6), but in this case for the matrix

element� @qN ��N�@ P � P � Z [ �� @qN�@ P � @+�»� E one gets a similar three-term recurrence relation with

respect to the variable @ [38]À Þ � MX��ßáà Þ � \ ��ßâà Þ � � \ � y \ � \ N<ßâà Þ � y M � � \ � ßâà Þ � M � y \ � � ßâà Þ � � \ � y M � \ N<ßâàÞ P � \ N<ßâà Þ P � MFN<ßâà Þ P � ß y à � @qN ��N�@ P � P � @ãi � �� Ep«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 àÞ P � \Yå ß àHÞ P � ß à Þ P � \ P ß y à Þ � � \ � y M � ß � �à � @ N � N @ P � P � @ p~� �� Epçæ Þ P � ß àHÞ P � � \ P ß à M Þ P ß à%Þ � � \ � y M � \ N<ß àHÞ � M � � \ � y ß àÞ P ß à%Þ P � ß àHÞ P � \ P ß à æ = M �y,¶ � \ N ¸ ] @ p �V`/E�i{= �y¶ � \ N ¸ ] @ãi��S`dE%èi E �Yé yyÞ P ß à æ = M �y¶ � � \ N ¸ ] @ N p � N `/E�i{= �y+¶ � � \ N ¸ ] @ N i�� N `dE%èXê � @ N � N @ P � P � @+�� E K�t �Finally, from (2.3) a straightforward calculation leads to

(2.8)� @ N � N @ P � P � Z�_�� @ N @ P � @+�»� E K À ] @ s �V`dE ] @ b � p7� `/E � @ N � N @ P � P � @+�� E �

3. Nikiforov-Uvarov approach. A polynomial sequencenIë�ì1� J %u�ì © [ orthogonal with

respect to a positive measure í on the real line is such thatë�ì

has degreew

and satisfies theconditions î.ï ë ì � J J ¨�ð í � J �K~t!�lñÌK~t��H�� H�� w i �+� ò~ó7ô �This defines the polynomial up to a multiplicative factor. In the case of discrete orthogonalpolynomials, we have a discrete measure í (with finite moments)í Köõ¯¨ ° [�÷ ¨ ¼HøIùÄ� ÷ ¨ ¬ t!� J ¨rú ô

and û ú nÄ��&�!� ��H� u�ü�nqpTý7u+�which is a linear combination of Dirac measures on the û pþ�

points J [ � �H�� J õ . Theorthogonality conditions of a discrete orthogonal polynomial

ë�ìon the lattice

n J � C �ÿR ô \��

ETNAKent State University [email protected](�),�� AND (*)��#"%$�"'� ASSOCIATED WITH � -HAHN POLYNOMIALS 29C K~t!�H�+� ��H� � û u

are usually written asõ¯ Q ° [ ëaì1� J � C � J ¨ � C ÷ � C �K�t!� ñzK7t!�� H�H� � w i � �There exist several mathematical methods and approaches to the study of classical poly-

nomials orthogonal with respect to a discrete measure, usually known as discrete orthogonalpolynomials [29]. Nevertheless, for physicists perhaps the more easy method to get in touchwith these kind of mathematical objects (discrete orthogonal polynomials) is via the dis-cretization of the first and second derivatives � ¹ � J and � ¹ ¹ � J involved in the hypergeometricdifferential equation [29] �� J �v¹ ¹ � J Yp �� J Yp�� J �K~t��where ��

, �� K �and

� K�� w C�� . This method is usually known as Nikiforov-Uvarov approach.

Thus, the corresponding hypergeometric-type difference equation is [29]

(3.1)

� � C �� J � C�i �y � � � C � J � C p � � C �� C � J � C p�� C �K~t�� C �K �� J � C � i �y �� J � C � � J � C�i �y %� � � C �K �� J � C '%�where � � � C TK � � C i�� C�i ��

and � � � C �K � � C pç�� i�� C denote the backward andforward 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 formJ � C �K�� N%= Q p�� P = M Q p�� å �U� = ú ô \�� n+�,u� or J � C �K�� C P p�� C p��q�where

� N , � P , � å , � ,�

and�

are constants. In fact, the lattice J � C «K E�2 M1NE M1N belongs to thisclass.

The polynomial solutions of (3.1) can be orthogonalized constructing a Sturm-Liouvilleproblem. First, the equation (3.1) is written in the self-adjoint form

(3.2)�� J � C�i �y � � � C ÷ � C � � � C � J � C �� p�� ÷ � C � � C �K~t��

for two different polynomial solution of degreew

and � , respectively. The function ÷ � C (the so-called symmetrization factor of (3.1)) is the solution of the Pearson-type differenceequation [29]

(3.3)�� J � C�i �y ] � � C ÷ � C ` K � � C ÷ � C where ÷ � C � J � C¾i �y ¬ t�� 7CT� � i � �

Second, the equation (3.2) for polynomial solution of degreew

is multiplied by the otherpolynomial solution of degree � of the same equation and vice versa, and then one subtractsthe resulting equations (one from the other). Finally, we sum over

� � CÌ� � i �for which

we obtain the orthogonality property (see [29, pages 70-72])

(3.4) � M1N¯Q ° � ë ì � J � C ��ë � � J � C ' ÷ � C � J � CWi �y �K�¼ ì ½ � �� ë ì �� P �


30 JORGEARVESU

under additional (boundary) conditions� � C ÷ � C J ¨ � C�i �y "!! Q ° � ½ � K7t��lñzK7t!�� H�H�Since we are interested in polynomial solutions the same procedure could be carried out

for theñ

-order finite difference of a solution of (3.1), defined by� ¨ � J � C '�K �� J ¨ MFN � C �� J ¨ M P � C $#�#�# �� J � C � � J � C '�% � ¶ ¨ ¸ � � C %�where J ¨ � C �K J � C p�& : , because this

ñ-th finite difference verifies a difference equation of

the same type (hypergeometric type) [29]

(3.5) � � C �� J ¨ � C�i NP � � � ¨ � C E� J ¨ � C '� p � ¨ � C � � ¨ � C E� J ¨ � C p í ¨ � ¨ � C E K~t��being� ¨ � C �K � � C p ñ! i � � C Yp � � C p ñ! � J � C p ñ i NP � J ¨ MFN � C and í ¨ K�� ì p ¨ MFN¯� ° [ � � � � C � J5� � C �

For (3.5) the symmetrization factor is

(3.6) ÷ ¨ � C �K ÷ � C p ñ! ¨§( ° N � � 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 coef-ficients for the quantum algebras and the = -Hahn polynomials. Now we will write some ofthem (see [29] for more details).

It is well known that the polynomial solutionsë�ìF� J � C � of (3.1) are determined up to

a normalizing factor * ì, by means of the discrete analog of the Rodrigues formula [29, Eq.

3.2.19, page 66]ëaìF� J � C ��K * ì÷ � C � ¶ ì ¸[ ] ÷ ì1� C ` � � ¶ ì ¸¨ % �� J ¨ \ N � C �� J ¨ \ P � C +##�# �� J ì � C �(3.7)

where ÷ ì � C is given in (3.6). Notice that these polynomial solutions correspond to certaineigenvalues

� ì(see (3.1)) which can be computed by simple substitution of

ë ì � J � C � intothe equation (3.1) and comparing the coefficients for the powers of J � C , i.e.,

(3.8)��ìzK i ] w ` E-, ��/. = ì M1N p = M ì \ N�0 �� ¹ p ] w i � ` E �� ¹ ¹� ê �

where (see (3.1)) �� C �K213 ¿ ¿P J � C P p �� ¹ �#tÄ J � C Yp �� #tÄ and �� C �K � ¹ J � C Yp � �Gt+ .Similarly, for the

ñ-th finite difference the following Rodrigues-type formula is also valid

(3.9) � ¨ ½ ì � J � C ��K � ¶ ¨ ¸ ë ì � J � C '�K54 ì ½ ¨ * ì÷ ¨ � C � ¶ ì ¸¨ ] ÷ ì � C ` where * ì K � ¶ ì ¸ ëaì4 ì ½ ì �and

(3.10) 4 ì ½ ¨ K ] w ` Eq�] w i ñ `dE � ¨ M1N§� ° [76 =Y�y¶ ì \ �¾M1N ¸ p = M �y¶ ì \ �¾M1N ¸� �� ¹ p ] w p � i � ` E �� ¹ ¹�/8 �


From (3.7) follows an explicit expression for the polynomialsë ì

[2]ë ì � J � C � K * ì = M ì Q \ x � ¶ ì \ N ¸� ì N � =�i �� ì ì¯� ° [ ] w `/E � = M é ya¶ ì MFN ¸ � i �� \ ì] �S` Eq� ] w i��S` Eq�(3.11) 9 ì M5�¾M1N§Å ° [ ] � � CWi;: ` �¾M1N§Å ° [ ] � � C p : Fp � � C p : � J � C p :Xi �y ` �with the assumption < M1NÅ ° [ ¤^� : =% �

. Notice that this expression depends only on the coeffi-cients of � and � given in (3.1).

4. = -Hahn polynomials in the non-uniform lattice J � C 9K E 2 M1NE MFN and = -Clebsch-Gordan coefficients. It is well known [29] the relation between Clebsch-Gordan coefficientand Hahn polynomials in the uniform lattice J � C �K C . Analogously, in [33] is used a rela-tion (see (4.24) below) for the = -case in the non-uniform lattice J � C K =P6Q . In fact, we willdeduce a similar relation but in the lattice J � C �K E 2 M1NE MFN . In such a way an useful and fruitfulparallelism between = -Clebsch-Gordan coefficients for C�DYE �G�+ and = -Hahn polynomials willbe constructed. Therefore, studying in details the relations and properties satisfied by the= -Hahn polynomials we have the corresponding partner among the relations and properties ofC�DXE �� Clebsch-Gordan coefficients and vice versa. For such a purpose let first to determinethe = -Hahn polynomials as well as their main characteristics based on the aforementionedNikiforov-Uvarov approach. Second, selecting a special choice of C , û , > , ? , and

wwe will

establish the connection between both mathematical objects.In [30] has been proved that the most general orthogonal polynomial solution of (3.1) in

the lattice J � C �K�� NH=,Q p�� å corresponds to the choice

(4.1)

� � C �KA@4 � =,Q MCB � i ��H� =,Q MDB y i �I%�� C Yp � � C � J � CWi �y �K @4 � =,Q MFEB � i ��H� =,Q MGEB y i �� From (3.8) considering (4.1) the following expression for the eigenvalues of (3.1)

(4.2)� ì K i @4� P N = M �y¶ B � \ B y \ EB � \ EB y ¸ ] w `dE ] H N p H P i @H N i @H P p w i � `/E �

yields. The corresponding weight function is, in this case,

(4.3) ÷ � C �K � E � C�i @H N � E � C�i @H P � E � CWi H N p~�� E � C�i H P p~�� 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]

(4.4)

ëaìF� J � C �öK * ì , @4� N � = �y i{= M �y ê ì = M x � x � �#�� M ì B � � = B � MFEB � ¥&= *ìF� = B � MGEB y ¥&= *ì9 åJI P , = M ì � = B � \ B y MFEB � MFEB y \ ì M1N � = B � M Q= B � MGEB � � = B � MFEB y ¥�= � = Q MCB y \ N ê �(4.5)

ë ì � J � C 'öK * ì , i @4� N � = �y i�= M �y ê ì = M+K x � x � �#�� M ì ¶ B � \ B y MFEB � ¸ � = B � MFEB � ¥'= ì9 � = B y MFEB � ¥'= ì å I P , = M ì � = B � \ B y MFEB � MFEB y \ ì M1N � =,Q MFEB �= B � MGEB � � = B y MFEB � ¥�= � =Iê �


32 JORGEARVESU

whereL I=M-NPO 4RQ O : Q ÜáÜáÜ Q O LS 4 Q S : Q ÜáÜáÜ Q S MUT Æ QWVJX ÚUYZ&\[^]`_ O 4 T Æba &FcRc\c _ O L T ÆJa &_ S 4 T ÆJa &Gc\cRc _ S M T ÆJa & V &_ Æ T Æba &ed _ Ê Í a & Æ ù yÄ× & 3�4GÙgf M 3 L Ø 4 Qand �h� ¥'= ¨ K ¨ MFN§� ° [ �*� i � = � �

Let the end points of the orthogonality interval�9K t

and�LK û . Moreover,

� N Ki � å K NE M1N , i.e., J � C �K E 26MFNE M1N , and

(4.6) 4 N K i =jilkWmy=�i � � * ì K � i �� M ì] w ` EI� � H N K�t!� H P K û p > � @H N K i=?�i �� @H P K û i ��with the restrictions > � ?~¬þi �

andw ª 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 � C�i �y �K i�= Q \ mkony ] C p ? p~� `dE ] C�i{û p7� `/EI�Consequently, from (3.8) or (4.2) the eigenvalues of (3.1) are given by the expression��ìzK = n�k y � iy ] w ` E ] w p > p ? p7� ` E �

The solution ÷ � C of the Pearson-type equation (3.3) is

÷ � C �K = ¶ mkony ¸#Q �� E � C p ? p~�� E � û p >�i{C �� E � C p~�� E � û i{C �(4.7)

Notice that the same result can be found if we use (4.3) and the relation �� E � C �K = M�� 2*� �#� � 2*� y �� E � C .Considering (4.5) the hypergeometric representation for the = -Hahn polynomials

(4.8)pCq ½ rì � C � û ¥'= �K � = r \ N ¥'= ì � = õ \ q \ r \ N ¥'= ì= x y!¶ ì \ q \ P r \ õ ¸ � =.¥'= ì åbI P , = M ì � = ì \ q \ r \ N � =,Q \ r \ N= r \ N � = õ \ q \ r \ N ¥!= � = ê �yields. From here follows other useful characteristic of the = -Hahn polynomials, i.e., thevalue of this polynomials at the end points of the orthogonality interval

] t�� û i � ` . Indeed,pDq rì �Gt!� û ¥'= Ko� i �� ì ] û i � ` E,� �� E ] ? p w p7� `] w ` Eq�� E ] ? p~� ` ] û i w i � ` EI� = �y ì ¶·P q \ r \ õ \ �y,¶ ì M1N ¸d¸ �p q rì � û i �� û ¥'= K ] û i � `/E � �� E ] > p w p~� `] w `dE � ] û i w i � `dE � �� E ] > p�� ` = �y ì ¶ q \ õ M �y¶ ì M1N ¸d¸ �By �� ë ì �� P we will denote the square norm of

ë ì. In [29, Chapter 3, Section 3.7.2, page 104]

the authors show a convenient way to compute it. In fact,

(4.9) �� ë ì �·� P KÝ� i �� ì 4 ì ½ ì * Pì � M ì MFN¯Q ° � ÷ ì � C � J ì � C�i �y �


TABLE 4.1Main data for the � -Hahn polynomials in the lattice -.�8(&��0 ��26354�6354 .ë ì � J � C ' p q ½ rì � C � ûj¥&=

Interval] t�� û i � `

÷ � C = ¶ mJk�ny ¸#Q �� E � C p ? p7�I �� E � û p >�i{C �� E � C p7�I �� E � û ijC � � C i�= M ilksmy J � C P p = M �y ] û p >Y`/EHJ � C � � C i�= n�k y � iy ] > p ? p � `dE6J � C ap = q \ r \ N ] ? p7� `/E ] û i � `/E� ìF� C i�= n�k y � iy J ìF� C i{= n�k y � iy ] � w p > p ? p � ` E��ì = n�k y � iy ] w ` E ] w p > p ? p~� ` E* ì � i �� M ì] w ` Eq�� ëÐì �� P = õ ¶ mJkony \ õ ¸ M m � m � K � k y� �� E � w p > p7�I �� E � w p ? p~�� E � w p > p ? p û p7�I= ì � m � ny \ õ \ N ] w `dE � ] û i w i � `dE � �� E � w p > p ? p~�� E �G� w p > p ? p �÷ ì1� C = Q6¶ mkony \ ì MFN ¸ \ x yv¶ q \ r \ ì M1N ¸ \ N �� E � C p w p ? p7�I �� E � û p >LijC �� E � C p7�I �� E � û ijC i w

Based on (4.9) after some cumbersome calculation we arrive to the expression,

(4.10) tt p q ½ rì � C � ûj¥'= tt P K = õ � mJk�ny \ õ M �y �� E � > p w p~��= ì � m � ny \ õ \ N \ m � m � K �� ] w `dE � ] û i w i � `dE �9 �� E � ? p w p~�� E � > p ? p û p w p~�� E � > p ? p w p~�� E � > p ? p � w p � �Finally, after the calculations of the very basic characteristics of the = -Hahn polynomials

like ÷ � C , � � C , � � C , ��ì and tt p q ½ rì � C � û ¥'= tt P we summarize in Table 4.1 the results of theremaining calculations by using the Nikiforov-Uvarov approach.

4.1. Three-term recurrence relation. A simple consequence of (3.4) is [29]

(4.11) J � C �ë ì � J � C ��K > ì ë ì \ N � J � C 'Yp ? ì ë ì � J � C �ap;u ì ë ì M1N � J � C �%� wwv t!�with

ë M1N � J � C ��K~tand

ë [ � J � C ��Ko�.

Denoting by� ì

and� ì

the coefficients in the expansionë ì � J � C �FKx� ì J ì � C 'p-� ì J ì MFN � C &p#�## we have the following explicit expressions for the recurrence coefficients [2, 29]

(4.12) > ì K � ì�.ì \ N � ? ì K � ì�Äì i � ì \ N�Äì \ N � u ì K � ì M1N�Äì �� ë ì �� P�·� ëaì M1N+�·� P �


34 JORGEARVESU

TABLE 4.2Three-term recurrence coefficients and leading coefficient for � -Hahn polynomials in the lattice -.�/('��0 ��2 3�4�6354

Coefficient Explicit expression> ì = M n�k y � iy ] w p~� `dE ] w p > p ? p7� `/E] � w p > p ? p � ` E ] � w p > p ? p~� ` E? ì = q \ õ \ x y M1N ] w p > p ? p7� `/E ] w p ? p~� `dE ] û i w i � `/E] � w p > p ? p � ` E ] � w p > p ? p~� ` Ep = M �y"y'P õ \ r \ ì \Få ¶ q \ N ¸{z ] w p >Y`/E ] w p > p ? p û»`dE ] û i w `dE ] w `/E] � w p > p ? p~� ` E ] � w p > p ?F` PE ] � w p > p ?�i � ` E ] û i w i � ` Eu.ì = M ilkWmy M P ] w p >Y` E ] w p ?1` E ] w p > p ? p û»` E ] û i w ` E] � w p > p ? p~� `dE ] � w p > p ?1` PE ] � w p > p ?»i � `/E� ì = ì n�k y � iy �� E �G� w p > p ? p~��] w ` Ev�� E � w p > p ? p~��To find the leading coefficient

� ìit is enough to note that� J ì � C � J � C K ] w `/E J ì MFN � C p �y Yp #�## � so , �� J � C ê ì J ì � C �K ] w `dE � �

On the other hand,æ}|| ø ¶�Q³¸ è ì ëÐì1� J � C '�K ] w ` Eq� �.ì , then using (3.9),

] w ` Eq� �.ìOK * ì 4 ì ½ ì , andtaking into account (3.10)�.ìÌK * ì ì M1N§¨ ° [ 6 = �y¶ ì \ ¨ M1N ¸ p = M �y¶ ì \ ¨ MFN ¸� �� ¹ p ] w p ñ i � ` E �� ¹ ¹�~8 �

Frequently, the computation of� ì

as well as ? ì is very tedious, then if we know > ì andu ì, and

ë ì �h�v��K�tfor all

w, we can use (4.11). Indeed,? ìÌK J �{�v*ëaìF�h�v i;> ìvëaì \ N �h�v i uvìvëÐì MFN �{�vë ì �{�v �

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� J � C '� J � C K ��ì] w `dE � ¹ì � � 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 relations for the corresponding polynomial solutions of (3.1)in the non-uniform lattice J � C SK�� N%=,Q p�� å . Thus, the reader could follow easily all thecalculations. For such a purpose we will substitute � ì1� C as follows� ìY� C �K � ¹ì J ìF� C Ðp � ì1�Gt+�K � ¹ì = x y J � C Ðp � ì1�#tÄ i � ¹ì � å � = x y i �I%�


in (4.13). Then, based on the three-term recurrence relation (4.11) we obtain the first structurerelation

(4.14) � � C � ëaì1� J � C �� J � C K ��1ìvëaì \ N � J � C �ap �� ìvëaì1� J � C �Ðp �� ìvëÐì MFN � J � C '%�where

(4.15)�� ì K ��ì] w ` E � = x y > ì i * ì� ¹ì * ì \ N � � �� ì K ��ì] w ` E � = x y ? ì p � ì1�Gt+� ¹ì i � å � = x y i �I � �� ìOK � ì = x y u ì] w ` E �

To deduce the second structure relation we will transform (4.14) with the help of theidentity

(4.16) � � ë ì � J � C '� J � C K � ë ì � J � C �� J � C i � ë ì � J � C �� J � C �Thus, using � J � C i �y �K | E J � C i � å | E , as well as (3.1) and (4.11), one gets] � � C Yp � � C � J � CWi �y ` � ëaì1� J � C �� J � C K � ì ë ì \ N � J � C �Ðp � ì ë ì � J � C 'Ðp � ì ë ì MFN � J � C '%�where�FìzK ��1ì ix> ì��ì | E � � ìzK �� ì iw? ì��ì | E p�� å ��ì | E � � ìOK �� ì i uvì��ì | E �or equivalently, using (4.15),� ì K ��ì] w ` E � = M x y > ì i * ì� ¹ì * ì \ N � � � ì K ��ì] w ` E � = M x y ? ì p � ì1�#tÄ� ¹ì i � å � = M x y i �� ìÌK � ì = M x y u ì] w ` E �

Now we will find a difference-recurrence relation of the form [2]

(4.17)ëaì1� J � C ��K��ì � ë ì \ N � J � C �� J � C p��jì � ë ì � J � C �� J � C p û ì � ë ì MFN � J � C �� J � C �

being��ì

,� ì

y û ìconstants. Notice that this relation is not include in [29].

Let describe briefly how to prove (4.17). Applying the operator|| ø ¶�Q³¸ on both sides of

(4.14), and using (3.1) as well as = M �y � J � C �K � J � C�i �y one obtains

(4.18)

� = �y � � � C � J � C i � � C � � ëaì1� J � C �� J � C i � ì ë ì � J � C '�KK = �y ��1ì � ë ì \ N � J � C �� J � C p = �y �� ì � ë ì � J � C �� J � C p = �y �� ì � ë ì M1N � J � C �� J � C �Since � � C �K � ¹ ¹� J P � C Ðp � ¹ �#tÄ J � C Yp � �#t+ � and � � C �K � ¹ J � C Ðp � �#tÄ%�


36 JORGEARVESU

and J � C pj�I�K =YJ � C i � å � =�i �Ithe expression

| 3 ¶�Q³¸| ø ¶8Q³¸ is a polynomials J � C of degree one.Hence, � = �y � � � C � J � C i � � C � K 4 J � C Yp * �where 4 K � ¹ ¹� ��p = = �y i � ¹ � and * K = �y � ¹ �#tÄ i � ¹ ¹� � å�= �y � =Wi �� i � �#tÄ �Thus, (4.18) becomes

(4.19)= M �y , 4 J � C � ëaìF� J � C �� J � C i ��ìvëÐìY� J � C � ê K�� ì � ë ì \ N � J � C '� J � C p �� ì � ë ì � J � C �� J � C p �� ì � ë ì M1N � J � C '� J � C i *= �y � ë ì � J � C �� J � C �

Now, if we apply the operator|| ø ¶�Q³¸ on both sides of (4.11) we can eliminate the termJ � C |�� x ¶ ø ¶8Q³¸d¸| ø ¶�Q*¸ in (4.19) by using the identity J � C p~��K =aJ � C i � å � =�i �I

, i.e.,=IJ � C s� ëaì1� J � C �� J � C K > ì�� ëÐì \ N � J � C �� J � C p ] ? ìUp�� å � =�i �I ` � ëaì1� J � C �� J � C p�u.ì`� ëaì M1N � J � C �� J � C i ëaì1� J � C � �Finally, multiplying (4.19) by = and using the above equation one getsëaì1� J � C � K æ > ì 4 i{= Ky �� ì è� 4 p = ��ì� � ë ì \ N � J � C '� J � C p æ�u ì 4 i�= K y�� ì è� 4 p = ��ì� � ë ì M1N � J � C �� J � C p æ ? ì 4 p�� å 4 � =�i ��Fp ="*oi�= K y¾�� ì è� 4 p = ��ì� � ë ì � J � C �� J � C �which is of the form (4.17) if 4 p = � ì �K�t

. In particular, we have4 K = n � ilk �y ] > p ?1` E � * K ] û p >Y` E i{= M mJksi � �y i{= q \ r \ N ] ? p~� ` E ] û i � ` E �consequently, 4 p = ��ìOK = n � ilk �y � ] > p ?F` E p ] w ` E ] w p > p ? 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 hypergeome-tric-type difference equation (3.1) for the = -Hahn polynomials is equivalent to the recurrencerelation (2.6) for the = -Clebsch-Gordan coefficients with respect to the projection ��N or � Pof the angular momentum @N or @ P , respectively. For such a purpose let to rewrite (3.1) as arecurrence relation in C , i.e.,

(4.20) � � C � � C p~��Fp��1� C � � C ap�� C � � CWi ��K7t��


TABLE 4.3Recurrence coefficients for � -Hahn polynomials in the lattice -.�8(&��0 ��2%3�4�6354

Coefficient Explicit expression1� x M E � � x ksmJk�n�k �G�y Þ ì \ N³ß à Þ ì ß à%Þ ì \ q \ r \ N<ß àÞ P ì \ q \ r \ P ß à Þ P ì \ q \ r \ N<ß àE n�k y � iy Þ ì \ q \ r \ N³ß à� Þ ì \ q \ r \ N³ß à%Þ ì \ r \ N³ß à Þ õ M ì M P ß àE � � x � m � i Þ P ì \ q \ r \ P ßáà Þ P ì \ q \ r \ N³ßâà1� x \ E � �yb� y ilkon�k K³� mJk �G�� Þ ì \ q ßâà Þ ì \ q \ r \ õ ßáà Þ õ M ì ßâà Þ ì ßáàÞ P ì \ q \ r \ N³ßáà Þ P ì \ q \ r,ß yà Þ P ì \ q \ r MTN³ßâà Þ õ M ì MTN³ßáà \ãE x � yy Þ x y ß à\YE i � x � n � Ky � E x k � k mkony Þ x y \ r \ N³ß à%Þ x y M õ \ N<ß à M Þ x y ß à Þ x y \ õ \ q ß à��Þ P ì \ q \ r \ P ß à �1� x E � x kon � m � y i � y �y Þ ì \ q ß à Þ ì \ r,ß à6Þ ì \ q \ r \ N³ß à%Þ ì \ q \ r \ õ ß à6Þ õ M ì ß àÞ P ì \ q \ r \ N<ßâà Þ P ì \ q \ r,ß yà Þ P ì \ q \ rWMTN³ßáà� x M E x kWmkon�k �y Þ ì \ N³ß à Þ ì ß à%Þ ì \ q \ r \ N<ß àÞ P ì \ q \ r \ P ßâà Þ P ì \ q \ r \ N<ßâàE n�k y � iy Þ ì \ q \ r \ N<ß à � Þ ì \ q \ r \ N³ß à%Þ ì \ r \ N³ß à Þ õ M ì M P ß àE � � m � i Þ P ì \ q \ r \ P ß à%Þ P ì \ q \ r \ N³ß à� x \ E6� �y"� y ilk�n�k y x k K³� mJk �#�� Þ ì \ q ß à Þ ì \ q \ r \ õ ß à%Þ õ M ì ß à Þ ì ß àÞ P ì \ q \ r \ N³ß à%Þ P ì \ q \ r,ß yà Þ P ì \ q \ r�MUN<ß à Þ õ M ì MTN³ß à M E � x k y�\YE i � x � n � Ky � E x k � k mkony Þ x y \ r \ N³ßâà Þ x y M õ \ N<ßâàFM Þ x y ßâà Þ x y \ õ \ q ßâà �Þ P ì \ q \ r \ P ßâà �� x E n � m � xy � i � � Þ ì \ q ßâà Þ ì \ r,ßâà Þ ì \ q \ r \ N<ßâà Þ ì \ q \ r \ õ ßâà Þ õ M ì ßáàÞ P ì \ q \ r \ N<ß à Þ ì \ q \ r,ß yà Þ P ì \ q \ r�MUN<ß àTABLE 4.4

Recurrence coefficients for � -Hahn polynomials in the lattice -.�8(&��0 ��2%3�4�6354Coefficient Explicit expression� x E i � x � n � yy Þ ì \ N³ß à%Þ ì \ q \ r \ N<ß à Þ ì \ q \ r,ß àÞ P ì \ q \ r \ P ß à%Þ P ì \ q \ r \ N³ß à ¶ Þ q \ rß à \ Þ ì ß à%Þ ì \ q \ r \ N<ß à ¸N¶ Þ q \ r,ßâà \ Þ ì ßâà Þ ì \ q \ r \ N<ßâà ¸s� M Þ ì \ N<ß àÞ P ì \ q \ r \ N<ßâà�� Þ ì \ q \ r \ N<ß àE � � m � ny � m � i � x Þ ì \ r \ N³ß à%Þ õ M ì M P ß àÞ P ì \ q \ r \ P ßâà \ E � i � m � � Þ ì \ q ß à Þ ì \ q \ r \ õ ß à%Þ õ M ì ß à%Þ ì ß àÞ P ì \ q \ r,ß yà Þ P ì \ q \ r MTN³ßâà Þ õ M ì MTN³ßáà�¡¢ x \ Þ õ \ q ß àE n�k y � iy \ NE mJk�n�k Ky M Þ r \ N<ß à Þ õ MUN<ß àE � m � ilk�ny M Þ q \ r \ P ß àE M Þ ì \ q \ r \ N³ßáà Þ x y ßâàE � x �M Þ ì \ q \ r \ N<ß à � E x k � k mJkony Þ x y \ r \ N³ß à Þ x y M õ \ N³ß à M Þ x y ß à%Þ x y \ õ \ q ß à��E x � ilkon�k yy Þ P ì \ q \ r \ P ßâà £õ x M E x � ilk�n � Ky Þ ì \ N<ßâà Þ ì \ q ßáà Þ ì \ r,ßâà Þ ì \ q \ r,ß8� �à Þ õ M ì ßâàÞ P ì \ q \ r \ N³ß à%Þ P ì \ q \ r�MTN³ß à ¶ Þ q \ rß à \ Þ ì ß à6Þ ì \ q \ r \ N³ß à ¸


38 JORGEARVESU

where

(4.21)� � C �K � � C Yp � � C � J � C�i �y � J � C i �y � J � C �¤�!� C �K � � C � J � C i �y � J � C ��1� C �K��ì i �� C ix� � C �

Let

(4.22)C K @qN�i��LN � û K @IN p @ P i�� p~�� > K � p @IN�i»@ P �? K � i�@qN p @ P � w K @Ti��

Therefore, substituting in the above equations (4.21) the choice (4.22) we have� � @IN�i��LN K i�=qP � � \ � y M � � \ �y ] @ P p � P p~� ` E ] � P i�@ P ` E ��1� @IN�i��LN K = � � M � � \ �y � = � y ] @ãi��V` E ] @ p � p7� ` E p ] @qN�i��LN&` E ] @IN p �LN p7� ` Ep = � � \ � y ] � P p @ P p7� ` E ] � P i»@ P ` E 1��!� @ N i�� N K = � � M � � \ �y ] @ N i�� N `dE ] � N p @ N p~� `dEI�Consequently, the equation (4.20) for the = -Hahn polynomials becomes

(4.23)� � @ N i�� N p � \ � � M � y ½ �¾M � � \ � y� M5� � @ N i�� N p~�� @ N p @ P i�� p7� ¥'= p��1� @qN�i��LN p � \ � � M � y ½ �¾M � � \ � y� M5� � @qN�i��LN � @qN p @ P i�� p~� ¥&= p-�� @ N i�� N p � \ � � M � y ½ �¾M � � \ � y� M5� � @ N i�� N i �� @ N p @ P i�� p7� ¥'= �K�t �

Comparing the recurrence relations (2.6) and (4.23) one deduces� @IN%��N�@ P � P � @+�»� E � � KÝ� i �� ì \ Qb¥ ÷ � C � J � CWi �y b¦pDq ½ rì � C � ûj¥&= %�(4.24)

where¦p q ½ rì � C � ûj¥&= denotes the orthonormal = -Hahn polynomials and C , û , > , ? ,

wcoincide

with (4.22). To efficiently carry out a comparison between these two recurrence relations weshould consider (4.10) as well as the following useful expressions÷ � @qN�i��LN ÷ � @ N i�� N p~�� K = M5� ] @IN p �LN&` E ] @qN�i��LN p~� ` E] @ P i�� P `dE ] @ P p � P p~� `dE � � J � C�i �y � J � C i åP K =÷ � @ N i�� N ÷ � @IN�i��LN�i �� K = � ] @ P p � P `dE ] @ P i�� P p~� `dE] @IN�i��LN&` E ] @qN p �LN p~� ` E � � J � C i �y � J � C p �y K = 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 knownrelation between the classical Hahn polynomials and Clebsch-Gordan coefficients. Further-more, this expression is in accordance with the results obtained in [33] (see also [1] for moredetails).

From (3.4) as well as from (4.8) the symmetry property for the = -Hahn polynomialspCq ½ rì � C � û ¥'= �KÝ� i �� ì = ì ¶ q \ r \ õ ¸ p r!½ qì � û ijCWi �+� ûj¥&= M1N �holds. It leads as a consequence the corresponding symmetry property for the = -Clebsch-Gordan coefficients given in (2.7) by a simple substitution of the parameters (4.22). Now, thesubstitution of the parameters (4.22) into the expression (3.11) with the help of (4.24) givesthe = -analog of the Racah formula for C�DFE �� Clebsch-Gordan coefficients� @qN%��N�@ P � P � @�� E K¦� i �I � � M5� �¨§©©ª Þ P � \ N³ß à Þ � MX��ß àR«áÞ � \ ��ß àR«áÞ � � MX� � ß à\«áÞ � y M5� y ß àR«áÞ � � \ � y M � ß à\«Þ � � \ � y \ � \ N³ß àR«áÞ � � M � y \ � ß à¬«áÞ � y M � � \ � ß à¬« Þ � � \ � � ß àR«áÞ � y \ � y ß à\«= M5� � ¶ � \ N ¸ \ �y,¶ � ¶ � \ N ¸ \ � � ¶ � � \ N ¸ M � y ¶ � y \ N ¸d¸9 ¯ ¨ � i �I ¨ = �y ¨ ¶ � \ � \ N ¸ ] @qN�i��LN p ñ ` Eq� ] @ p @ P i��LN�i ñ ` E,�] ñ ` Eq� ] @Ui�� i ñ ` Eq� ] @IN�i��LN�i ñ ` Eq� ] @ P i»@ p ��N�i ñ ` Eq� �(4.25)


Two particular cases are immediately derived from (4.25):i) First, when @ N K � N� @ N @ N @ P � P � @�� E K §©©ª Þ P � \ N<ß àHÞ � \ ��ß à\«áÞ P � � ß à\« Þ � y MX� y ß à\«áÞ � M � � \ � y ß àR«Þ � � \ � y \ � \ N<ß à¬«áÞ � � \ � y M � ß àR«áÞ � � M � y \ � ß àR«áÞ � y \ � y ß à\«áÞ � M5��ß à= � � ¶ � M5� ¸ M �y,¶ � � \ � y M � ¸<¶ � M � � \ � y MFN ¸ �

which corresponds to the choice C K~tfor the = -Hahn polynomials.

ii) Second,� @qN ��N�@ P � P � @�@.� E� i �I � � M5� � K §©©ª Þ P � \ N<ßâà «áÞ � � \ � � ßáà «áÞ � y \ � y ßâà «áÞ � � \ � y M � ßâà «Þ � � \ � y \ � \ N³ßáà «áÞ � � M � y \ � ßâà « Þ � y M � � \ � ßâà «áÞ � � M5� � ßâà «áÞ � y MX� y ßâà «= ¶ � \ N ¸<¶ � � M5� � ¸d¸ M �y¶ � � \ � y M � ¸³¶ � M � � \ � y \ N ¸#¸ �which coincides with the expression (4.24) for � K @� @qN6�LN*@ P � P � @I@.� E¦p � \ � � M � y ½ �¾M � � \ � y[ � @qN�i��LN � @qN p @ P i�� p~� ¥&= K¦� i �� Qb¥ ÷ � C � J � CWi �y �

From the Rodrigues-type formula (3.7) and the identity (4.16) it is easy to obtainp q MFN6½ r!MFNì \ N � C p~�� û ¥'= i p q M1N%½ r!M1Nì \ N � C � ûj¥&= �K = Q \ N \ mkon � xy p q ½ rì \ N � C � ûj¥&= �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,¥ � @ N b � N H� @ N s � N p7�I = é yy � @ N � N s7� @ P � P � @�� Ep ¥ � @ P b � P H� @ P s � P p~�� = M é �y � @ N � N @ P � P s � @+�� E K¥ � @ s � H� @ b � p~��F� @qN%�LN*@ P � P � @+� p~� � Ewhich can be also obtained from (2.8).

4.4. Relation between the Clebsch-Gordan coefficients for the quantum algebrasC�D E �� and C�D E ��H�I . It is well known [21] that the quantum algebra C�D E �*�+�H�I can be gen-erated from the operators]® [ � _a` Kcb _ � ]� \ � M ` K i ] � [ `/E � e[ K [ � e_ K f �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 with the Section 2 we use the notation ‘prime’ overthe basic vectors � @ ¹ � ¹ � E , � ¹ K @ ¹ po�+� @ ¹ p �v� ��H� These basic vectors of the irreduciblerepresentations of the discrete positive series � � ¿ \ are determined by � @ ¹ @ ¹ p~� � E , such that M � @Ä¹ @+¹ p7� � E K~t�� [�� @Ä¹ @+¹ p7� � E K¦� @Ä¹ p~�� @+¹ @Ä¹ p~� � E �Ð� @Ä¹ @+¹ p7� � @Ä¹ @+¹ p7� � E K¦��and � @ ¹ � ¹ � E KÝ� ] � @ ¹ p7� ` Eq�] @ ¹ p � ¹ ` EI� ] � ¹ i�@ ¹ i � ` EI� � \ � ¿ M � ¿ M1N � @ ¹ @ ¹ p7� � E �


40 JORGEARVESU

Analogously, the tensor product of the irreducible representation � ¿� \ and � ¿y \ of thequantum algebra C�DFE �*�+�H�I can be decomposed into the direct sum � ¿� \ ® � ¿y \ K ¯¯� ¿ °5� ¿� \ � ¿y \ N ² � � ¿ \ �Its generators (co-products) are given by the expressions [ �*�+�&��K [ �*�IYp [ �� _ �*�+�&�+^K = �yW° µ ¶·P&¸ _ �*��Fp = M �yW° µ ¶ N ¸ _ �G�+ �

The explicit form of the irreducible representation± @Ä¹ �� ¹ � [ � @Ä¹â�»¹h² E K�¼ 1��¿d½ ��¿ �± @Ä¹ �� ¹ � _ � @Ä¹��»¹ ² E K À ] � ¹ s @ ¹ ` E ] � ¹ b @ ¹ b7� ` E ¼ 1� ¿ ½ � ¿ _ N �as well as the Casimir operator

(4.26) Á P K i \ M p ]� [ ` E ]® [ i � ` E �Á P � @ ¹ � ¹ � E K ] @ ¹ ` E ] @ ¹ p~� ` E � @ ¹ � ¹ � E �are given in a quite similar form to those presented in Section 2. Analogously, the definitionof the = -Clebsch-Gordan coefficients is� @Ä¹N @Ä¹P � @Ä¹·�º¹�� E K ¯� ¿ � ½ � ¿y � @Ä¹N �»¹N @Ä¹P �º¹P � @Ä¹·�º¹8� E � @Ä¹N �»¹N � E � @Ä¹P �º¹P � E �

Á P ��+ � @ ¹N @ ¹P � @ ¹ � ¹ � E K ] @ ¹ `/E ] @ ¹ p~� `dE�� @ ¹N @ ¹P � @ ¹ � ¹ � E �(4.27)

Now, using the Casimir operator (4.26) as well as the expression (4.27) the matrix ele-ments

� @ ¹N � ¹ N @ ¹P � ¹P � Á P ��&�+ � @ ¹N @ ¹P � @ ¹ � ¹ � E can be found. From these matrix elements (see [16])the following three-term recurrence relation in � ¹ N � � ¹P for the = -Clebsch-Gordan coefficients

(4.28)

Ç È É´³ : Ê«Ë ³: ÊLÍ'Î � È Ë ³: Ï Éµ³ : Î � È É´³ 4 ÊÌË ³4 Î � È Ë ³4 Ï É´³ 4 ÏjÍ'Î �ÐÑ Ë ³4 É ³ 4 ÏjÍ³Ë ³: É ³ : Ê�ÍIÒ Ë ³ É ³ Ó �Ï Æ Ç È Éµ³ : ÊzË ³: Î � È Ë ³: Ï Éµ³ : Ï{Í&Î � È Ë ³ 4 Ï Éµ³ 4 Î � È Éµ³ 4 Ê«Ë ³ 4 ÊLÍ&Î �aÑ Ë ³4 É ³ 4 Ê�Í*Ë ³: É ³ : Ï{ÍqÒ Ë ³ É ³ Ó �Ï�¶ Æ 3vÕ ¿ � È Ë ³: Ï É ³ : Ï{Í&Î � È É ³ : Ê«Ë ³: Î � Ï Æ Õ ¿y È Ë ³ 4 Ï É ³ 4 ÏjÍ'Î � È É ³ 4 ÊÌË ³4 Î �Ï È Ë ³ Ï �y Î :� Ê È É ³ Ï �y Î :� Ö Æ 3 �y× Õ ¿y 3vÕ ¿ � 354GÙ Ñ Ë ³4 É ³ 4 Ë ³: É ³ : Ò Ë ³ É ³ Ó � ÚjÛ Qyields.

Finally, to conclude this section let establish a useful relation between the = -Clebsch-Gordan coefficients for the C�D E �G�+ and C�D E �*�+�H�I algebras, i.e.,

(4.29)� @qN6�LN'@ P � P � @+�� Q¸· à ¶·P&¸ KÝ� @ ¹N � ¹ N @ ¹P � ¹P � @ ¹ � ¹ � Q¸· à ¶ N6½ N ¸ �

where @ N K � ¿ \ � ¿� M � ¿y MFNP � � N K � ¿ � M5� ¿y \ � ¿� \ � ¿y \ NP � @ K @ ¹@ P K � ¿ M � ¿� \ � ¿y MFNP � � P K � ¿y M5� ¿ � \ � ¿� \ � ¿y \ NP � � K @ ¹N p @ ¹P p7� �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].


TABLE 5.1Three-term recurrence coefficients for � -Hahn polynomials in the lattice -.�8(&��0º� ; [1]

Coefficient Explicit expression> ì | E = M �y¶ q \ r \ N ¸ ] w p~� ` E ] > p ? p w p7� ` E] > p ? p � w p � ` E ] > p ? p � w p7� ` E= m � n�ksi � �y] > p ? p � w `/E ] > p ? p � w p � `/EF¹ � ] û i w ` E ] w ` E ] > p ? p � w p � `? ì i ] û i w i � ` E ] w p7� ` E ] > p ? p � w ` E p = n � my , ] > p ? p û p w p7� ` E ] w p7� ` E] > p ? p � w ` M1NE i ] > p ? p û p w ` E ] w ` E] > p ? p � w p � ` M1NE êPºu.ì | E = q \ õ M �y ] > p w ` E ] ? p w ` E ] > p ? p û p w ` E ] û i w ` E] > p ? p � w `/E ] > p ? p � w p7� `/E5. Remark on the = -Hahn polynomials for J � C ãK = Q . Here we will present a brief

description on some basic characteristics of these polynomials from the point of view ofthe Nikiforov-Uvarov approach [29], specially, the three-term recurrence relation. The im-portance of this relation in connection with the = -CGC was already discussed in the abovesection. We focus our attention in this relation since when ={R �

the parameters > ì andu.ìclearly 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, thiscomparative analysis reveals that the three-term recurrence relation –for instance– is not a = -analogue relation because it has not the corresponding partner property for the classical Hahnpolynomials, which does not happen in the investigated case J � C �K E 2%M1NE M1N (see Section 4).

The = -Hahn polynomials in the lattice J � C �K =+Q have been studied in [1, 2, 33]. Despitethe fact that we can use the same scheme presented in Section 4 to obtain them, we will omitit. Below we will summarize few results. Let chose J � C ¾K = Q , i.e.,

� N K �and

� å K¦t. In

this case, with the parameters * ìÌK � i �� ì= x y | ì E ] w ` Eq� �H N K�t!� H P K û p > � @H N K i=?ºi �� @H P K û i �+� 4 K i�= �y¶ õ \ q ¸ �we obtain the = -Hahn polynomials [24, 33]. Indeed, from (4.4)-(4.5) we havepCq ½ rì � C � û E K = x y!¶ q \ õ ¸ � = r \ N ¥'= �ì1� = N%M õ ¥'= *ì| ì E � =.¥'= �ì å I P , = M ì � = M Q � = ì \ q \ r \ N= r \ N � = N%M õ ¥&= � =IêK � = r \ N ¥'= �ì1� = õ \ q \ r \ N ¥'= �ì= M x y!¶ õ \ q \ P r \ ì \ N ¸ | ì E � =.¥&= *ì åJI P , = M ì � =,Q \ r \ N � = ì \ q \ r \ N= r \ N � = õ \ q \ r \ N ¥&= � =Iêj�

In addition, for�5ì

we find��ìOK = �y,¶ q \ r \ P'¸ ] w ` E ] w p > p ? p~� ` E �


42 JORGEARVESU

6. Conclusions and remarks. The method presented in this paper connects in a preciseway (see equation (4.24)) the = -Hahn polynomials with = -CGC, i.e., the Nikiforov-Uvarovapproach and the quantum group representation one. The Pearson type approach to = -Hahnpolynomials used here is based on a specific kind of lattices J � C . In fact, this approach is veryeffective 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 differenceequation, structure relation, difference-recurrence relation, and orthogonality relation whichcould not occur in other approaches. For example, it could happen that the difference equationis given on the lattice J � C ãK =Q and the orthogonality relation on J � C rK = M Q . Thus, ourapproach avoids this and unifies the election of the lattice.

As the reader has observed the limit =LR �do not transform directly the polynomials

on J � C VK =,Q into the classical discrete Hahn polynomials since a rescaling factor beforetaking limits is needed. However, one of the contributions contained in the paper is preciselyto avoid this. In such a way we avoid to answer the following questions: Is the rescalingfactor the same for all the aforementioned relations/characterizations? or different rescalingfactor should be introduced depending on the specific relation for which one is looking for theclassical analogue? With the lattice J � C �K E 2 M1NE M1N proposed in the paper one can avoid this;therefore the = -Hahn polynomials on J � C just becomes directly into the classical Hahn poly-nomials. Moreover, there is also another reason. When one computes all the characteristicsof the = -Hahn polynomials on J � C �K =+Q such as the three-term recurrence relation, structurerelations, 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 thehigher (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 newones despite the fact they are included in the Nikiforov-Uvarov approach. Regarding this toavoid confusions let us comment a couple of things. First, any change of variable carried outto transform the weight (measure) which leads to a non-linear change of the support of themeasure never will produce up to a constant depending on the degree the same polynomials.For instance, if the orthogonality condition of a discrete orthogonal polynomial

ë ìare given

on the latticen J ¨ K J ��ñ! ÿR ô \�� ñºKçt��H�+� �H�� û i �u

, being the support of the measurethe closure of

n J ¨ u õ ¨ ° [ . Then, for such a support the orthogonality conditionõ M1N¯ Q ° [ ëÐìF� J � C ' J ¨ � C ÷ � C � J � C�i �y �K7t��lñÌK~t!�� H� � w i ��by replacing C by û i � ie� leads us to a completely different polynomial family. Notice thatthe support is not the set

n�t�� H� � û i �,uto which belongs the variable C .

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 this equation is linear one can always make a change of variable to convert the latticeJ � C ¾K»� N =,Q pP� P into J � C �K =Q , in fact if one writes the solution of the equation in termsof the basic series it looks very similar (see [29], or the paper by Nikiforov and Uvarov [31]).Actually, this representation does not depend on

� P . Nevertheless, the polynomials are quitedifferent since they are polynomials on different lattice. This means that when one writes theexpansions of

ëaì1� C as polynomials in J � C the coefficients are completely different, that isnot evident when one compares the basic hypergeometric function —the reason is that in bothcases 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.


2. A simple limit =«R �without a rescaling factor is computed wherever is needed to

obtain any classical property.

Acknowledgements. The author was supported by the ’Ramon y Cajal’ grant of theMinisterio de Ciencia y Tecnologıa of Spain as well as for the grant BFM2003-06335-C03-02. The author is grateful to Professor R. Alvarez-Nodarse from the Departamento de AnalisisMatematico de la Universidad de Sevilla (Spain) for several helpful and insightful commentson an earlier draft of this contribution.

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