Applied Mathematics and Computation 223 (2013) 520–536
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Linear partial q-difference equations on q-linear lattices and their bivariate q-orthogonal polynomial solutions q I. Area a,⇑, N. Atakishiyev b, E. Godoy c, J. Rodal a a
Departamento de Matemática Aplicada II, E.E. Telecomunicación, Universidade de Vigo, 36310 Vigo, Spain Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, C.P. 62251 Cuernavaca, Morelos, Mexico c Departamento de Matemática Aplicada II, E.E. Industrial, Universidade de Vigo, 36310 Vigo, Spain b
a r t i c l e
i n f o
a b s t r a c t
Keywords: q-Derivative operator q-Integral Partial q-difference equations q-Pearson system Bivariate big q-Jacobi polynomials Bivariate q-orthogonal polynomials Generalized bivariate basic hypergeometric series
Orthogonal polynomial solutions of an admissible potentially self-adjoint linear secondorder partial q-difference equation of the hypergeometric type in two variables on q-linear lattices are analyzed. A q-Pearson’s system for the orthogonality weight function, as well as for the difference derivatives of the solutions are presented, giving rise to a solution of the q-difference equation under study in terms of a Rodrigues-type formula. The monic orthogonal polynomial solutions are treated in detail, giving explicit formulae for the matrices in the corresponding recurrence relations they satisfy. Lewanowicz and Woz´ny [S. Lewanowicz, P. Woz´ny, J. Comput. Appl. Math. 233 (2010) 1554–1561] have recently introduced a (non-monic) bivariate extension of big q-Jacobi polynomials together with a partial q-difference equation of the hypergeometric type that governs them. This equation is analyzed in the last section: we provide two more orthogonal polynomial solutions, namely, a second non-monic solution from the Rodrigues’ representation, and the monic solution both from the recurrence relation that govern them and also explicitly given in terms of generalized bivariate basic hypergeometric series. Limit relations as q " 1 for the partial q-difference equation and for the all three q-orthogonal polynomial solutions are also presented. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction As recalled in [1], Hahn [20] considered the operator
Lq;x f ðxÞ ¼ ðDq;x f ÞðxÞ ¼
f ðqx þ xÞ f ðxÞ ; ðq 1Þx þ x
x2Rn
x
1q
;
ð1Þ
for all q 2 R n f1; 0g, x 2 R and ðq; xÞ – ð1; 0Þ, which for q ¼ 1 becomes the finite difference operator Dx , and if q – 1 and x ¼ 0 then Lq;0 is the q-derivative operator [[20], Eq. (2.3)]
Lq;0 f ðxÞ ¼ ðDq f ÞðxÞ ¼
f ðqxÞ f ðxÞ ; ðq 1Þx
x – 0; q – 1;
and ðDq f Þð0Þ :¼ f 0 ð0Þ by continuity, provided f 0 ð0Þ exists. Note that limq"1 ðDq f ÞðxÞ ¼ f 0 ðxÞ if f is differentiable. q The referee deserves special thanks for helpful and constructive comments. This work has been partially supported by the Ministerio de Economía y Competitividad of Spain under grants MTM2009-14668-C02-01 and MTM2012-38794-C02-01, co-financed by the European Community fund FEDER. The participation of NA in this work has been supported by the DGAPA-UNAM IN105008-3 and SEP-CONACYT 79899 projects ‘‘Óptica Matemática’’. ⇑ Corresponding author. E-mail addresses:
[email protected] (I. Area),
[email protected] (N. Atakishiyev),
[email protected] (E. Godoy),
[email protected] (J. Rodal).
0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.08.018
I. Area et al. / Applied Mathematics and Computation 223 (2013) 520–536
521
Hahn seems to have been the first to realize that the characterizations of classical orthogonal polynomial sequences (OPS) based on derivatives and differential equations are too restrictive [8]. He posed and solved the following problems: find all OPS such that one of the following holds. 1. fLq;0 Pn ðxÞg is also OPS. 2. fP n ðxÞg satisfy the functional equation
rðxÞL2q;0 Pn ðxÞ þ sðxÞLq;0 Pn ðxÞ þ kn Pn ðxÞ ¼ 0: 3. P n ðxÞ has the representation
Pn ðxÞ ¼
1
.ðxÞ
Lnq;0 ff0 ðxÞf1 ðxÞ fn1 ðxÞ.ðxÞg;
where fk ðxÞ ¼ fkþ1 ðqxÞ. P 4. If Pn ðxÞ ¼ ank xk then ank =an;k1 is a rational function of qn and qk . 5. The moments associated with fP n ðxÞg satisfy n
Mn ¼
a þ bq n M n1 ; c þ dq
a d b c – 0;
ð2Þ
where M n are either the power moments (moments against xk ) or the generalized moments (moments against the q-shifted Qk1 factorial ðx; qÞk ¼ j¼0 ð1 xqj ). Hahn’s investigation led him to the most general set of polynomials belonging to this class [8], now called q-Hahn polynomials:
Q n ðx; a; b; N; qÞ ¼ 3 /2
! nþ1 qn ; abq ; x q; q ; aq; qN
n ¼ 0; 1; . . . ; N:
ð3Þ
Following the works of Nikiforov and Uvarov [40,39,41], a review of the hypergeometric-type difference equation for a function yðxðsÞÞ on a non-uniform lattice xðsÞ has been given in [8]. Note that the difference-derivatives of yðxðsÞÞ satisfy similar (to the initial ones for yðxðsÞÞ) equations if and only if the lattice xðsÞ has the form
xðsÞ ¼ c1 qs þ c2 qs þ c3 ; or
ð4Þ
xðsÞ ¼ c4 s2 þ c5 s þ c6 ;
ð5Þ
where q – 1; c1 ; . . . ; c6 are constants. Depending on the particular choice of constants, the lattices are commonly referred to as 1. 2. 3. 4.
Linear lattices if we choose in (5) c4 ¼ 0 and c5 – 0. Quadratic lattices if we choose in (5) c4 – 0. q-Linear lattices (or q-exponential lattices) if we choose in (4) c2 ¼ 0 and c1 – 0. q-Quadratic lattices if we choose in (4) c1 c2 – 0.
The Askey tableau of hypergeometric orthogonal polynomials contains the classical orthogonal polynomials, which can be written in terms of a hypergeometric function, starting at the top with Wilson and Racah polynomials on quadratic lattices and ending at the bottom with Hermite polynomials [33]. Hahn [20] actually studied the q-analogue of this scheme. So, there are q-analogues of all the families in the Askey tableau, often several q-analogues for one classical family. The master class of all these q-analogues is formed by the Askey–Wilson polynomials [7] on the q-quadratic lattice xðsÞ ¼ ðqs þ qs Þ=2, which contain all other families as special or limit cases [3]. In [28] Koornwinder gave a q-Hahn tableau: a q-analogue of the part of the Askey tableau dominated by q-Hahn polynomials, in the q-linear lattice xðsÞ ¼ qs . Basic hypergeometric functions and q-orthogonal polynomials for arbitrary (including complex) values of q are connected with quantum algebras and groups [54]. Recently, Koekoek et al. [26] presented a classification of all families of classical orthogonal polynomials and their q-analogues, the classical q-orthogonal polynomials, as orthogonal polynomial solutions of the eigenvalue problem
/ðxÞðDq;w Þ2 yn ðxÞ þ wðxÞðDq;w Þyn ðxÞ ¼ kn yn ðqx þ wÞ;
ð6Þ
where Dq;w is the Hahn’s operator (1), /ðxÞ is a polynomial of at most degree 2, wðxÞ is a polynomial of exact degree 1, and kn is the spectral parameter. Besides well-known three-term recurrences, that q-orthogonal polynomial solutions of the latter equation do satisfy [11,41,49], these solutions can be characterized in a number of ways, e.g., kth q-derivatives of each family are again orthogonal and belong to the same family [1,41]. Moreover, the orthogonality weight functions satisfy q-Pearson equations [9,40],
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giving rise to Rodrigues’ formulae [1,41] for the corresponding orthogonal polynomials and their derivatives of any order. Also, the orthogonal polynomials posess a number of algebraic and q-difference properties, expressed as q-derivative representations [1,2,9] and structure relations [1,11,29]. The list of the above references is not exhaustive but only indicative for the kind of references that could be consulted on this topic. It is quite remarkable that in these classical settings the coefficients, appearing in all of the aforementioned algebraic and differential characterizations, can be explicitly computed in terms of the polynomial coefficients /ðxÞ and wðxÞ of the hypergeometric-type q-difference equation (6) [26,41], which governs those q-classical families. Orthogonal polynomials in several variables have been analyzed since a long time ago [22] and we refer to the books of Suetin [48] and Dunkl and Xu [13], as basic references on this topic. Various multivariate extensions have been used in many applications such as image description and pattern recognition [59], or ternary drug mixtures [38], among others. In 1991 Tratnik introduced some multivariable extensions of univariate orthogonal polynomials (see [51,52] and references therein). Moreover, q-analogues of these systems have been constructed by Gasper and Rahman [15–17], yielding systems of multivariable orthogonal Askey–Wilson polynomials and their special and limit cases. Bispectrality of multivariable Racah–Wilson and Askey–Wilson polynomials has been studied in [18,19], respectively. As indicated in [18], a beautiful extension of univariate orthogonal polynomials to the multivariate case is exemplified by symmetric Macdonald-Koornwinder polynomials, see, for instance, [21,27,37,53]. In more recent papers the second-order linear partial differential equations of the hypergeometric type [6] and their discretization on uniform lattices [5,4,45,46], as well as a general way of introducing orthogonal polynomial families in two discrete variables on the simplex [44], have been analyzed. Therefore, it is possible to generalize the univariate classical orthogonal polynomials to the bivariate and multivariate versions by requiring that they obey a second-order partial differential equation of the hypergeometric type (continuous case) [32,48], or a second-order partial difference equation of the hypergeometric type (discrete case), as indicated before. Thus, the ‘‘continuous’’ polynomials can be analyzed as limits of the ‘‘discrete’’ ones [4]. Likewise, the corresponding differential operator will appear as a scaling limit of an appropriate difference operator, and the continuous distribution (the weight measure for the bivariate continuous polynomials) is obtained through a scaling limit from the discrete distribution (the weight for the bivariate discrete polynomials). The main goal of this paper is to extend the latter results on continuous and discrete bivariate cases to an admissible potentially self-adjoint linear second-order partial q-difference equation of the hypergeometric type on particular non-uniform lattices, and to study their orthogonal polynomial solutions. The paper is organized as follows. In Section 2 the linear second-order partial q-difference equations of the hypergeometric type are introduced, giving explicitly the coefficients of the equation for the partial q-derivatives (of arbitrary order) of any solution in terms of the coefficients of the initial equation. In Section 3 we study the admissibility conditions for partial q-difference equations of the hypergeometric type. Next, in Section 4, the partial q-difference equation is written in the self-adjoint form, which gives a number of useful identities for the orthogonality weight function for the polynomial solutions (q-Pearson’s system), as well as of the q-difference derivatives of the polynomial solutions. The key point is to determine the orthogonality weight function from the polynomial coefficients of the initial equation, which is also explicitly worked out. In the remaining part of the paper we deal with admissible potentially self-adjoint linear second-order partial q-difference equations of the hypergeometric type. In Section 5 an analogue of the well-known Rodrigues’ formula for classical orthogonal polynomials is presented for orthogonal polynomial solutions of the partial qdifference equation. The monic orthogonal polynomial solutions of the partial q-difference equation are analyzed in detail in Section 6, where we give explicitly the matrices of the corresponding three-term recurrence relations for the most general equation, which belongs to the class under study. Section 7 is related with bivariate big q-Jacobi polynomials and a partial q-difference equation, that govern them [34]. Two novel bivariate q-orthogonal polynomial solutions of this equation are explicitly given. The first (non-monic) one is constructed from the Rodrigues’ representation, derived in Section 5. The second novel (monic) solution of the equation is obtained from the general analysis, given in Section 6, i.e. by employing in this particular case the matrices of the three-term recurrence relations for the vector column of polynomials. Besides, this monic solution is also explicitly given in terms of generalized bivariate basic hypergeometric series. Finally, limit relations as q " 1 for the partial q-difference equation, as well as for the three above-mentioned solutions, are analyzed in detail. 2. A linear second-order partial q-difference equation of the hypergeometric type In what follows we shall assume that 0 < q < 1 and we shall consider disconnected or independent non-uniform lattices [40] as
x ¼ xðsÞ ¼ qs ;
y ¼ yðtÞ ¼ qt :
ð7Þ
Related with these q-linear lattices (7), let us introduce the following partial q-difference operators
D1q f ðx; yÞ ¼
f ðqx; yÞ f ðx; yÞ ; ðq 1Þx
D2q f ðx; yÞ ¼
f ðx; qyÞ f ðx; yÞ ; ðq 1Þy
ð8Þ
I. Area et al. / Applied Mathematics and Computation 223 (2013) 520–536
D1q1 f ðx; yÞ ¼
qðf ðx; yÞ f ðx=q; yÞÞ ; ðq 1Þx
D2q1 f ðx; yÞ ¼
qðf ðx; yÞ f ðx; y=qÞÞ : ðq 1Þy
523
ð9Þ
The rules for the partial q-derivatives of a product of two functions f ðx; yÞ and gðx; yÞ are given by
D1q ðfgÞðx; yÞ ¼ f ðx; yÞD1q gðx; yÞ þ gðqx; yÞD1q f ðx; yÞ;
ð10Þ
D2q ðfgÞðx; yÞ ¼ f ðx; yÞD2q gðx; yÞ þ gðx; qyÞD2q gðx; yÞ:
ð11Þ
The following readily verified relations will also be used
8 1 2 Dq Dq1 f ðx; yÞ ¼ D2q1 D1q f ðx; yÞ; D2q D1q1 f ðx; yÞ ¼ D1q1 D2q f ðx; yÞ; > > > > > < D1 D2 f ðx; yÞ ¼ D2 D1 f ðx; yÞ; D11 D21 f ðx; yÞ ¼ D21 D11 f ðx; yÞ; q q q q q q q q > D1q1 f ðx; yÞ ¼ D1q f ðx; yÞ þ ð1 qÞxD1q D1q1 f ðx; yÞ; > > > > : 2 Dq1 f ðx; yÞ ¼ D2q f ðx; yÞ þ ð1 qÞyD2q D2q1 f ðx; yÞ:
ð12Þ
The following linear second-order partial differential equation has been considered in [6,36]
~11 ðx; yÞ a
@ 2 uðx; yÞ 2
@ x
~12 ðx; yÞ þa
@ 2 uðx; yÞ @ 2 uðx; yÞ ~ @uðx; yÞ ~ @uðx; yÞ ~22 ðx; yÞ þa þ b1 ðx; yÞ þ b2 ðx; yÞ þ kuðx; yÞ ¼ 0: 2 @x@y @x @y @ y
ð13Þ
Among many methods of approximating (13), we shall discuss a linear partial q-difference equation, obtained from (13) via the simplest q-difference schemes of the second-order precision [42,43]:
pffiffiffi pffiffiffi a11 ðx; yÞ qD1q D1q1 uðx; yÞ þ a22 ðx; yÞ qD2q D2q1 uðx; yÞ þ a12a ðx; yÞD1q D2q uðx; yÞ þ a12d ðx; yÞD1q1 D2q1 uðx; yÞ þ b1 ðx; yÞD1q uðx; yÞ þ b2 ðx; yÞD2q uðx; yÞ þ kuðx; yÞ ¼ 0:
ð14Þ
It is important to note here that from the cross second partial derivative we have obtained two second-order q-difference operators. As is shown below through an example (see Section 7), the associated polynomial coefficients a12a ðx; yÞ and a12d ðx; yÞ can be distinct. Definition 2.1. We shall refer to ðkÞ
ð‘Þ
kÞ
‘Þ
uðk;‘Þ ðx; yÞ :¼ ½D1q ½D2q uðx; yÞ ¼ D1q D1q D2q D2q uðx; yÞ as generalized difference of order ðk; ‘Þ for the function uðx; yÞ. Definition 2.2. We shall say that Eq. (14) is a partial q-difference equation of the hypergeometric type if all the generalized differences uðk;‘Þ ðx; yÞ for any solution u ¼ uðx; yÞ of (14) are also solutions of equations of the same type. In a similar way as Lyskova [36] introduced the so-called basic class in the continuous case, we have: Lemma 2.3. Eq. (14) is a partial q-difference equation of the hypergeometric type if and only if it has the form
q a1 x2 þ b1 x þ c1 D1q D1q1 uðx; yÞ þ q a2 y2 þ b2 y þ c2 D2q D2q1 uðx; yÞ þ ða3a xy þ b3a x þ c3a y þ d3a ÞD1q D2q uðx; yÞ þ ða3d xy þ b3d x þ c3d y þ d3d ÞD1q1 D2q1 uðx; yÞ þ ðf1 x þ g 1 ÞD1q uðx; yÞ þ ðf2 y þ g 2 ÞD2q uðx; yÞ þ kuðx; yÞ ¼ 0;
ð15Þ
that is,
a11 ðx; yÞ ¼ a11 ðxÞ ¼
pffiffiffi 2 q a1 x þ b1 x þ c1 ;
a12a ðx; yÞ ¼ a3a xy þ b3a x þ c3a y þ d3a ; b1 ðx; yÞ ¼ b1 ðxÞ ¼ f1 x þ g 1 ;
a22 ðx; yÞ ¼ a2 ðyÞ ¼
pffiffiffi 2 q a2 y þ b2 y þ c2 ;
a12d ðx; yÞ ¼ a3d xy þ b3d x þ c3d y þ d3d ;
b2 ðx; yÞ ¼ b2 ðyÞ ¼ f2 y þ g 2 :
Proof. Apply the operator D1q to (14) in order to reveal that the lemma is simply a consequence of the following seven partial results, based on the relations (10)–(12):
ð1Þ D1q ½a11 ðx; yÞq1=2 D1q D1q1 uðx; yÞ ¼ a11 ðx; yÞq1=2 D1q D1q1 uð1;0Þ ðx; yÞ þ D1q a11 ðx; yÞq1=2 D1q uð1;0Þ ðx; yÞ; ð2Þ D1q ½a22 ðx; yÞq1=2 D2q D2q1 uðx; yÞ ¼ a22 ðx; yÞq1=2 D2q D2q1 uð1;0Þ ðx; yÞ; provided that a22 ðx; yÞ does not depend on x (in order to preserve the same structure of the equation);
ð3Þ D1q ½a12a ðx; yÞD1q D2q uðx; yÞ ¼ D1q a12a ðx; yÞD2q uð1;0Þ ðx; yÞ þ a12a ðqx; yÞD1q D2q uð1;0Þ ðx; yÞ;
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ð4Þ D1q ½a12d ðx; yÞD1q1 D2q1 uðx; yÞ ¼ D1q a12d ðx; yÞD2q1 uð1;0Þ ðx; yÞ þ a12d ðx; yÞq1=2 q1=2 D1q1 D2q1 uð1;0Þ ðx; yÞ; ð5Þ D1q ½b1 ðx; yÞD1q uðx; yÞ ¼ b1 ðqx; yÞq1=2 q1=2 D1q uð1;0Þ ðx; yÞ þ D1q b1 ðx; yÞuð1;0Þ ðx; yÞ; ð6Þ D1q ½b2 ðx; yÞD2q uðx; yÞ ¼ b2 ðx; yÞD2q uð1;0Þ ðx; yÞ; provided that b2 ðx; yÞ does not depend on x; and finally,
ð7Þ D1q ½kuðx; yÞ ¼ k uð1;0Þ ðx; yÞ: Repeating this process k times in x and ‘ times in y, one obtains the following partial q-difference equation for the generalized difference of order ðk; ‘Þ of the function uðx; yÞ:
pffiffiffi pffiffiffi ðk;‘Þ ðk;‘Þ ðk;‘Þ ðk;‘Þ a11 ðxÞ qD1q D1q1 uðk;‘Þ ðx; yÞ þ a22 ðyÞ qD2q D2q1 uðk;‘Þ ðx; yÞ þ a12a ðx; yÞD1q D2q uðk;‘Þ ðx; yÞ þ a12d ðx; yÞD1q1 D2q1 uðk;‘Þ ðx; yÞ ðk;‘Þ
ðk;‘Þ
þ b1 ðxÞD1q uðk;‘Þ ðx; yÞ þ b2 ðyÞD2q uðk;‘Þ ðx; yÞ þ lðk;‘Þ uðk;‘Þ ðx; yÞ ¼ 0; where
ð16Þ
8 ðkþ1;‘Þ ðkþ1;‘Þ ðk;‘Þ ðk;‘Þ 1 ðk;‘Þ 1=2 > ð1 qÞyD1q a12d ðx; yÞ; > > a11 ðxÞ ¼ q a11 ðxÞ; a22 ðyÞ ¼ a22 ðyÞ þ q > > ðkþ1;‘Þ ðk;‘Þ ðkþ1;‘Þ ðk;‘Þ > 1 > > a12a ðx; yÞ ¼ a12a ðqx; yÞ; a12d ðx; yÞ ¼ q a12d ðx; yÞ; < ðkþ1;‘Þ ðk;‘Þ 1=2 1 ðk;‘Þ b1 ðxÞ ¼ b1 ðqxÞ þ q Dq a11 ðxÞ; > > ðkþ1;‘Þ ðk;‘Þ > 1 ðk;‘Þ 1 ðk;‘Þ > b ðyÞ ¼ b ðyÞ þ D a > 2 2 q 12a ðx; yÞ þ Dq a12d ðx; yÞ; > > > : lðkþ1;‘Þ ¼ lðk;‘Þ þ D1 bðk;‘Þ ðxÞ; q 1
and
8 ðk;‘þ1Þ ðk;‘Þ ðk;‘Þ ðk;‘þ1Þ ðk;‘Þ > a ðxÞ ¼ a11 ðxÞ þ q1=2 ð1 qÞxD2q a12d ðx; yÞ; a22 ðyÞ ¼ q1 a22 ðyÞ; > > 11 > ðk;‘þ1Þ ðk;‘Þ ðk;‘þ1Þ ðk;‘Þ > 1 > > < a12a ðx; yÞ ¼ a12a ðx; qyÞ; a12d ðx; yÞ ¼ q a12d ðx; yÞ; ðk;‘þ1Þ ðk;‘Þ ðk;‘Þ ðk;‘Þ b1 ðxÞ ¼ b1 ðxÞ þ D2q a12a ðx; yÞ þ D2q a12d ðx; yÞ; > > ðk;‘þ1Þ ðk;‘Þ ðk;‘Þ >b > ðyÞ ¼ b2 ðqyÞ þ q1=2 D2q a22 ðyÞ > 2 > > : ðk;‘þ1Þ 2 ðk;‘Þ ðk;‘Þ l ¼l þ Dq b2 ðyÞ: If one computes the action of D1q D2q ¼ D2q D1q on the Eq. (14), then one obtains that
D1q D2q a12i ðx; yÞ ¼ 0; or equivalently, the polynomials a12a ðx; yÞ and a12d ðx; yÞ should not contain the terms x2 and y2 . It is not hard to prove by induction that
a11 ðx; yÞ ð1 q‘ Þxðc3d þ a3d xÞ ðk;‘Þ þ ; a12a ðx; yÞ ¼ a12a ðqk x; q‘ yÞ; qk qkþ‘1=2 a22 ðx; yÞ ð1 qk Þyðb3d þ a3d yÞ ðk;‘Þ ðk;‘Þ þ ; a12d ðx; yÞ ¼ qðkþ‘Þ a12d ðx; yÞ; a22 ðx; yÞ ¼ q‘ qkþ‘1=2 ðk;‘Þ
a11 ðx; yÞ ¼
ð18Þ
½kq ðb1 þ a1 ðqk þ 1ÞxÞ ½‘q ðc3d þ a3d x þ q‘1 ðc3a þ a3a xÞÞ þ ð19Þ q‘1 qk1 kþ‘1 ‘ k1 ð1 qÞ½kq ½‘q ðb3d þ ða3d a3a q ÞyÞ ½‘q ðb2 þ a2 ðq þ 1ÞyÞ ½kq ðb3d þ a3d y þ q ðb3a þ a3a yÞÞ ðk;‘Þ b2 ðx;yÞ ¼ b2 ðqk x;q‘ yÞ þ þ þ ; ð20Þ q‘1 qkþ‘1 qk1 k2 ‘2 kþ‘2 ½k ðf1 q þ a1 ½k 1q Þ ½‘q ðf2 q þ a2 ½‘ 1q Þ ½kq ½‘q ða3d þ a31 q Þ lðk;‘Þ ¼ k þ q þ þ ; q‘2 qk2 qkþ‘2 ðk;‘Þ
b1 ðx;yÞ ¼ b1 ðqk x;q‘ yÞ þ
ð1 qÞ½kq ½‘q ðc3d þ ða3d a3a qkþ‘1 ÞxÞ
ð17Þ
qkþ‘1
;þ
where the q-number is
½zq ¼
qz 1 ; q1
z 2 C:
ð21Þ
3. Admissible equations Definition 3.1. The partial q-difference equation of the hypergeometric type (14) will be called admissible if there exists an infinite sequence fkn g (n ¼ 0; 1; . . .) such that for each k ¼ kn , there are precisely n þ 1 linearly independent polynomial solutions of total degree n and no non-trivial solutions in the form of polynomials, whose total degree is less than n.
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This concept was introduced by Krall and Sheffer [32] in the case of second-order partial differential equations and also by Xu in [[58], Section 2] for the case of second-order partial difference equations (without assuming that equations are of the hypergeometric type), and analyzed later on in [6] and [5,45,43], for the continuous and discrete cases, respectively. In the case n ¼ 0, the Eq. (14) also implies that a non-trivial solution can only exist when k0 ¼ 0. Observe that this definition of admissibility of Eq. (14) implies that all numbers
k0 ¼ 0; k1 ; k2 ; . . . ; kn ; . . . ; are distinct, km – kn , m – n. From Lemma 2.3 one can deduce Theorem 3.2. The partial q-difference equation of the hypergeometric type (15) is admissible if and only if
f2 ¼ f1 ;
a2 ¼ a1 ;
a3a ¼ a1 q þ f1 ðq 1Þ;
a3d ¼ a1 ;
ð22Þ
and
kn ¼ ½nq f1 a1 q½1 nq ;
ð23Þ
and the numbers a1 and f1 are such that for any non-negative integer m the following condition holds
f1 a1 q½1 mq – 0: Proof. A proof can be given in a similar way as in [48, pp. 93–97] for the multivariate continuous situation. It is therefore plain that with the notations of Lemma 2.3, Eq. (15) can be written as
h
q a1 x2 þ b1 x þ c1 D1q D1q1 uðx; yÞ þ q a1 y2 þ b2 y þ c2 D2q D2q1 uðx; yÞ þ ðða1 q þ f1 ðq 1ÞÞxy þ b3a x þ c3a y þ d3a ÞD1q D2q uðx; yÞ þ ða1 xy þ b3d x þ c3d y þ d3d ÞD1q1 D2q1 uðx; yÞ þ ðf1 x þ g 1 ÞD1q uðx; yÞ þ ðf1 y þ g 2 ÞD2q uðx; yÞ þ kn uðx; yÞ ¼ 0;
ð24Þ
i.e.,
8 pffiffiffi 2 > < a11 ðxÞ ¼ q a1 x þ b1 x þ c1 ; b1 ðxÞ ¼ f1 x þ g 1 ; b2 ðyÞ ¼ f1 y þ g 2 ; a12a ðx; yÞ ¼ ða1 q þ f1 ðq 1ÞÞxy þ b3a x þ c3a y þ d3a ; > pffiffiffi : a22 ðyÞ ¼ q a1 y2 þ b2 y þ c2 ; a12d ðx; yÞ ¼ a1 xy þ b3d x þ c3d y þ d3d :
ð25Þ
4. Potentially self-adjoint operator From the admissible linear second-order partial q-difference equation of the hypergeometric type (24) we introduce the following second-order partial q-difference operator:
pffiffiffi pffiffiffi Dq ½f ðx; yÞ ¼ a11 ðxÞ qD1q D1q1 f ðx; yÞ þ a22 ðyÞ qD2q D2q1 f ðx; yÞ þ a12a ðx; yÞD1q D2q f ðx; yÞ þ a12d ðx; yÞD1q1 D2q1 f ðx; yÞ þ b1 ðxÞD1q f ðx; yÞ þ b2 ðyÞD2q f ðx; yÞ:
ð26Þ
This enables us to write (24) as
Dq f ðx; yÞ þ kn f ðx; yÞ ¼ 0:
ð27Þ
Lemma 4.1. The adjoint operator Dyq of Dq , defined by (26), is given by
Dyq ½f ðx; yÞ ¼
pffiffiffi 1 1 pffiffiffi qDq Dq1 ða11 ðxÞf ðx; yÞÞ þ qD2q D2q1 ða22 ðyÞf ðx; yÞÞ þ q2 D1q D2q ða12d ðx; yÞf ðx; yÞÞ þ
1 1 2 1 1 D 1 D 1 ða12a ðx; yÞf ðx; yÞÞ D1q1 ðb1 ðxÞf ðx; yÞÞ D2q1 ðb2 ðyÞf ðx; yÞÞ: q2 q q q q
ð28Þ
Proof. The result is a direct consequence of
h
Diq
iy
1 ¼ Diq1 ; q
h
Diq1
iy
¼ qDiq ;
i ¼ 1; 2:
Definition 4.2. The operator Dq is potentially self-adjoint in a domain R, if there exists a positive real function .ðxÞ ¼ .ðx; yÞ in this domain, such that the operator .ðxÞDq is self-adjoint in the domain R, i.e., ð.ðxÞDq Þy ¼ .ðxÞDq (see [48, Chapter V]).
526
I. Area et al. / Applied Mathematics and Computation 223 (2013) 520–536
In order that Dq be potentially self-adjoint, we multiply (24) through by a positive function .ðxÞ ¼ .ðx; yÞ in some domain R, to be chosen later, to arrive at
pffiffiffi pffiffiffi a11 ðxÞ.ðx; yÞ qD1q D1q1 f ðx; yÞ þ a22 ðyÞ.ðx; yÞ qD2q D2q1 f ðx; yÞ þ a12a ðx; yÞ.ðx; yÞD1q D2q f ðx; yÞ þ a12d ðx; yÞ.ðx; yÞD1q1 D2q1 f ðx; yÞ þ b1 ðxÞ.ðx; yÞD1q f ðx; yÞ þ b2 ðyÞ.ðx; yÞD2q f ðx; yÞ þ k.ðx; yÞf ðx; yÞ ¼ 0;
ð29Þ
which can be written in the self-adjoint form if the following q-Pearson’s system of equations is satisfied:
8 2 > < .ðx; yÞa12a ðx; yÞ ¼ q .ðqx; qyÞa12d ðqx; qyÞ; .ðx; yÞ/1 ðx; yÞ ¼ .ðqx; yÞx1 ðqx; yÞ; > : .ðx; yÞ/2 ðx; yÞ ¼ .ðx; qyÞx2 ðx; qyÞ;
ð30Þ
8 pffiffiffi x1 ðqx; yÞ ¼ qya11 ðqxÞ xq2 a12d ðqx; yÞ; > > > < x ðx; qyÞ ¼ pffiffiffi qxa22 ðqyÞ yq2 a12d ðx; qyÞ; 2 pffiffiffi > /1 ðx; yÞ ¼ qya11 ðxÞ xa12a ðx; yÞ þ ðq 1Þxyb1 ðxÞ; > > : pffiffiffi /2 ðx; yÞ ¼ qxa22 ðyÞ ya12a ðx; yÞ þ ðq 1Þxyb2 ðyÞ:
ð31Þ
where
The q-Pearson’s system (30) can be also written as
8 pffiffiffi 1 qD ð.ðx; yÞa11 ðxÞÞ þ q1 D2q1 ð.ðx; yÞa12a ðx; yÞÞ ¼ .ðx; yÞb1 ðx; yÞ; > > < pffiffiffi q qD2q ð.ðx; yÞa22 ðyÞÞ þ q1 D1q1 ð.ðx; yÞa12a ðx; yÞÞ ¼ .ðx; yÞb2 ðx; yÞ; > > : 4 1 2 q Dq Dq ð.ðx; yÞa12d ðx; yÞÞ ¼ D1q1 D2q1 ð.ðx; yÞa12a ðx; yÞÞ;
ð32Þ
or equivalently,
8 1 D ðx1 ðx; yÞ.ðx; yÞÞ ¼ 1q D1q1 ð/1 ðx; yÞ.ðx; yÞÞ; > > < q D2q ðx2 ðx; yÞ.ðx; yÞÞ ¼ 1q D2q1 ð/2 ðx; yÞ.ðx; yÞÞ; > > : 4 1 2 q Dq Dq ð.ðx; yÞa12d ðx; yÞÞ ¼ D1q1 D2q1 ð.ðx; yÞa12a ðx; yÞÞ:
ð33Þ
4.1. Computation of the weight function By introducing the functions
G1 ðx; yÞ ¼
/1 ðx; yÞ
x1 ðqx; yÞ
;
G2 ðx; yÞ ¼
/2 ðx; yÞ
x2 ðx; qyÞ
;
ð34Þ
where /j ðx; yÞ and xj ðx; yÞ are defined in (31), j ¼ 1; 2, and using the q-Pearson’s system (30), one obtains that
.ðqx; yÞ ¼ G1 ðx; yÞ.ðx; yÞ; .ðx; qyÞ ¼ G2 ðx; yÞ.ðx; yÞ; ;
ð35Þ
q2 G1 ðx; yÞG2 ðqx; yÞa12d ðqx; qyÞ ¼ a12a ðx; yÞ ¼ q2 G1 ðx; qyÞG2 ðx; yÞa12d ðqx; qyÞ;
ð36Þ
yG2 ðx; yÞD2q ðG1 ðx; yÞÞ ¼ xG1 ðx; yÞD1q ðG2 ðx; yÞÞ:
ð37Þ
From (35) it follows then that
ln ½.ðqx; yÞ ln ½.ðx; yÞ ln ½G1 ðx; yÞ ¼ ; ðq 1Þx ðq 1Þx or, equivalently,
h i D1q ½lnð.ðx; yÞÞ ¼ ln ðG1 ðx; yÞÞ1=ððq1ÞxÞ ; and therefore
D1q ½lnð.ðx; yÞÞ D1q ½ln .ðx; y0 Þ ¼
1 G1 ðx; yÞ : ln ðq 1Þx G1 ðx; y0 Þ
Upon using the q-integral due to Thomae [50] and Jackson [24] (see also [14,23,26]), this yields
Z
x
x0
D1q ½lnð.ðs; yÞÞdq s ¼ ln ½.ðx; yÞ ln ½.ðx0 ; yÞ;
ð38Þ
I. Area et al. / Applied Mathematics and Computation 223 (2013) 520–536
and
ln ½.ðx; yÞ ln ½.ðx0 ; yÞ ¼
Z
x
h i ln ðG1 ðs; yÞÞ1=ððq1ÞsÞ dq s
x0
¼ ð1 qÞx
1 1 X X 1=ððq1Þqj xÞ 1=ððq1Þqj x0 Þ qj ln ðG1 ðqj x; yÞÞ ð1 qÞx0 qj ln ðG1 ðqj x0 ; yÞÞ j¼0
¼
1 X j¼0
G1 ðqj x0 ; yÞ þ c1 ðyÞ: ln G1 ðqj x; yÞ
In a similar way, one has
ln ½.ðx; yÞ ln ½.ðx; y0 Þ ¼
1 X
527
ln
j¼0
j¼0
G2 ðx; qj y0 Þ þ c2 ðxÞ: G2 ðx; qj yÞ
ð39Þ
ð40Þ
From (37) we deduce that
ðq 1ÞxD1q ðG2 ðx; yqj ÞÞ ðq 1Þyqj D2q ðG1 ðx; yqj ÞÞ : ¼ G2 ðx; yqj Þ G1 ðx; yqj Þ Upon using
"
ðq
1ÞxD1q ðln
# ðq 1ÞxD1q f ðx; yÞ j f jÞ ¼ ln þ1 ; f ðx; yÞ
and then applying the operator D1q to (40), from (38) one obtains that
ln
X 1 G1 ðx; yÞ ¼ ðq 1ÞxD1q ½lnðG2 ðx; qj y0 ÞÞ ðq 1ÞxD1q ½lnðG2 ðx; qj yÞÞ þ ðq 1ÞxD1q ðc2 ðxÞÞ G1 ðx; y0 Þ j¼0 " # " # 1 X ðq 1ÞxD1q G2 ðx; qj y0 Þ ðq 1ÞxD1q G2 ðx; qj yÞ ln ¼ þ 1 ln þ 1 þ ðq 1ÞxD1q ðc2 ðxÞÞ G2 ðx; qj y0 Þ G2 ðx; qj yÞ j¼0 " # " # 1 X ðq 1Þy0 qj D2q G1 ðx; qj y0 Þ ðq 1Þyqj D2q G1 ðx; qj yÞ þ 1 ln þ 1 þ ðq 1ÞxD1q ðc2 ðxÞÞ ln ¼ G1 ðx; qj y0 Þ G1 ðx; qj yÞ j¼0
¼
1 X ðq 1Þy0 qj D2q ðlnðG1 ðx; qj y0 ÞÞÞ ðq 1Þyqj D2q ðlnðG1 ðx; qj yÞÞÞ þ ðq 1ÞxD1q ðc2 ðxÞÞ j¼0
¼
Z
y
y0
D2q ðlnðG1 ðx; tÞÞÞdq t þ ðq 1ÞxD1q ðc2 ðxÞÞ ¼ lnðG1 ðx; yÞÞ lnðG1 ðx; y0 ÞÞ þ ðq 1ÞxD1q ðc2 ðxÞÞ;
and therefore
ðq 1ÞxD1q ðc2 ðxÞÞ ¼ 0; which implies that c2 ðxÞ ¼ c3 is a constant. In a similar way one verifies that c1 ðyÞ ¼ c4 is also constant. Substituting in (39) and (40),
ln ½.ðx; yÞ ln ½.ðx0 ; yÞ ¼
1 X
ln
G1 ðqj x0 ; yÞ þ c4 ; G1 ðqj x; yÞ
ln
G2 ðx; qj y0 Þ þ c3 ; G2 ðx; qj yÞ
j¼0
ln ½.ðx; yÞ ln ½.ðx; y0 Þ ¼
1 X j¼0
we finally obtain (up to a multiplicative constant) the explicit expression for the weight function solution of the q-Pearson’s system of Eqs. (30),
.ðx; yÞ ¼
1 Y G1 ðqj x0 ; yÞ G2 ðx0 ; qj y0 Þ ; G1 ðqj x; yÞ G2 ðx0 ; qj yÞ j¼0
ð41Þ
where G1 ðx; yÞ and G2 ðx; yÞ are defined in (34). In a similar way one can obtain the following representation for the orthogonality weight function, associated with the qderivatives of any order:
.ðk;‘Þ ðx; yÞ ¼
ðk;‘Þ ðk;‘Þ 1 Y G1 ðqj x0 ; yÞ G2 ðx0 ; qj y0 Þ ðk;‘Þ
j¼0
ðk;‘Þ
G1 ðqj x; yÞ G2 ðx0 ; qj yÞ
:
ð42Þ
528
I. Area et al. / Applied Mathematics and Computation 223 (2013) 520–536 ðk;‘Þ
ðk;‘Þ
ðk;‘Þ
ðk;‘Þ
ðk;‘Þ
Here G1 ðx; yÞ and G2 ðx; yÞ are defined by inserting into (34) the polynomial coefficients a11 ðx; yÞ; a22 ðx; yÞ, a12a ðx; yÞ, ðk;‘Þ ðk;‘Þ ðk;‘Þ a12d ðx; yÞ, b1 ðx; yÞ, and b2 ðx; yÞ, introduced in Section 2 and given explicitly in terms of the coefficients of the initial equation (24) in (17)–(20). It is important to note here that, for example, .ð1;1Þ ðx; yÞ can be computed in two ways: as the D1q derivative of the D2q derivative or vice versa. The following relation ensures that one arrives at the same result: ðkþ1;‘Þ ðk;‘Þ x1ðk;‘þ1Þ ðqx; yÞxðk;‘Þ ðx; qyÞx1 ðqx; qyÞ; 2 ðqx; qyÞ ¼ x2
k; ‘ P 0:
ð43Þ
We shall refer to the latter equation as the coupling hypergeometric condition, analogous to [[45], Eq. (52)]. 5. Rodrigues’ formula Rodrigues’ formula for classical orthogonal polynomials in one variable is an important tool for analyzing the fundamental properties of these polynomials [10,12,40]. The great advantage of the Rodrigues’ formula is its form as nth derivative of the orthogonality weight function. In [48], an analogue of the Rodrigues’ formula for orthogonal polynomials over a domain in two variables, which are solutions of admissible and potentially self-adjoint equations, is presented. Kwon et al. [25] succeeded in deriving a (functional) Rodrigues-type formula for multivariable orthogonal polynomial solutions of a second-order partial differential equation. In recent papers appropriate Rodrigues’ formulae for polynomials solutions of second-order admissible, hypergeometric and potentially self-adjoint partial differential and difference equations have been presented [6,45]. By using the results of the previous sections in a similar vein as was elaborated by Suetin [[48], Theorem 6.2, p. 151] for the continuous case, it is not hard to arrive at an explicit expression for a polynomial solution of an admissible potentially self-adjoint second-order partial q-difference equation of the hypergeometric type (24). The expression
e n;m ðx; yÞ ¼ Kn;m ½D11 ðnÞ ½D21 ðmÞ .ðn;mÞ ðx; yÞ P q .ðx; yÞ q
qnð1nÞ=2þmð1mÞ=2 Kn;m 1 ðnÞ 2 ðmÞ ðn;mÞ n ¼ . ðq x; qm yÞ ½Dq ½Dq .ðx; yÞ " # n1 m1 Y Y qnð1nÞ=2þmð1mÞ=2 Kn;m 1 ðnÞ 2 ðmÞ k s .ðx; yÞ x1 ðq x; yÞ x2 ðx; q yÞ ½Dq ½Dq ¼ .ðx; yÞ s¼0 k¼0
ð44Þ
defines an algebraic polynomial of total degree n þ m in the variables x and y, called Rodrigues’ formula for the bivariate qe n;m ðx; yÞ, that are solutions of (24). In (44) the Kn;m are normalizing constants, .ðx; yÞ and .ðn;mÞ ðx; yÞ orthogonal polynomials P are defined by (41) and (42), respectively, and x1 ðx; yÞ and x2 ðx; yÞ are defined in (31). In the limit as q tends to 1, the Rodrigues’ formula (44) reduces to the one derived in [48] for the continuous case. Moreover, in the bivariate discrete case a Rodrigues’ formula has been also given in [45] upon employing the same approach as in [48]. 6. Monic orthogonal polynomial solutions One essential difference between polynomials in one variable and in several variables is the lack of an obvious basis in the latter [13]. One possibility to avoid this problem is to consider graded lexicographical order and use the matrix vector representation, first introduced by Kowalski [30,31] and later on studied by Xu [55–57]. Let x ¼ ðx; yÞ 2 R2 , and let xn (n 2 N0 ) denote the column vector of the monomials xnk yk , whose elements are arranged in graded lexicographical order (see [13, p. 32]):
xn ¼ ðxnk yk Þ;
0 6 k 6 n; n 2 N0 :
ð45Þ
fPnnk;k ðx; yÞg
2 n
Let be a sequence of polynomials in the space P of all polynomials of total degree at most n in two variables, x ¼ ðx; yÞ, with real coefficients. Such polynomials are finite sums of terms of the form axnk yk , where a 2 R. From now on Pn will denote the (column) polynomial vector T
Pn ¼ ðP nn;0 ðx; yÞ; Pnn1;1 ðx; yÞ; . . . ; Pn1;n1 ðx; yÞ; Pn0;n ðx; yÞÞ :
ð46Þ
Then, each polynomial vector Pn can be written in terms of the basis (45) as:
Pn ¼ G n;n xn þ G n;n1 xn1 þ þ G n;0 x0 ;
ð47Þ
where G n;j are matrices of the size ðn þ 1Þ ðj þ 1Þ and G n;n is a nonsingular square matrix of the size ðn þ 1Þ ðn þ 1Þ. b n;n is the identity matrix (of the size b n is said to be monic if its leading matrix coefficient G A polynomial vector P ðn þ 1Þ ðn þ 1Þ), that is,
b n;n1 xn1 þ þ G b n;0 x0 : b n ¼ xn þ G P
ð48Þ
I. Area et al. / Applied Mathematics and Computation 223 (2013) 520–536
529
b n ðx; yÞ are of the form: Then each of its polynomial entries P nk;k
b n ðx; yÞ ¼ xnk yk þ terms of lower total degree: P nk;k
ð49Þ
b n will represent monic polynomials. In what follows the ‘‘hat’’ notation P The following existence theorem has been proved in [13]: Theorem 6.1. Let L be a positive definite moment linear functional acting on the space P2n of all polynomials of total degree at most n in two variables, and fPn gnP0 be an orthogonal family with respect to L. Then, for n P 0, there exist unique matrices An;j of the size ðn þ 1Þ ðn þ 2Þ, Bn;j of the size ðn þ 1Þ ðn þ 1Þ, and C n;j of the size ðn þ 1Þ n, such that
xj Pn ¼ An;j Pnþ1 þ Bn;j Pn þ C n;j Pn1 ;
j ¼ 1; 2;
ð50Þ
with the initial conditions P1 ¼ 0 and P0 ¼ 1. Here the notation x1 ¼ x and x2 ¼ y is used. In this section we give explicit expressions for the matrices An;j , Bn;j and C n;j , which appear in the three-term recurrence relations (50), in terms of the coefficients of aii , a12j and bi in (25). These matrices enable one to compute the monic orthogonal polynomial solutions of an admissible potentially self-adjoint second-order partial q-difference equation of the hypergeometric type. The weight function (41) determines the moment linear functional L, defined in the space P2n of all polynomials of total degree at most n in two variables, in terms of a double q-integral
LðPÞ ¼
ZZ
Pðx; yÞ.ðx; yÞdq xdq y; R
b ng in an appropriate domain R R2 , which can be applied to polynomial vectors. Thus, in what follows f P n2N0 denotes a monic vector polynomial family solution of (24), that is orthogonal with respect to .ðx; yÞ,
b TÞ ¼ Lðxm P n
(
ZZ R
b T .ðx; yÞdq xdq y ¼ xm P n
0 2 Mðmþ1;nþ1Þ ;
if n > m;
Hn 2 Mðnþ1;nþ1Þ ; if m ¼ n;
ð51Þ
where Hn ðwith size ðn þ 1Þ ðn þ 1ÞÞ is nonsingular. Let us first introduce the matrices Ln;j of the size ðn þ 1Þ ðn þ 2Þ
0 Ln;1
1
B ¼B @
.. .
1
1 0 1 0 0 1 C B C .. C and L ¼ B .. . . C; n;2 . A . @. A 0 0 1
ð52Þ
so that
x xn ¼ Ln;1 xnþ1 ;
y xn ¼ Ln;2 xnþ1 :
ð53Þ
Observe that
x2 xn ¼ Ln;1 Lnþ1;1 xnþ2 ;
y2 xn ¼ Ln;2 Lnþ1;2 xnþ2 ;
ð54Þ
Ln;2 Lnþ1;1 ¼ Ln;1 Lnþ1;2 ; and for j ¼ 1; 2,
Ln;j LTn;j ¼ I nþ1 ;
ð55Þ
where I nþ1 denotes the identity matrix of the size n þ 1. From the definition of the partial q-difference operators in (8) and (9), one obtains that
Djq xn ¼ E n;j xn1 ;
Djq1 xn ¼ Kn;j xn1 ;
j ¼ 1; 2;
where the matrices E n;j of the size ðn þ 1Þ n are given by
0 E n;1
B B B B ¼B B B @
½nq
C C C C .. C; . C C 1 A
0
½n 1q
0
1
0
0
E n;2
0 B1 B B B ½2q ¼B B B @
0
1
C C C C C; C .. C . A ½nq
ð56Þ
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I. Area et al. / Applied Mathematics and Computation 223 (2013) 520–536
the matrices Kn;j of the size ðn þ 1Þ n are given by
0 Kn;1
B B B B ¼B B B @
q1n ½nq
0
C C C C .. C; . C C 1 A
0
q2n ½n 1q
0
1
0
Kn;2
0 B1 B B B q1 ½2q ¼B B B @
0 ..
. q1n ½nq
1 C C C C C; C C A
ð57Þ
and the q-number is defined in (21). Substitute the expansion (48) into (24) and then equate the coefficients of xn1 and xn2 to arrive at the following explicit b n;n1 and G b n;n2 : expressions for the matrices G
b n;n1 ¼ G
1 Sn ; kn kn1
ð58Þ
b n;n2 ¼ G
1 b n;n1 S n1 ; TnþG kn kn2
ð59Þ
where kn is given in (23) and the matrix S n of the size ðn þ 1Þ n is given in terms of the coefficients of the polynomials aii , a12i and bi from the partial q-difference equation (24), explicitly written down in (25), as
0
1
sn;n1
sn;n
C C C C C C ðn P 1Þ: C C C A
0
snþ1;n
s1;1
B s2;1 B B B B Sn ¼ B B B B @
s2;2 .. .
..
.
sn1;n2
sn1;n1
ð60Þ
Here, for 1 6 i 6 n,
si;i ¼ ½n i þ 1q g 1 þ q1þin b1 ½n iq þ c3a ½i 1q þ q2n c3d ½i 1q ; siþ1;i ¼ ½i 1q b2 q3i ½i 2q þ ½n þ 1 iq b3a þ b3d q2n þ g 2 : Besides, the matrix T n of the size ðn þ 1Þ ðn 1Þ, which appears in (59), is given in terms of the coefficients of the partial qdifference equation (24) as
T n ¼ d3a E n;2 E n1;1 þ c1 qKn;1 E n1;1 þ c2 qKn;2 E n1;2 þ d3d Kn;2 Kn1;1
ðn P 2Þ;
ð61Þ
where the matrices E n;i and Kn;i ; i ¼ 1; 2, are defined by (56) and (57), respectively. Now, in this monic situation, it is possible to generalize the well-known explicit expressions for the coefficients in the three-term recurrence relation for the one variable case [[40], p. 14] to the q-bivariate case. This is achieved with the aid of the auxiliary matrices Ln;j , defined in (52) and (53), and the following result, proved in [6] in the continuous bivariate situation and therefore valid also in this q-bivariate situation, because it is a consequence of the three-term recurrence relations (50). Theorem 6.2. In the monic case, the explicit expressions of the matrices An;j , Bn;j and C n;j (j ¼ 1; 2), that appear in (50) in terms of b n;n1 and G b n;n2 (see (58) and (59), respectively), are given by the values of the leading coefficients G
8 An;j ¼ Ln;j ; n P 0; > > > > < B0;j ¼ L0;j G b 1;0 ; Bn;j ¼ G b n;n1 Ln1;j Ln;j G b nþ1;n ; n P 1; b 2;0 þ B1;j G b 1;0 Þ; > > C1;j ¼ ðL1;j G > > : b n;n2 Ln2;j Ln;j G b nþ1;n1 Bn;j G b n;n1 ; n P 2; Cn;j ¼ G
ð62Þ
where the matrices Ln;j have been introduced in (52). It is of interest to remark here that, as it is described in [13], since
rankðLn;j Þ ¼ n þ 1 ¼ rankðC nþ1;j Þ;
j ¼ 1; 2;
n P 0;
ð63Þ
531
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the columns of the joint matrices
T Ln ¼ LTn;1 ; LTn;2
T and C n ¼ C Tn;1 ; CTn;2 ;
of the size ð2n þ 2Þ ðn þ 2Þ and ð2n þ 2Þ n, respectively, are linearly independent, that is,
rankðLn Þ ¼ n þ 2;
rankðCn Þ ¼ n:
ð64Þ
Therefore the matrix Ln has full rank, so that there exists a unique matrix Dyn of the size ðn þ 2Þ ð2n þ 2Þ, called the generalized inverse of Ln ,
1 Dyn ¼ ðDn;1 jDn;2 Þ ¼ LTn Ln LTn ;
ð65Þ
such that
Dyn Ln ¼ I nþ2 : Moreover, using the left inverse Dyn of the joint matrix Ln ,
0
1
B B B B y Dn ¼ B B B @
1
0
C C C C C; C C A
1=2 1=2 .. .. . . 1=2 1=2 0
1
one can write a recursive formula for the monic orthogonal polynomials
b nþ1 ¼ Dy P n
x b n Dy C n P b n1 ; I nþ1 Bn P n y
n P 0;
ð66Þ
b 1 ¼ 0, P b 0 ¼ 1. In (66) the symbol denotes the Kronecker product and with the initial conditions P
T Bn ¼ BTn;1 ; BTn;2 ;
T Cn ¼ C Tn;1 ; C Tn;2 ;
ð67Þ
are matrices of the size ð2n þ 2Þ ðn þ 1Þ and ð2n þ 2Þ n, respectively, which can be obtained by using (62) in terms of the coefficients of the partial q-difference equation (24), explicitly given in (25). This means that the recurrence (66) gives another realisation of [13, (3.2.10)], already appeared in the bivariate discrete case in [46]. Therefore, from (66) it is possible to compute a monic orthogonal polynomial solution of an admissible potentially selfadjoint linear second-order partial q-difference equation of the hypergeometric type (24). 7. Illustrative example In this section we discuss in detail an example, which is related to the admissible potentially self-adjoint linear secondorder partial q-difference equation of the hypergeometric type, satisfied by a (non-monic) bivariate extension of big q-Jacobi polynomials, recently introduced in [34,35]. The monic orthogonal polynomial solutions are expressed by means of the three-term recurrence relations, that govern them, and also explicitly in terms of generalized bivariate basic hypergeometric series. Moreover, a third (non-monic) solution of the same partial q-difference equation is provided by using the Rodrigues’ representation (44). In the limit when q " 1 the partial q-difference equation reduces to a second-order partial differential equation of the hypergeometric type, with monic Appell polynomials as solutions. Limit relations between the three orthogonal polynomial solutions of the partial q-difference equation and corresponding orthogonal polynomial solutions of the partial differential equation are explicitly given. Besides, the corresponding orthogonality weight functions are also linked by appropriate limit relations as q " 1. Example. Lewanowicz and Woz´ny [34] have recently introduced the following bivariate extension of the big q-Jacobi polynomials 2kþ1
Pn;k ðx; y; a; b; c; d; qÞ :¼ Pnk ðy; a; bcq
k
; dq ; qÞyk ðdq=y; qÞk P k ðx=y; c; b; d=y; qÞ; n 2 N;
2 ð0; 1Þ; 0 < aq; bq; cq < 1; d < 0;
k ¼ 0; 1; . . . ; n; q ð68Þ
where the univariate big q-Jacobi polynomials [3] (see also [[26], Eq. (19.5.1)]) are defined by means of the 3/2 basic hypergeometric series as
Pm ðt; A; B; C; qÞ :¼ 3 /2
! qm ; ABqmþ1 ; t q; q ; Aq; Cq
0 < A < 1=q; 0 < B < 1=q; C < 0:
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I. Area et al. / Applied Mathematics and Computation 223 (2013) 520–536
They have demonstrated that the polynomials (68) satisfy a linear second-order partial q-difference equation of the form (24), with coefficients given by
8 pffiffiffi pffiffiffi a ðxÞ ¼ q ðd q xÞ a c q2 x ; a22 ðyÞ ¼ q ða q yÞ ðd q yÞ; > > > 11 > > > > a12a ðx; yÞ ¼ a c q4 ðd b xÞ ð1 yÞ; a12d ðx; yÞ ¼ ðd q xÞ ða q yÞ; > > > > > > q ðq ðda c d q2 þa c q ð1þb q ð1þxÞÞÞxÞ < b1 ðxÞ ¼ ; 1þq > > > q ðd qþa q ð1d qþb c q2 ð1þyÞÞyÞ > > > ; b2 ðyÞ ¼ > 1þq > > > > > > ð1abcqnþ2 Þ : kn ¼ q2n ½nq : q1
ð69Þ
Thus this q-difference equation is admissible, potentially self-adjoint and of the hypergeometric type. In [34] it has been proved that the polynomials (68) satisfy the following orthogonality relation
Z
aq
dq
Z
cqy
Wðx; y; a; b; c; d; qÞPn;k ðx; y; qÞPm;l ðx; y; qÞdq xdq y ¼ Hn;k ða; b; c; d; qÞdn;m dk;l ;
ð70Þ
dq
where 0 < aq; bq; cq < 1; d < 0, and the weight function is defined by
Wðx; y; a; b; c; d; qÞ :¼
ðdq=y; c1 x=y; x=d; y=a; y=d; qÞ1 : yðc1 d=y; cqy=d; x=y; bx=d; y; qÞ1
ð71Þ
Here the q-shifted factorial ða; qÞk is equal to
ða; qÞ0 ¼ 1;
ða; qÞk ¼
k1 Y
ð1 aqj Þ ðk ¼ 1; 2; . . . ; or 1Þ;
j¼0
and we have employed the conventional notation
ða1 ; . . . ; ar ; qÞk ¼ ða1 ; qÞk ðar ; qÞk for products of q-shifted factorials.It is easy to check that the Wðx; y; a; b; c; d; qÞ is a solution of the q-Pearson’s system (30), which has been presented in a different form in [35, Lemma 3.1] for this particular example. From the explicit expression for the weight function (41), we have (up to a normalization constant), upon taking into account that x0 ¼ dq,
.ðx; yÞ ¼
ðy=a; x=d; dq=y; x=ðcyÞ; y=d; qÞ1 ; ðy; x=y; bx=d; dq=ðcyÞ; cy=d; qÞ1
ð72Þ
which coincides with the weight function, given in (71), up to the positive multiplicative constant d=c. Observe that Eqs. (36), (37), and (43) also hold. The partial q-difference (24) with polynomial coefficients (69) has another (non-monic) orthogonal polynomial solution which can be computed from the Rodrigues’ formula (44) as
e n;m ðx; y; a; b; c; d; qÞ ¼ Kn;m ½D1 ðnÞ ½D2 ðmÞ .ðx; yÞx2n y2m ðdq=x; qÞ ðaq=y; qÞ ðx=y; qÞ ðcqy=x; qÞ ; P q n m m n .ðx; yÞ q
ð73Þ
where .ðx; yÞ is given in (72) and Kn;m is a normalizing constant.The partial q-difference equation (24), with coefficients given in (69), has a third (monic) orthogonal polynomial solution, which can be computed recursively from Theorem 6.2. From (62) it follows that the matrix Bn;1 has entries
8 iþnþ2 > dq acq2i1 qiþnþ1 b acqiþnþ1 þ qiþnþ2 q 1 q 1 þ 1 þ 1 > > > > 2nþ1 2nþ3 > > 1 abcq 1 abcq > > > > > > nþ2 ni 2i > acq bðq þ 1Þq acq þ q þ abðb þ 1Þcq2nþ2 þ b þ 1 > > ; ði ¼ jÞ; <þ 2nþ1 2nþ3 1 abcq 1 abcq > > > i1 2nþ2 > > q 1 aqi1 1 qiþnþ2 abcdq abcðq þ 1Þqnþ1 þ d > > > > ; ði ¼ j þ 1Þ; > 2nþ1 2nþ3 > > 1 abcq 1 abcq > > > > : 0; otherwise;
ð74Þ
I. Area et al. / Applied Mathematics and Computation 223 (2013) 520–536
533
the matrix Bn;2 has entries
8 0 1 nþ1 n > i q½iq aqi 1 dq 1 > q½i 1 q aq 1 dq > q > > qi @ þ þ qA; ði ¼ jÞ; > 2nþ3 2nþ1 > > abcq 1 1 abcq > < iþnþ1 2ðnþ1Þ acqiþ1 qiþnþ1 1 bq 1 abcq dðq þ 1Þqn þ 1 > > > ; ði ¼ j 1Þ; > > 2nþ1 2nþ3 > 1 abcq 1 abcq > > > : 0; otherwise;
ð75Þ
the matrix C n;1 has entries
8 n iþnþ1 > acqnþ2 dq 1 qiþnþ1 1 bq 1 > > > > 2 > > 2n 2nþ1 2nþ2 > > 1 abcq 1 abcq 1 abcq > > > > > nþ1 iþn > acqiþn 1 abcq d abcq 1 ; ði ¼ jÞ; > > > > n > nþ1 > > d < ac qi1 1 aqi1 1 dq 1 qiþ2nþ3 abcq 2 2n 2nþ1 2nþ2 > > 1 abcq 1 abcq 1 abcq > > > > > > ðbðq þ 1Þqiþn1 acq2i þ q2 þ abðb þ 1Þcq2nþ1 þ b þ 1 ; ði ¼ j þ 1Þ; > > > > > abc aqi q aqi q2 dqn 1 q2iþ3nþ2 ðq; qÞi1 ðabcqnþ1 dÞ > > ; ði ¼ j þ 2Þ; > 2 > > 2n 2nþ1 2nþ2 > > 1 abcq 1 ðq; qÞi3 ðabcq 1Þ abcq > > : 0; otherwise;
ð76Þ
and the matrix C n;2 has entries
8 n nþ1 nþ1 > acðq 1Þ dq 1 qni ½i þ n þ 1q bq qi abcq d > > > > ; 2 > > 2n 2nþ1 2nþ2 > > 1 1 abcq 1 abcq abcq > > > > > 2nþ1 > > ða þ 1Þqiþ1 abcq þ 1 abcðq þ 1Þq2nþ2 aðq þ 1Þq2i ; ði ¼ jÞ; > > > > n > iþ2nþ2 > a qi1 1 qnþ1 aqi1 1 dq 1 bcq 1 > > < 2 2n 2nþ1 2nþ2 1 abcq 1 abcq 1 abcq > > > > > nþ1 iþ2nþ2 > > d abcq 1 ; ði ¼ j þ 1Þ; abcq > > > > n > n nþ1 nþ1 2 2 nþ2 > dq 1 qi bq qi bq dÞ ðq; qÞniþ1 ðabcq > > a c q > > ; ði ¼ j 1Þ; > 2 > > 2n 2nþ1 2nþ2 > ðabcq 1Þ abcq 1 abcq 1 ðq; qÞ > ni1 > > : 0; otherwise:
ð77Þ
From (66) each column polynomial vector (46) can be obtained by using the above matrices Bn;i and C n;i ; i ¼ 1; 2. Remark 1. In [34] the authors obtained the matrices of the three-term recurrence relations (50), satisfied by the non-monic solution (68), by using the recurrence relation that governs the univariate q-Jacobi polynomials. Notice that we have derived the matrices in the monic case from our approach given in Section 6, with different shapes as in the non-monic case. b n;m ðx; y; a; b; c; d; qÞ of the partial q-difference equation (24), with Remark 2. Observe that the monic polynomial solution P coefficients, given in (69) and obtained from Theorem 6.2 (with the matrix coefficients that are given above), can be also written in terms of generalized bivariate basic hypergeometric series as
b n;m ðx; y; a; b; c; d; qÞ ¼ P
dn b
ðaq; qÞm ðbq; qÞn ðdq
mþ2
; qÞm ðabcq
=d;qÞn
ðabcq ; qÞnþm
n m 1 mþnþ2 ð1Þij q2ðiði2nþ1Þþjðj2mþ1ÞÞ ðabcq ; qÞiþj n X m X i q j q i¼0 j¼0
dn ¼
nþ1
mþnþ2
b
ðaq; qÞj ðbq; qÞi ðdq
ðaq; qÞm ðbq; qÞn ðdq
nþ1
mþnþ2
ðabcq
nþ1
mþ2
; qÞj ðabcq mþ2
; qÞm ðabcq
; qÞnþm
=d;qÞn
=d; qÞi 1:2;2 0:2;2
U
"
ðbx=d; qÞi ðy; qÞj
# : qn ;bx=d;qm ;y q : q;q ; mþ2 nþ1 0; 0;0 : bq; abcq =d; aq;dq nþmþ2
abcq
ð78Þ
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where the generalized bivariate basic hypergeometric series is defined by [47]
a ; . . . ; a : a ; . . . ; a ; c ; . . . ; c q : x; y
1 k 1 r 1 s Ulk:r;s :u;v b ; . . . ; b : b ; . . . ; b ; d ; . . . ; d i; j; k 1 u 1 v 1 l
m n 1 X ða1 ; . . . ; ak ; qÞmþn ða1 ; . . . ; ar ; qÞm ðc1 ; . . . ; cs ; qÞn xm yn qi 2 þj 2 þkmn ¼ : ðq; qÞm ðq; qÞn ðb1 ; . . . ; bl ; qÞmþn ðb1 ; . . . ; bu ; qÞm ðd1 ; . . . ; dv ; qÞn m;n¼0
Limit relations. Let us consider the case when a ¼ qa ; b ¼ qb , c ¼ qc and d ¼ qd . As q " 1 the second-order partial q-difference equation goes formally to the following second-order partial differential equation of the hypergeometric type
2 @2 @2 @2 x 1 f ðx; yÞ þ y2 1 f ðx; yÞ þ 2ððx þ 1Þðy 1ÞÞ f ðx; yÞ 2 2 @x @y @x@y @ @ þ ðxða þ b þ c þ 3Þ þ a b þ c þ 1Þ f ðx; yÞ þ ðyða þ b þ c þ 3Þ þ a b c 1Þ f ðx; yÞ @x @y nða þ b þ c þ n þ 2Þf ðx; yÞ ¼ 0:
ð79Þ
An orthogonality weight function for the polynomial solutions of the above equation can be computed in the same way as in [6], giving rise to
.ða;b;cÞ ðx; yÞ ¼ ð1 yÞa ðx þ 1Þb ðy xÞc ;
ð80Þ
in the triangular domain
R ¼ fðx; yÞ 2 R2 j x 6 y 6 1; 1 6 x 6 1g:
ð81Þ
It is important to note that
lim Wðx; y; qa ; qb ; qc ; qd ; qÞ ¼ ð1 yÞa ðx þ 1Þb ðy xÞc ¼ .ða;b;cÞ ðx; yÞ: q"1
The monic polynomial solutions of (79) satisfy a three-term recurrence relation of the form (50), where the matrix coefficients can be easily computed by considering the limit as q " 1 in (74)–(77) for a ¼ qa , b ¼ qb , c ¼ qc and d ¼ qd , or, eventually, from [5]. The monic orthogonal polynomial solutions of (79) can be written in terms of generalized Kampé de Fériet hypergeometric series as
b ða;b;cÞ ðx; yÞ ¼ ð1Þn 2nþm A n;m
a þ b þ c þ m þ n þ 2 : n; m x þ 1 1 y ða þ 1Þm ðb þ 1Þn : F 1:1;1 ; 0:1;1 2 2 ða þ b þ c þ m þ n þ 2Þnþm : b þ 1; a þ 1
ð82Þ
Remark 3. We would like to mention the following limit relation between the monic bivariate big q-Jacobi polynomials in (78) and the monic bivariate Jacobi polynomials, defined in (82),
b ða;b;cÞ ðx; yÞ: b n;m ðx; y; qa ; qb ; qc ; qd ; qÞ ¼ A lim P n;m
ð83Þ
q!1
Remark 4. Note the following limit relation for the non-monic bivariate big q-Jacobi polynomials, defined in (68),
m; b þ c þ m þ 1 y x lim Pn;m ðx; y; qa ; qb ; qc ; qd ; qÞ ¼ ðy þ 1Þm 2 F 1 q!1 yþ1 cþ1 m n; a þ b þ c þ m þ n þ 2 1 y ¼: J n;m ðx; y; a; b; cÞ; 2F1 2 aþ1
0 6 m 6 n:
The polynomials J n;m ðx; y; a; b; cÞ are a non-monic polynomial solution of (79) and they are orthogonal on the same domain (81) with respect to the weight (80). This non-monic polynomial solution can be written as
J n;m ðx; y; a; b; cÞ ¼
m!ðy þ 1Þm ðn mÞ! ðc;bÞ 2x y þ 1 ða;bþcþ2mþ1Þ P nm Pm ðyÞ; ðc þ 1Þm ða þ 1Þnm yþ1
where
Pða;bÞ ðxÞ ¼ n
n; n þ a þ b þ 1 1 x ða þ 1Þn ; 2F1 2 n! aþ1
are the Jacobi polynomials [26, Eq. (9.8.1)].
a > 1; b > 1;
I. Area et al. / Applied Mathematics and Computation 223 (2013) 520–536
535
There exists at least a third family of orthogonal polynomial solutions of the partial differential equation (79) (on the same domain R, defined in (81), and with respect to the weight function .ða;b;cÞ , given in (80)). The non-monic polynomials which can be computed from the Rodrigues’ formula [[5], Eq. (38)],
e ða;b;cÞ ðx; yÞ ¼ A n;m
1
.
ða;b;cÞ ðx; yÞ
i @ nþm h ðx þ 1Þbþn ð1 yÞaþm ðy xÞcþnþm : n m @x @y
ð84Þ
Remark 5. Observe the following limit relation between the non-monic bivariate big q-Jacobi polynomials, derived from the Rodrigues’ formula (44), and the non-monic bivariate Jacobi polynomials, defined by the Rodrigues’ formula (84),
e ða;b;cÞ ðx; yÞ: e n;m ðx; y; qa ; qb ; qc ; qd ; qÞ ¼ A lim P n;m q!1
ð85Þ
8. Conclusions and an outlook of future research In the present work we have initiated a general approach to the study of solutions of bivariate linear second-order partial q-difference equations on non-uniform lattices and concentrated our efforts on the particular case of q-linear lattices of the form xðsÞ ¼ qs and yðtÞ ¼ qt . We have dealt with those bivariate polynomials, written in vector representation (and graded lexicographical order), that are solutions of admissible potentially self-adjoint linear second-order partial q-difference equation of the hypergeometric type. In this context, we have proved that (similar to the one variable hypergeometric-type case) the coefficients of the three-term recurrence relations, obeyed by the vector polynomials, can be written explicitly in terms of the coefficients of the partial q-difference equation, they satisfy. 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