A high order q -difference equation for q -Hahn multiple orthogonal polynomials
aa r X i v : . [ m a t h . C A ] O c t A high order q -Difference equation for q -Hahnmultiple orthogonal polynomials Jorge Arves´u ∗ Department of Mathematics, Universidad Carlos III de Madrid,Avenida de la Universidad, 30, 28911, Legan´es, Spain;Chiara Esposito † Department of Mathematics, Copenhagen University,Universitetsparken 5, DK-2100 Copenhagen, DenmarkMay 29, 2018
Abstract
A high order linear q -difference equation with polynomial coefficientshaving q -Hahn multiple orthogonal polynomials as eigenfunctions is given.The order of the equation is related to the number of orthogonality condi-tions that these polynomials satisfy. Some limiting situations when q → The relevance of the special functions and classical orthogonal polynomials aseigenfunctions of a second order differential equation is a well established fact[9]. For instance, many fundamental problems of quantum mechanics as theharmonic oscillator, the solution of the Schr¨odinger, Dirac and Klein-Gordonequations for a Coulomb potential, the motion of a particle in homogeneouselectric or magnetic field lead to the study of the eigenfunctions for the gener-alized equation of hypergeometric type (see [9]). Recall that the spherical andcylinder (Bessel) functions -perhaps the most popular special functions- as wellas the classical orthogonal polynomials are particular solutions of this equation(see also equation (1) below).Recently, the appearance of new special functions, namely, the multiple or-thogonal polynomials also having the property of being eigenfunctions of a dif-ferential equation have attracted the interest of several researchers -see [5], and[7] for a discrete case-. In an effort to consider more general situations, thenotion of q -Hahn multiple orthogonal polynomials was introduced in [3]. Fur-thermore, a particular situation in which a third order q -difference equationhaving the aforementioned multiple orthogonal polynomials as eigenfunctionswas considered. Here we complete the study initiated in [3] by obtaining a ∗ Electronic address: [email protected] † Electronic address: [email protected] q -difference equation having q -Hahn multiple orthogonal polynomi-als as eigenfunctions. This q -difference equation gives a relation for the poly-nomial with a given degree evaluated at non-uniformed distributed points x ( s ), x ( s + 1) , . . . , x ( s + r + 1); where x ( s ) denotes the q -exponential lattice (4) and r , the number of orthogonality conditions (11). In addition, other multipleorthogonal polynomial families like some studied in [4, 12] as well as their cor-responding differential and difference equations (see [5, 7]) can be obtained asa limiting case.The contents of this paper are as follows. In Section 2, some well-knownresults on classical and multiple orthogonality are summarized. In Section 3we give an explicit expression for the multiple orthogonal polynomials studiedhere as well as a high order q -difference equation that these polynomials satisfy.Indeed, the notion of lowering and raising operators as well as the auxiliarylemma 3.1 play an essential role in the accomplishment of the q -difference equa-tion (theorem 3.1). As corollary we get some previous results [3, 7]. Lastly,Section 4 comprises some limiting cases known in the literature as multiple Ja-cobi and Hahn polynomials, respectively [4, 5, 12]. We also show how thesefamilies of polynomials can be obtained from the q -Hahn multiple orthogonalpolynomials based on the Rodrigues-type formula. One of the most remarkable features of the classical orthogonal polynomials onthe real line, namely, Jacobi, Laguerre, and Hermite polynomials is that theyare eigenfunctions of the second order differential operator [1, 6] (cid:18) a ( x ) d dx + a ( x ) ddx + λ I (cid:19) y ( x ) = 0 , (1)where deg a ( x ) ≤
2, deg a ( x ) = 1, λ is a constant independent on x and I denotes the identity operator. The above equation is known as hypergeome-tric equation. An exhaustive study of their solutions in term of the polynomialcoefficients a ( x ) and a ( x ) for each classical family can be found in [9].The notion of orthogonality for these classical families is assumed to be withrespect to a Borel measure µ on the real line R (with infinitely many pointsof increase) supported on a subset Ω ⊂ R , where Ω = ( a, b ), with | a | < ∞ , isthe smallest interval on the real line that contains supp µ . This orthogonalityconcept can be briefly described as follows: If R Ω p ( x ) dµ ( x ) converges for everypolynomial p , then one can define an inner product h p, q i = Z Ω p ( t ) q ( t ) dµ ( t ) , where p, q are polynomials. For such an inner product, which we will say tobe standard, a sequence of polynomials ( p n ) n ≥ is said to be orthogonal withrespect to the above inner product if(i) deg p n ( x ) = n ,(ii) h p n , p m i = δ n,m k p n k , m, n ∈ N , and δ m,n denotes the Kronecker deltafunction. 2 difference analog of the equation (1) on the lattice x ( s ), { x ( s ) R + : a ≤ s ≤ b − } , is the hypergeometric-type difference equation [8] (cid:18) σ ( s ) △△ x ( s − ) ▽▽ x ( s ) + τ ( s ) △△ x ( s ) + λ I (cid:19) y ( s ) = 0 ,y ( s ) = y ( x ( s )) , σ ( s ) = a ( x ( s )) − a ( x ( s )) △ x ( s − ) , τ ( s ) = a ( x ( s )) , (2)being ▽ y ( s ) = △ y ( s − △ and ▽ are the forward and backward differ-ence operators, respectively. Observe that △ f ( s ) = f ( s + 1) − f ( s ).Analogously, the classical orthogonal polynomials of a discrete variable canbe obtained as the corresponding eigenfunctions of (2). Indeed, the polynomialfamily that verifies (2) can be orthogonalized by constructing a Sturm-Liouvilleproblem with orthogonalizing weight ω ( s ) as the solution of a Pearson-typedifference equation (see [8, pp. 70-72]). Indeed, the following orthogonalityproperty b − X s = a P n ( x ( s )) P m ( x ( s )) ω ( s ) △ x ( s − ) = δ n,m || P n || , (3)yields under the additional (boundary) conditions σ ( s ) ω ( s ) x k ( s − ) (cid:12)(cid:12) s = a,b = 0, k = 0 , , . . . From now on we will denote any polynomial P n ( x ( s )) simply as P n ( s ).In [2] was studied the q -Hahn orthogonal polynomials P α,βn ( s ), n = 0 , , . . . ,with parameters α, β > −
1, on the lattice x ( s ) = q s − q − , q = 1 . (4)In particular, the orthogonalizing weight and the coefficients of the second orderdifference equation (2) are as follows: ω ( s ) = v α,β,Nq ( s ) = ˜Γ q ( s + β + 1)˜Γ q ( N + α − s + 1) q − α + β s ˜Γ q ( s + 1)˜Γ q ( N + 1 − s ) s = 0 , , . . . , N ∈ N , , otherwise , (5)where N denotes the set of natural numbers, ˜Γ( s ) = q − ( s − s − f ( s ; q ) if 0 1, or ˜Γ( s ) = f ( s ; q − ) if q > 1, being f ( s ; q ) = (1 − q ) − s Q k ≥ (1 − q k +1 ) Q k ≥ (1 − q s + k ) , and σ ( s ) = − q − N + α x ( s ) + q − [ N + α ] q x ( s ) , where [ x ] q = q x − q − x q − q − ,τ ( s ) = − q β +2 − N [ α + β + 2] q x ( s ) + q α + β +1 [ β + 1] q [ N − q , (6) λ n = q β +2 − N [ n ] q [ n + α + β + 1] q . For the classical orthogonal polynomials there is a characterization due toSonin (and independently obtained by W. Hahn) in the sense that they are thefamilies of orthogonal polynomials such that the sequence of their first deriva-tives constitutes again a sequence of orthogonal polynomials with a shifted set ofparameters. For classical orthogonal polynomials of a discrete variable there is a3imilar characterization. When one takes the forward difference of the classicalorthogonal polynomials of a discrete variable, one can show that these polyno-mials are again orthogonal polynomials of the same family, but with a differentset of parameters. As such, the △△ x ( s − ) operator acts as a lowering operator on these families of polynomials. As an example for the q -Hahn polynomials △△ x ( s − ) P α,β,Nn ( s ) = q − n [ n ] q P α +1 ,β +1 ,N − n − ( s ) . (7)Also for the classical orthogonal polynomials of a discrete variable we have raising operators which can also be found using the orthogonality property (3)and summation by parts. Hence, v α − ,β − ,N +1 q ( s ) ▽▽ x ( s ) v α,β,Nq ( s ) ! P α,β,Nn ( s )= − [ n + α + β ] q q N + n + β P α − ,β − ,N +1 n +1 ( s ) . (8)By combining the lowering and raising operators (7)-(8) one can obtain theequation (2) for the q -Hahn orthogonal polynomials with the above polynomialcoefficients (6). This fact is a general feature of the classical orthogonal poly-nomials of a discrete variable, i.e., the corresponding combination of lowering araising operators leads to the second order difference equation (2).In section 3 we will generalize the above procedure for a non-standard orthog-onality. Indeed, a new situation appears, namely, a third order q -difference equa-tion having q -orthogonal polynomials as eigenfunctions. This situation keepscertain similarity with the previous accomplishment of the hypergeometric equa-tion via raising and lowering operators; however two orthogonality conditionsmust be considered instead (see [3]). Hence, we need to deal with the concept ofmultiple orthogonal polynomial. This notion appears naturally in simultaneousPad´e approximation -which is often known as Hermite-Pad´e approximation- inorder to get simultaneous rational approximants to a vector function. As wewill see below they can be interpreted as an extension of ordinary orthogonalpolynomials.Let µ , µ , . . . , µ r ( r ≥ 2) be Borel measures on R with infinitely manypoints of increase such that supp µ i ⊂ Ω i , i = 1 , . . . , r , where Ω i = ( a i , b i ), with | a i | < ∞ , is the smallest interval on the real line that contains supp µ i . TheCauchy transforms of the corresponding measuresˆ µ i ( z ) = Z Ω i dµ i ( x ) z − x , z / ∈ Ω i i = 1 , . . . , r, also known as Markov (or Stieltjes) functions, can be simultaneously approxi-mated by rational functions with prescribed order near infinity. For studyingthis problem [10] one needs to introduce a multi-index ~n = ( n , n , . . . , n r )of nonnegative integers, and finds a polynomial P ~n ( z ) | ~n | = n + · · · + n r , such that the expressions P ~n ( z )ˆ µ i ( z ) − Q ~n,i ( z ) = ζ i z n i +1 + · · · = O ( z − n i − ) , i = 1 , . . . , r, P ~n ( z ) is the common denominator of thesimultaneous rational approximants Q ~n,i ( z ) /P ~n ( z ), to the Markov (Stieltjes)functions ˆ µ i ( z ), i = 1 , , . . . , r .Indeed, P ~n ( z ) is a type II multiple orthogonal polynomial of degree ≤ | ~n | defined by the following orthogonality conditions (see [4, 10, 12]) Z Ω i P ~n ( x ) x k dµ i ( x ) = 0 , k = 0 , , . . . , n i − , i = 1 , . . . , r. (9)These conditions (9) give a linear system of | ~n | homogeneous equations for the | ~n | + 1 unknown coefficients of P ~n ( z ). If the multi-index ~n is normal [10] thesolution is a unique polynomial P ~n ( z ) (up to a multiplicative factor) of degreeexactly | ~n | . In this situation throughout the paper we consider always monicmultiple orthogonal polynomials.If the measures in (9) are positive discrete measures on R , i.e., µ i = N i X k =0 ω i,k δ x i,k , ω i,k > , x i,k ∈ R , N i ∈ N ∪ { + ∞} , i = 1 , . . . , r, where x i ,k = x i ,k , k = 0 , . . . , N i , whenever i = i , the corresponding polyno-mial solution is then a discrete multiple orthogonal polynomial P ~n ( z ) [4]. Herewe have that supp µ i is the closure of { x i,k } N i k =0 and that Ω i is the smallestclosed interval on R that contains { x i,k } N i k =0 . If the above system of measuresforms an AT system then every multi-index is normal (see [10] for a detailedexplanation on the concept).In [4] for several AT systems of measures was studied the correspondingdiscrete multiple orthogonal polynomials of type II on the linear lattice x ( s ) = s (those of Charlier, Kravchuk, Meixner of first and second kind, and Hahn). Evenmore, it was obtained rising operators and then the Rodrigues-type formula.Here we will deal with an AT system formed by different discrete measuressupported on the same interval. This situation was analyzed in [3] for ~n =( n , . . . , n r ) ∈ N r and the set of parameters N , α and ~α = ( α , . . . , α r ). Indeed,the orthogonality conditions (9) were considered with respect to the followingpositive discrete measures on R µ i = N X s =0 v α i ,α ,Nq ( k ) △ x ( k − ) δ ( k − s ) , i = 1 , . . . , r, (10)where v α i ,α ,Nq ( k ) is defined in (5) and α , α i > − α i − α j / ∈ { , , . . . , N − } when i = j . Definition 2.1. A polynomial P ~α,α ,N~n ( s ) that verifies the orthogonality con-ditions (9) with respect to the measures (10) is said to be the type-II q -Hahnmultiple orthogonal polynomial of a multi-index ~n ∈ N r , i.e., the conditions N X s =0 P ~α,α ,N~n ( s )( s ) [ k ] q v α i ,α ,Nq ( s ) ▽ x ( s + ) = 0 , k ≤ n i − , i = 1 , . . . , r, (11)5 old, where the symbol ( s ) [ k ] q denotes the following polynomial of degree at most k in the variable x ( s )( s ) [ k ] q = k − Y j =0 q s − j − q − x ( s ) x ( s − · · · x ( s − k + 1) . Notice that ( s ) [ k ] q is a polynomial of degree k in x ( s ), and the orthogonal-ity conditions (9) have been written more conveniently as (11). When r = 1definition 2.1 gives the standard orthogonality (3). In addition, { , ( s ) [1] q , . . . , ( s ) [ k ] q } , constitutes a basis of the linear space of polynomials of degree at most k in x ( s ).Indeed, we consider this basis as the canonical one. When q goes to 1, ( s ) [ k ] q converges to ( − k ( − s ) k , where ( s ) k denotes the usual Pochhammer symbol( s ) k = s ( s + 1) · · · ( s + k − s ) = 1.An important feature for P ~α,α ,N~n ( s ) is the existence of raising operators [3] D α i ,α ,N P ~α,α ,N~n ( s ) = − q − N + | ~n | + α [ | ~n | + α i + α ] q P ~α − ~e i ,α − ,N +1 ~n + ~e i ( s ) , (12) D α i ,α ,N def = v α i − ,α − ,N +1 q ( s ) ∇ v α i ,α ,Nq ( s ) ! , where the multi-index ~e i denotes the standard r dimensional unit vector withthe i th entry equals 1 and 0 otherwise, and ∇ def = ▽▽ x ( s ) . Indeed, D α i ,α ,N f ( s ) = A α ,α i ,N ( s ; q ) ▽ f ( s ) + B α ,α i ,N ( s ; q ) f ( s ) , (13)where A α ,α i ,N ( s ; q ) = q − ( αi + α [ s ] q [ N + α − s + 1] q (14) B α ,α i ,N ( s ; q ) = [ s + α i ] q [ N − s + 1] q − [ s ] q [ N + α − s + 1] q q αi + α ! = (cid:16) [ N + 1] q [ α i ] q − x ( s )[ α i + α ] q q − ( α N )2 (cid:17) . (15) q -difference equation The next theorem 3.1 extends the result of [3] concerning the q -difference equa-tion. We will combine the lowering and raising operators to get a ( r + 1)-order q -difference equation. As a corollary, an explicit difference equation for theHahn multiple orthogonal polynomials is given (see [7]).An explicit expression for the type-II monic q -Hahn multiple orthogonalpolynomials can be obtained as a direct consequence of a summation by partsand formula (12). 6 roposition 1. The following finite-difference analog of the Rodrigues formula P ~α,α ,N~n ( s ) = ( − | ~n | q ( N + α | ~n | + Q ri =1 ni + P ri =1 ( ni ) Q rk =1 ( | ~n | + α + α k + 1 | q ) n k q − α s ˜Γ q ( s + 1)˜Γ q ( N − s + 1)˜Γ q ( α + N − s + 1) r Y i =1 q − αi s ˜Γ q ( α i + s + 1) ∇ n i ˜Γ q ( α i + n i + s + 1) q − αi + ni s ! q α | ~n | s ˜Γ q ( α + N − s + 1)˜Γ q ( s + 1)˜Γ q ( N − | ~n | − s + 1) , (16) holds, where ( a | q ) k = Q k − m =0 [ a + m ] q = ˜Γ q ( a + k ) / ˜Γ q ( a ) is the q -analogue of thePochhammer symbol. Notice that the above Rodrigues-type formula characterizes the type-II q -Hahn multiple orthogonal polynomials in terms of a finite-difference property. Proof. Replacing ( s ) [ k ] q in (11) by the following finite-difference expression( s ) [ k ] q = q ( k − [ k + 1] q ∇ ( s + 1) [ k +1] q , the orthogonality conditions can be written in a more convenient way as follows N X s =0 P ~α,α ,N~n ( s ) ∇ ( s + 1) [ k +1] q v α i ,α ,Nq ( s ) ▽ x ( s + ) = 0 ,k = 0 , , . . . , n i − , i = 1 , , . . . , r. (17)From here, using the summation by parts one gets (12). Recursively using thisraising operator gives the Rodrigues-type formula (16).In the next theorem we will need the following auxiliary lemma. Lemma 3.1. Let A be the following r -dimensional matrix A = n ] q n + α − α ] q · · · n + α − α r ] q n + α − α ] q n ] q · · · n + α − α r ] q ... ... . . . ... n r + α r − α ] q n r + α r − α ] q · · · n r ] q = ( a i,j ) ri,j =1 , a i,j = 1[ n i + α i − α j ] q , then the determinant of A is det A = r − Y k =1 r Y l =1 [ α k − α l ] q [ n l − n k + α l − α k ] qr Y k =1 r Y l =1 [ n l + α l − α k ] q . (18)Here for proving (18) we will follow the operations indicated in [7, Lemma2.8, p. 18]. 7 roof. Let us proceed by column and row operations on the matrix A . Observethat, for k = 1 , . . . , r and i = 2 , . . . , r the following relation a k,i − a k, = ˜ λ i, a k,i a k, q − nk + αk (cid:16) q αi + α + q α k + n k (cid:17) , ˜ λ i, = [ α i − α ] q q αi + q α , (19)yields.Therefore, based on (19) if A k denotes the k th column of A ( k = 1 , . . . , r )one gets det A = det( A , A − A , . . . , A r − A )= r Y k =1 a k, ! r Y i =2 ˜ λ i, ! a , · · · ˜ a ,r a , · · · ˜ a ,r ... ... . . . ...1 ˜ a r, · · · ˜ a r,r , where ˜ a k,i = a k,i q − nk + αk (cid:16) q αi + α + q α k + n k (cid:17) . Now, if one substracts the firstrow from the other ones, and takes into account that˜ a k,i − ˜ a ,i = a k,i a ,i µ k, (cid:16) q αi + q α (cid:17) , µ k, = [ n − n k + α − α k ] q i, k = 2 , . . . , r, thendet A = r Y k =1 a k, ! r Y j =2 a ,j r Y i =2 λ i, µ i, ! a , a , · · · a ,r a , a , · · · a ,r ... ... . . . ... a r, a r, · · · a r,r , where λ i, = [ α i − α ] q .Finally, repeating the previous column and row operations -but on lowerdimensional matrices- the expression (18) can be inductively proved. Theorem 3.1. The type-II monic q -Hahn multiple orthogonal polynomial P ~α,α ,N~n ( s ) is an eigenfunction of the following ( r + 1) -order q -difference equation r Y i =1 D α i +1 ,α +1 ,N − ! △△ x ( s ) P ~α,α ,N~n ( s ) = − q − ( N + | ~n | + α − r X i =1 ξ i [ | ~n | + α + α i + 1] q D α i +1 ,α +1 ,N − ! P ~α,α ,N~n ( s ) , (20) where ξ i = r X k =1 ( − k + l [ n k + α k + α + 1] q Q ri =1 ,i = l [ n k + α k − α i ] q Q r − i =1 ,i = k [ n i + α i − n k − α k ] q Q rj = k +1 [ n k + α k − n j − α j ] q ! q | ~n |− Q r − k =1 Q rl =1 [ α k − α l ] q [ n l − n k + α l − α k ] q Q rk =1 Q rl =1 [ n l + α l − α k ] q , i = 1 , . . . , r. (21)8 roof. Taking into account the expressions (12)-(15), the q -Hahn multiple or-thogonal polynomial can be expressed in term of the raising operator as follows P ~α,α ,N~n ( s ) = − q N + | ~n | + α − [ | ~n | + α i + α ] q (cid:0) B α +1 ,α i +1 ,N − ( s ; q ) I + A α +1 ,α i +1 ,N − ( s ; q ) ▽ (cid:1) P ~α + ~e i ,α +1 ,N − ~n − ~e i ( s ) , where i = 1 , , . . . , r . Hence, for k ≤ r , one gets the following relation N X s =0 P ~α,α ,N~n ( s )( s ) [ n k − q v α k +1 ,α ,Nq ( s ) △ x ( s − )= − q N + | ~n | + α − [ | ~n | + α i + α ] q N X s =0 ( s ) [ n k − q v α k +1 ,α ,Nq ( s ) △ x ( s − ) (22) (cid:0) B α +1 ,α i +1 ,N − ( s ; q ) I + A α +1 ,α i +1 ,N − ( s ; q ) ▽ (cid:1) P ~α + ~e i ,α +1 ,N − ~n − ~e i ( s ) . Now, transforming (22) by doing a suitable combination of the orthogonalizingweight v α k +1 ,α ,Nq ( s ) with the terms A α +1 ,α i +1 ,N − ( s ; q ) and B α +1 ,α i +1 ,N − ( s ; q ),and summing by parts N X s =0 P ~α,α ,N~n ( s )( s ) [ n k − q v α k +1 ,α ,Nq ( s ) △ x ( s − ) = q θ [ n k + α k − α i ] q [ | ~n | + α i + α + 1] q × N X s =0 P ~α + ~e i ,α +1 ,N − ~n − ~e i ( s )( s ) [ n k − q v α k +1 ,α +1 ,N − q ( s ) △ x ( s − ) , (23)yields, where θ = N + | ~n | + n k + α k + α . Here, aimed to shift the parameters α and N in the orthogonalizing weight v α k +1 ,α ,Nq ( s ) we have used the relations v α k +1 ,α +1 ,N − q ( s ) v α k +1 ,α ,Nq ( s ) = q s [ N − s ] q ,v α k +1 ,α +1 ,N − q ( s ) v α k +1 ,α ,Nq ( s + 1) = ( s + 1) [ n k − q [ N + α − s ] q q αk + α − ( s ) [ n k − q [ α k + s + 2] q . By using recursively relation (23) one gets N X s =0 P ~α + ~e l ,α +1 ,N − ~n − ~e l ( s )( s ) [ n k − q v α k +1 ,α +1 ,N − q ( s ) △ x ( s − )= ˜ a k,l N X s =0 P ~α + ~e,α + r,N − r~n − ~e ( s )( s ) [ n k − q v α k +1 ,α + r,N − rq ( s ) △ x ( s − ) , (24)where ~e = P ri =1 ~e i and˜ a k,l = [ | ~n | + α + α l + 1] q q − ( r − θ − − [ α k − α l + n k ] q r Y j =1 [ α k − α j + n k ] q [ | ~n | + α + α j + 1] q , k, l = 1 , . . . , r. { ξ l } rl =1 (not allzero) such that the relation∆ P ~α,α ,N~n ( s ) = r X l =1 ξ l P ~α + ~e l ,α +1 ,N − ~n − ~e l ( s ) , ∆ = △△ x ( s ) , (25)is valid. Thus, for finding explicitly ξ , . . . , ξ r one takes into account (24) and(25) to get N X s =0 (cid:16) ∆ P ~α,α ,N~n ( s ) (cid:17) ( s ) [ n k − q v α k +1 ,α +1 ,N − q ( s ) △ x ( s − )= r X l =1 ξ l ˜ a k,l ! N X s =0 P ~α + ~e,α + r,N − r~n − ~e ( s )( s ) [ n k − q v α k +1 ,α + r,N − rq ( s ) △ x ( s − ) . (26)Now, the left hand side of this equation can be easily transformed by means ofa summation by parts and orthogonality relation (11). Indeed, N X s =0 (cid:16) ∆ P ~α,α ,N~n ( s ) (cid:17) ( s ) [ n k − q v α k +1 ,α +1 ,N − q ( s ) △ x ( s − )= [ n k + α + α k + 1] q q θ − | ~n |− N X s =0 P ~α,α ,N~n ( s )( s ) [ n k − q v α k +1 ,α ,Nq ( s ) △ x ( s − ) . Based on (23) and (24) the right hand side of this expression transforms intothe equation N X s =0 (cid:16) ∆ P ~α,α ,N~n ( s ) (cid:17) ( s ) [ n k − q v α k +1 ,α +1 ,N − q ( s ) △ x ( s − )= ˜ b k N X s =0 P ~α + ~e,α + r,N − r~n − ~e ( s )( s ) [ n k − q v α k +1 ,α + r,N − rq ( s ) △ x ( s − ) , (27)where ˜ b k = q | ~n | +( r − θ − [ n k + α k + α + 1] q r Y i =1 [ n k + α k − α i ] q [ | ~n | + α + α i + 1] q . From equations (26) and (27) leads the following linear system of equation forthe unknown coefficients ξ , . . . , ξ r , b k = r X l =1 ξ l s k,l , k = 1 , . . . , r, ⇐⇒ Sξ = b, ξ = ( ξ , . . . , ξ r ) , (28)where the entries of the vector b and matrix S are as follows b k = q | ~n |− [ n k + α k + α + 1] q , s k,l = a k,l [ | ~n | + α + α l + 1] q . By the Cramers rule, the above system (28) has a unique solution if and only ifthe determinant of S is different from zero. Observe that S = A · D , where D denotes the diagonal matrix D = ( d k,l ) rk,l =1 , d k,l = [ | ~n | + α + α l + 1] q δ k,l . S = det( A · D ) = r Y i =1 [ | ~n | + α + α i + 1] q ! det A = 0 . Accordingly, if C i,j is the cofactor of the entry s i,j , and S j ( b ) denotes the matrixobtained from S replacing its j th column by b , then ξ l = det S l ( b )det S , l = 1 , . . . , r, wheredet S l ( b ) = r X k =1 b k C k,l = r X k =1 b k ( − k + l Q ri =1 ,i = l [ n k + α k − α i ] q Q r − i =1 ,i = k [ n i + α i − n k − α k ] q Q rj = k +1 [ n k + α k − n j − α j ] q . Consequently, expression (21) yields.Finally, applying the following product of r operators (cid:0)Q ri =1 D α i +1 ,α +1 ,N − (cid:1) on both sides of the equation (25) and considering that these raising operatorsare commuting, the expression (20) holds.In particular when r = 2, the q -Hahn multiple orthogonal polynomial verifya third order q -difference equation (see [3, theorem 2.1, pp. 7-8] and nextcorollary). Corollary 3.2. The type-II monic q -Hahn multiple orthogonal polynomial P α ,α ,α ,Nn ,n ( s ) verify the following third order q -difference equation a ( s )∆ ∇ y + a ( s )∆ ∇ y + a ( s )∆ y + a ( s ) ∇ y + a ( s ) y = 0 , (29) where a ( s ) = q s − A α +1 ,α +1 ,N − ( s ; q ) A α +1 ,α +1 ,N − ( s − q ) ,a ( s ) = A α +1 ,α +1 ,N − ( s ; q ) (cid:18) B α +1 ,α +1 ,N − ( s − q ) q − s + ▽ A α +1 ,α +1 ,N − ( s ; q ) q − s (cid:19) + q s − B α +1 ,α +1 ,N − ( s ; q ) A α +1 ,α +1 ,N − ( s ; q ) ,a ( s ) = B α +1 ,α +1 ,N − ( s ; q ) B α +1 ,α +1 ,N − ( s ; q )+ A α +1 ,α +1 ,N − ( s ; q ) ▽ B α +1 ,α +1 ,N − ( s ; q ) ,a ( s ) = q − N + | ~n | + α − s +32 ξ A α +1 ,α +1 ,N − ( s ; q )[ | ~n | + α + α + 1] − q + ξ A α +1 ,α +1 ,N − ( s ; q )[ | ~n | + α + α + 1] − q ! ,a ( s ) = q − N + | ~n | + α ξ B α +1 ,α +1 ,N − ( s ; q )[ | ~n | + α + α + 1] − q + ξ B α +1 ,α +1 ,N − ( s ; q )[ | ~n | + α + α + 1] − q ! , and ξ = q n n − [ n ] q [ n + α − α ] q [ α − α ] q , ξ = q n n − [ n ] q [ n + α − α ] q [ α − α ] q . q -difference equation can be considered as an extension of thehypergeometric-type difference equation (2). Again here the combination oflowering and rising operators (13) was the key fact to obtain a third orderdifference equations having q -Hahn multiple orthogonal polynomials as eigen-functions [3]. Remark 3.3. Notice that from the q -difference operator (20) can be obtainedthe difference equation studied in [7] for the Hahn multiple orthogonal polyno-mials since when q goes to the lattice x ( s ) transforms into a linear one s . Inparticular, for the above third order q -difference equation (29) one gets the sametype of third order difference equation with the following polynomial coefficients a ( s ) = s ( s − α + N − s + 1)( α + N − s + 2) ,a ( s ) = s ( α + N − s + 1) [2 α + α + 4 + ( α + α + 3) N − (2 α + α + α + 6) s ] ,a ( s ) = [( α + 1) N − ( α + α + 2) s ][( α + 1) N − ( α + α + 2) s ] − ( α + α + 2)( α + N − s + 1) s,a ( s ) = [ n ( α + α + N + 1) + ( α + α + N + 1) n − n n ]( α + N − s + 1) s,a ( s ) = N [( α + 1)( α + α + N + 1) n + ( α + 1)( α + α + N + 1) n +( α + N ) n n ] − [( α + α + 2)( α + α + N + 1) n +( α + α + 2)( α + α + N + 1) n + ( N − n n ] s. Observe that when q goes to 1 the expression (16) transforms into the Rodri-gues-type formula for Hahn multiple orthogonal polynomials in the linear lattice x ( s ) = s (see [4]) H ~α,α ,N~n ( s ) = ( − | ~n | Q rk =1 ( | ~n | + α + α k + 1) n k Γ( s + 1)Γ( N − s + 1)Γ( α + N − s + 1) r Y k =1 α i + s + 1) ∇ n i Γ( α i + n i + s + 1) ! Γ( α + N − s + 1)Γ( s + 1)Γ( N − | ~n | − s + 1) . (30)Now, based on this Rodrigues-type formula another limiting transition be-tween discrete and continuous multiple orthogonal polynomials can also be ob-tained. The basic tool for establishing this limiting transition is the usual ap-proximation of derivatives by means of finite-differences.Suppose that f ( s ) is a function defined on an interval of the real line, whichcontains the linear lattice { s i } Ni =0 . Furthermore, f ( s ) possesses first derivativeon { s i } Ni =0 , and second derivative for every χ i ∈ ( s i − h, s i ), i = 1 , . . . , N . Thus, ∇ f ( s i ) = f ( s i ) − f ( s i − h ) h = f ′ ( s i ) − h f ′′ ( χ i )= f ′ ( s i ) + O ( h ) , yields. In general, if f ( s ) has n derivatives at points s i and 2 n derivatives forany χ i one gets ∇ n f ( s i ) = f ( n ) ( s i ) + O ( h n ) . (31)Notice that the change of variable s = N x , transforms the interval [0 , N ]into [0 , N tends to infinity, i.e., the step h = △ x = 1 /N in the new variable x tends to 0, the Hahn multiple orthogonal polynomials (30) will tend to theaforementioned monic multiple Jacobi polynomials P ( α ,~α ) ~n ( x ) (see below theexplicit expression (41)). Proposition 2. The following limiting relation is valid: lim N →∞ N −| ~n | H ~α,α ,N~n ( N x ) = P ~α,α ~n ( x ) . (32) Proof. For simplicity let us consider the multi-index ~n = ( n , n ) as well as theinterval [0 , N − 1] as the support of the orthogonality measures (10). The prooffor r > 2, i.e., ~n = ( n , n , . . . , n r ) follows the same steps described below.Firstly, let us show that for N large enough the term Γ( N − N x ) / Γ( α + N − N x ) contained in (30) behaves like(1 − x ) − α N − α . (33)This estimation follows immediately from the well known asymptotic relationfor the gamma-function [9]Γ( z + a )Γ( z ) = z a (cid:20) O (cid:18) z (cid:19)(cid:21) , | arg z | ≤ π − δ, δ > . (34)Second, for N large enough we will prove that the expression N −| ~n | r Y k =1 Γ( N x + 1)Γ( α i + N x + 1) ∇ n i Γ( α i + n i + N x + 1)Γ( N x + 1) ! Γ( N + α − N x )Γ( N − | ~n | − N x ) , (35)behaves like N α Y i =1 x − α i d n i dx n i x α i + n i ! (1 − x ) α + | ~n | . (36)Indeed, using the relation for the n th backward difference ▽ n y ( x ) = n X k =0 ( − k n ! k !( n − k )! y ( x − k ) = n X k =0 ( − n ) k k ! y ( x − k ) , (37)the expression (35) becomes N −| ~n | Γ( N x + 1)Γ( N x + α + 1) n X j =0 n X k =0 ( − n ) j j ! ( − n ) k k ! Γ( N ( x − k ) + α + n + 1)Γ( N ( x − k ) + 1)Γ( N ( x − k ) + 1)Γ( N ( x − j − k ) + α + n + 1)Γ( N + α − N ( x − j − k ))Γ( N ( x − k ) + α + 1)Γ( N ( x − j − k ) + 1) N − | ~n | − N ( x − j − k )) . (38)Hence, taking into account (34) the above expression (38) can be rewritten as( N x ) − α n X j =0 n X k =0 ( − n ) j j ! ( − n ) k k ! ( N ( x − k )) α + n ( N ( x − k )) − α ( N ( s − j − k )) α + n ( N − N ( x − j − k )) α + | ~n | + O (cid:18) N (cid:19) . (39)13sing again (37), but this time to express (39) in terms of the backward differ-ence operators, i.e., N α Y i =1 x − α i ∇ n i x α i + n i ! (1 − x ) α + | ~n | + O (cid:18) N (cid:19) . (40)Now, according to (31) one can express the backward difference operators in-volved in (40) in terms of the ordinary derivatives. Thus, for N large enoughone verifies that indeed (35) behaves like (36).Finally, from (30) and the above estimations (33) and (36), the propositionholds.Recall that the multiple Jacobi polynomials are given explicitly by theRodrigues-type formula [12] P ~α,α ~n ( x ) = ( − | ~n | (1 − x ) − α Q ri =1 ( | ~n | + α + α i + 1) n i r Y i =1 x − α i d n i dx n i x α i + n i ! (1 − x ) α + | ~n | . (41)In fact, P ~α,α ~n ( x ) verifies the following orthogonality conditions Z P ~α,α ~n ( x ) x α i (1 − x ) α x k dx = 0 , k = 0 , , . . . , n i − , i = 1 , , . . . , r. Regarding the differential equation that these polynomials satisfy we referto [5]. Observe that this case constitutes a special limiting case of (20). To the best of our knowledge there is not in the literature any other high order q -difference equation different from (20) having multiple orthogonal polynomials-with q -discrete orthogonality- as eigenfunctions. Furthermore, the q -differenceequation obtained here is quite versatile since other difference and differen-tial equations can be simply obtained as a limiting case. For instance, thedifference and differential equations for the multiple Hahn and Jacobi polyno-mials, respectively are examples of these limiting cases. Also the well knownhypergeometric-type difference equation for Hahn polynomials as well as the hy-pergeometric equation for Jacobi polynomials are particular cases when r = 1.However, more general situations demand our attention. Firstly, the multipleorthogonal polynomials on the lattice x ( s ) = c q s + c q − s + c , where c , c and c are constants independent on s must be considered in the same fashionthat here. Finally, more general systems of measures such that under certain re-strictions on their elements (measures) one can recover the q -difference equation(20) must be analyzed. In this direction, the Askey-Wilson multiple orthogonalpolynomials could be an interesting challenge to be considered.In closing, this paper outlines the important points and techniques to befollowed in future investigations aimed to deduce the high order difference equa-tions for q -Charlier, q -Kravchuk and q -Meixner multiple orthogonal polynomials.14 eferences [1] M. Abramowitz, I. A. 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