On analytic properties of Meixner-Sobolev orthogonal polynomials of higher order difference operators
aa r X i v : . [ m a t h . C A ] A p r On analytic properties of Meixner-Sobolev orthogonalpolynomials of higher order difference operators
A. Soria-Lorente , Jean-Marie Vilaire Facultad de Ciencias T´ecnicas, Universidad de Granma,Km. 17.5 de la carretera de Bayano-Manzanillo, Bayamo, [email protected], [email protected] Institut des Sciences, des Technologies et des ´EtudesAvanc´ees d’Ha¨ıti (ISTEAH), Port-au-Prince, [email protected] (April 9, 2018)
Abstract
In this contribution we consider sequences of monic polynomials orthogonal with respectto Sobolev-type inner product h p, q i λ = X x ≥ p ( x ) q ( x ) µ x Γ ( γ + x )Γ ( γ ) Γ ( x + 1) + λ D ji p ( α ) D ji q ( α ) , i = 1 , , where λ ∈ R + , j ∈ N , α ≤ γ >
0, 0 < µ < D , D denote the forward andbackward difference operators, respectively. We derive an explicit representation for thesepolynomials. The ladder operators associated with these polynomials are obtained, and thelinear difference equation of second order is also given. In addition, for these polynomials wederive a (2 j + 3)-term recurrence relation. Finally, we find the Mehler-Heine type formulafor the particular case α = 0. AMS Subject Classification:
Primary 33C47; Secondary 39A12
Key Words and Phrases:
Meixner polynomials, Meixner-Sobolev orthogonal polyno-mials, Discrete kernel polynomials.
The study of the sequences of polynomials orthogonal with respect to the Sobolev innerproduct h p, q i λ = Z Ω p ( x ) q ( x ) ρ ( x ) dx + λp ( j ) ( c ) q ( j ) ( c ) , (1)where Ω = R , λ ∈ R + , c ∈ R and ρ ( x ) is a weight function being introduced by Marcell´an andRonveaux in [24]. In this work, they obtained the second order linear differential equation thatsuch polynomials satisfy. Thenceforwards, many researchers have achieved remarkable resultsin this area. For example, the zero distribution of the polynomials orthogonal with respectto the inner product (1) when Ω = (0 , ∞ ) and c = 0 was studied by Meijer in [25]. In [16],the authors analyzed the asymptotic behavior of polynomials orthogonal with respect to inner1product expressed in (1) when Ω = (0 , ∞ ), λ ∈ R + , c = 0 and ρ ( x ) = x α e − x with α > − j = 1 is considered in [18]. Recently, in [21]the authors established the asymptotic behavior of sequences of polynomials orthogonal withrespect to the inner product h p, q i S,n = Z − p ( x ) q ( x ) (1 − x ) α (1 + x ) β dx + λ n p ( j ) (1) q ( j ) (1) , where α, β > − j ≥ n →∞ λ n n γ = λ > , with γ a fixed real number. They also deduced the Mehler–Heine type formula for these poly-nomials.Indeed, the study of orthogonal polynomials with respect to the inner product involvingdifferences instead of derivatives h p, q i λ = Z R p ( x ) q ( x ) dψ ( x ) + λ ∆ p ( c )∆ q ( c ) , (2)where λ ∈ R + , c ∈ R and ψ is a distribution function with infinite spectrum, was introducedby H. Bavinck in [11, 12]. Moreover, in these works Bavinck obtained algebraic properties andsome results connected to the location of the zeros of the orthogonal polynomials with respectto the inner product (2). On the other hand, in [12] he proved that the orthogonal polynomialswith respect to inner product defined in equation (2) satisfy a five term recurrence relation.Furthermore, in [13] he considered the inner product h p, q i = (1 − µ ) γ X x ≥ p ( x ) q ( x ) µ x Γ ( γ + x )Γ ( γ ) Γ ( x + 1) + λp (0) q (0) , p, q ∈ P , (3)where γ >
0, 0 < µ < λ > P denote the linear space of all polynomials with realcoefficients. Here, he obtained a second order difference equation with polynomial coefficients,which the orthogonal polynomials with respect to (3) satisfy. Then, in [14] he showed that theSobolev type Meixer polynomials orthogonal with respect to the inner product h p, q i = (1 − µ ) γ X x ≥ p ( x ) q ( x ) µ x Γ ( γ + x )Γ ( γ ) Γ ( x + 1) + M p (0) q (0) + N ∆ p (0)∆ q (0) , where γ >
0, 0 < µ <
M, N ≥
0, are eigenfunctions of a difference operator. Other resultsconnected with the Sobolev type Meixer polynomials can be found in [7, 8, 9, 22, 26].In this contribution we will focus our attention on the sequence { Q i,λn } n ≥ of monic orthogonalpolynomials with respect to the following inner product on P involving differences of higher order h p, q i λ = X x ≥ p ( x ) q ( x ) µ x Γ ( γ + x )Γ ( γ ) Γ ( x + 1) + λ D ji p ( α ) D ji q ( α ) , i = 1 , , (4)where λ ∈ R + , j ∈ N , α ≤ γ >
0, 0 < µ < D i with i = 1 ,
2, denotes the forwarddifference operator D ≡ ∆ and backward difference operator D ≡ ∇ defined by D f ( x ) = f ( x + 1) − f ( x ) and D f ( x ) = f ( x ) − f ( x − PRELIMINARY RESULTS j + 3)-termrecurrence relation that the studied polynomials satisfy.The structure of the paper is the following: In Section 2, we introduce some preliminaryresults about Meixner polynomials which will be very useful in the analysis presented. In Section3, we obtain the connection formula between the Meixner polynomials and the polynomialsorthogonal with respect to (4), as well as we deduce the hypergeometric representation of suchpolynomials. In Section 4, we find the ladder (creation and annihilation) operators for thesequence of orthogonal polynomials of Sobolev type. As a consequence, the second order lineardifference equation associated with them is deduced. And on the other hand, in Section 5,we determine the (2 j + 3)-term recurrence relation that these polynomials satisfy. Finally, inSection 6, we determine the Mehler-Heine type formula for the especial case α = 0. Indeed, thetechniques used in Sections 3, 4 and 5 are based on those used in [2, 19, 20], respectively. Let { M γ,µn } n ≥ be the sequence of monic Meixner polynomials [1, 28], orthogonal with respectto the inner product on P h p, q i = X x ≥ p ( x ) q ( x ) µ x Γ ( γ + x )Γ ( γ ) Γ ( x + 1) , γ > , < µ < , which can be explicitly given by M γ,µn ( x ) = ( γ ) n (cid:18) µµ − (cid:19) n F − n, − x − µ − γ , (5)where γ > < µ <
1, please refer to [1, 28, 20]. Here, r F s denotes the ordinaryhypergeometric series defined by r F s a , . . . , a r xb , . . . , b s = X k ≥ ( a , . . . , a r ) k ( b , . . . , b s ) k x k k ! , (6)where ( a , . . . , a r ) k := Y ≤ i ≤ r ( a i ) k , and ( · ) n denotes the Pochhammer symbol [10, 17], also called the shifted factorial, defined by( x ) n = Y ≤ j ≤ n − ( x + j ) , n ≥ , ( x ) = 1 . Moreover, { a i } ri =1 and { b j } sj =1 are complex numbers subject to the condition that b j = − n with n ∈ N \ { } for j = 1 , , . . . , s . PRELIMINARY RESULTS Theorem 1 [6, p. 62] The series ( ) converges absolutelty of all x if r ≤ s and for | x | < if r = s + 1 , and it diverges for all x = 0 if r > s + 1 and the series does not terminate. Next, we summarize some basic properties of Meixner orthogonal polynomials to be used inthe sequel.
Proposition 1
Let { M γ,µn } n ≥ be the classical Meixner sequence monic orthogonal polynomials.The following statements hold.1. Three term recurrence relation xM γ,µn ( x ) = M γ,µn +1 ( x ) + α γ,µn M γ,µn ( x ) + β γ,µn M γ,µn − ( x ) , n ≥ , (7) where α γ,µn = n (1 + µ ) + µγ − µ , β γ,µn = nµ ( n + γ − − µ ) , with initial conditions M γ,µ − ( x ) = 0 , and M γ,µ ( x ) = 1 .2. Structure relations. For every n ∈ N , ( x + γδ i, ) D i M γ,µn ( x ) = nM γ,µn ( x ) + nµ δ i, ( n + γ − − µ M γ,µn − ( x ) , (8) where i = 1 , , and δ i,j denotes the Kronecker delta function.3. Squared norm. For every n ∈ N , k M γ,µn k = n ! ( γ ) n µ n (1 − µ ) γ +2 n . (9)
4. Orthogonality relation. For a < X x ≥ M γ,µm ( x ) M γ,µn ( x ) µ x Γ ( γ + x )Γ ( γ ) Γ ( x + 1) = k M γ,µn k δ m,n , where by δ i,j we denote the Kronecker delta function.5. Value in the initial extreme of the orthogonality interval, M γ,µn (0) = ( γ ) n (cid:18) µµ − (cid:19) n . (10)
6. Forward and backward shift operators. D ki M γ,µn ( x ) = [ n ] k M γ + k,µn − k ( x − δ i, k ) , i = 1 , . (11) Here, [ · ] n denotes the Falling Factorial [10, p. 6], defined by [ x ] n = ( − n ( − x ) n , n ≥ , [ x ] = 1 . PRELIMINARY RESULTS
7. Mehler–Heine type formula [15, eq. 35] lim n →∞ ( µ − n M γ,µn ( x )Γ ( n − x ) = 1(1 − µ ) γ + x Γ ( − x ) . (12)Furthermore, we denote the n -th reproducing kernel by K n ( x, y ) = X ≤ k ≤ n M γ,µk ( x ) M γ,µk ( y ) (cid:13)(cid:13) M γ,µk (cid:13)(cid:13) . (13)Then, for all n ∈ N , K n ( x, y ) = 1 k M γ,µn k M γ,µn +1 ( x ) M γ,µn ( y ) − M γ,µn +1 ( y ) M γ,µn ( x ) x − y . (14)Provided D ki f ( x ) = D i (cid:16) D k − i f ( x ) (cid:17) with i = 1 ,
2, for the partial finite difference of K n ( x, y ) wewill use the following notation K ( l,j ) n ( x, y ) = D li,x (cid:16) D ji,y K n ( x, y ) (cid:17) = X ≤ k ≤ n D ji M γ,µk ( x ) D ji M γ,µk ( y ) (cid:13)(cid:13) M γ,µk (cid:13)(cid:13) . (15)For abbreviation we denote h x i ni = ( [ x ] n , i = 1 , ( x ) n , i = 2 . Proposition 2
Let { M γ,µn } n ≥ be the sequence of monic Meixner orthogonal polynomials. Then,the following statement holds, for all n ∈ N , K (0 ,j ) n − ( x, y ) = j ! (cid:13)(cid:13) M γ,µn − (cid:13)(cid:13) h x − y i j +1 i × (cid:16) M γ,µn ( x ) X ≤ k ≤ j D ki M γ,µn − ( y ) k ! h x − y i ki − M γ,µn − ( x ) X ≤ k ≤ j D ki M γ,µn ( y ) k ! h x − y i ki (cid:17) , i = 1 , . (16) Proof.
In fact, applying the j -th finite difference to (13) with respect to y we obtain K (0 ,j ) n − ( x, y ) = 1 (cid:13)(cid:13) M γ,µn − (cid:13)(cid:13) " M γ,µn ( x ) D ji,y (cid:18) M γ,µn − ( y ) x − y (cid:19) − M γ,µn − ( x ) D ji,y (cid:18) M γ,µn ( y ) x − y (cid:19) . (17)Using a analogue of the Leibnitz’s rule [1, 3] D ni [ f ( x ) g ( x )] = X ≤ k ≤ n (cid:18) nk (cid:19) D ki [ f ( x )] D n − ki [ g ( x ± k )] , i = 1 , , (18) PRELIMINARY RESULTS D ni,y (cid:18) x − y (cid:19) = n ! h x − y i n +1 i , we deduce D ji,y (cid:18) M γ,µn − ( y ) x − y (cid:19) = X ≤ k ≤ j (cid:18) jk (cid:19) D ki (cid:2) M γ,µn − ( y ) (cid:3) D j − ki,y (cid:18) x − y ∓ k (cid:19) = X ≤ k ≤ j D ki M γ,µn − ( y ) k ! j ! h x − y ∓ k i j +1 − ki . Since h x i ni h x i ki = h x ∓ k i n − ki if n ≥ k, we deduce h x − y ∓ k i j +1 − ki = h x − y i j +1 i h x − y i ki . Thus, D ji,y (cid:18) M γ,µn − ( y ) x − y (cid:19) = j ! h x − y i j +1 i X ≤ k ≤ j D ki M γ,µn − ( y ) k ! h x − y i ki . Therefore, from the above and (17) we get (16). Evidently K (0 ,j ) n − ( x, y ) = j ! (cid:13)(cid:13) M γ,µn − (cid:13)(cid:13) h x − y i j +1 i (cid:2) M γ,µn ( x ) M j (cid:0) x, y, M γ,µn − (cid:1) − M γ,µn − ( x ) M j ( x, y, M γ,µn ) (cid:3) , where M j (cid:0) x, y, M γ,µn − (cid:1) and M j ( x, y, M γ,µn ) denote the Taylor polynomials of degree j [5, 4],around the point x = y , of the polynomials M γ,µn − ( x ) y M γ,µn ( x ), respectively. Proposition 3
Let { M γ,µn } n ≥ be the sequence of monic Meixner orthogonal polynomials. Then,the following statement holds, for all n ∈ N , K ( j,j ) n − (0 ,
0) = j ! (1 − µ ) γ +2 j µ j ( γ ) j X ≤ k ≤ n − j − ( j + 1) k ( γ + j ) k (1) k µ k k ! . Proof.
In fact, having (10)-(13) into account, as well as( x ) n ( x ) m = ( x + m ) n − m , if n ≥ m, (19)we deduce K ( j,j ) n − (0 ,
0) = (1 − µ ) γ +2 j µ j ( γ ) j X ≤ k ≤ n − [ k ] j ( γ ) k µ k k != (1 − µ ) γ +2 j µ j − ( γ ) j X ≤ k ≤ n − [ k + 1] j ( γ ) k +1 µ k ( k + 1)! . CONNECTION FORMULA AND HYPERGEOMETRIC REPRESENTATION OF M I,λN ( X )7Thus, applying again (19) and using the identity a + ka = ( a + 1) k ( a ) k , (20)we arrived to the desired result then a straightforward by tedious verification. Corollary 1
Let { M γ,µn } n ≥ be the sequence of monic Meixner orthogonal polynomials. Then,the following limit lim n →∞ K ( j,j ) n − (0 ,
0) = j ! (1 − µ ) γ +2 j µ j ( γ ) j F j + 1 , j + γ µ , hold. M i,λn ( x ) In this section, we first express the Meixner-Sobolev type orthogonal polynomials M i,λn ( x ) interms of the monic Meixner orthogonal polynomials M γ,µn ( x ) and the Kernel polynomial (16).Taking into account the Fourier expansion we have M i,λn ( x ) = M γ,µn ( x ) + X ≤ k ≤ n − a n,k M γ,µk ( x ) . Then, from the properties of orthogonality of M γ,µn and M i,λn respectively, we arrived to a n,k = − λ D ji M i,λn ( α ) D ji M γ,µk ( α ) (cid:13)(cid:13) M γ,µk (cid:13)(cid:13) , ≤ k ≤ n − . Thus, we get M i,λn ( x ) = M γ,µn ( x ) − λ D ji M i,λn ( α ) K (0 ,j ) n − ( x, α ) , After some manipulations, we deduce D ji M i,λn ( α ) = D ji M γ,µn ( α )1 + λ K ( j,j ) n − ( α, α ) . (21)Consequently, from the previous we have M i,λn ( x ) = M γ,µn ( x ) − λ D ji M γ,µn ( α )1 + λ K ( j,j ) n − ( α, α ) K (0 ,j ) n − ( x, α ) , i = 1 , . (22)Next we will focus our attention in the representation of M i,λn ( x ) as hypergeometric functions.Clearly, from (16) and (22) we have M i,λn ( x ) = A ( i )1 ,n ( x ) M γ,µn ( x ) + B ( i )1 ,n ( x ) M γ,µn − ( x ) , (23) CONNECTION FORMULA AND HYPERGEOMETRIC REPRESENTATION OF M I,λN ( X )8where A ( i )1 ,n ( x ) = 1 + 1 h x − α i j +1 i X ≤ k ≤ j a ( k ) i,n h x − α i ki , (24)and B ( i )1 ,n ( x ) = 1 h x − α i j +1 i X ≤ k ≤ j b ( k ) i,n h x − α i ki , (25)with a ( k ) i,n = − λj ! D ji M γ,µn ( α ) D ki M γ,µn − ( α ) (cid:13)(cid:13) M γ,µn − (cid:13)(cid:13) (cid:16) λ K ( j,j ) n − ( α, α ) (cid:17) k ! , (26)and b ( k ) i,n = λj ! D ji M γ,µn ( α ) D ki M γ,µn ( α ) (cid:13)(cid:13) M γ,µn − (cid:13)(cid:13) (cid:16) λ K ( j,j ) n − ( α, α ) (cid:17) k ! . (27) Theorem 2
The monic Meixner-Sobolev orthogonal polynomials M i,λn ( x ) have the followinghypergeometric representation for i = 1 , , M i,λn ( x ) = ( γ ) n − (cid:18) µµ − (cid:19) n − h ( i ) n ( x ) F − n, − x, f ( i ) n ( x ) 1 − µ − γ, f ( i ) n ( x ) − , (28) where f ( i ) n ( x ) is given in ( ) and h ( i ) n ( x ) = − (cid:16) µ ( γ + n −
1) (1 − µ ) − A ( i )1 ,n ( x ) − B ( i )1 ,n ( x ) (cid:17) . Proof.
In fact, having into account( − x ) k = 0 , if x < k. as well as (5) and (23) we deduce M i,λn ( x ) = ( γ ) n (cid:18) µµ − (cid:19) n A ( i )1 ,n ( x ) X ≤ k ≤ n ( − n ) k ( − x ) k ( γ ) k (cid:0) − µ − (cid:1) k k !+ ( γ ) n − (cid:18) µµ − (cid:19) n − B ( i )1 ,n ( x ) X ≤ k ≤ n − (1 − n ) k ( − x ) k ( γ ) k (cid:0) − µ − (cid:1) k k ! . Then, using the identity a + ka = ( a + 1) k ( a ) k , we get M i,λn ( x ) = ( γ ) n (cid:18) µµ − (cid:19) n A ( i )1 ,n ( x ) X ≤ k ≤ n ( − n ) k ( − x ) k ( γ ) k (cid:0) − µ − (cid:1) k k !+ ( γ ) n − (cid:18) µµ − (cid:19) n − B ( i )1 ,n ( x ) n X ≤ k ≤ n ( n − k ) ( − n ) k ( − x ) k ( γ ) k (cid:0) − µ − (cid:1) k k ! . LINEAR DIFFERENCE EQUATION OF SECOND ORDER M i,λn ( x ) = ( γ ) n − (cid:18) µµ − (cid:19) n − X ≤ k ≤ n g ( i ) n ( x ) ( − n ) k ( − x ) k ( γ ) k (cid:0) − µ − (cid:1) k k ! , where g ( i ) n ( x ) = − B ( i )1 ,n ( x ) n (cid:16) f ( i ) n ( x ) + k − (cid:17) , with f ( i ) n ( x ) = nµ ( γ + n −
1) (1 − µ ) − A ( i )1 ,n ( x ) B ( i )1 ,n ( x ) − n + 1 , (29)A trivial verification shows that g ( i ) n ( x ) = − B ( i )1 ,n ( x ) (cid:16) f ( i ) n ( x ) − (cid:17) n (cid:16) f ( i ) n ( x ) (cid:17) k (cid:16) f ( i ) n ( x ) − (cid:17) k = − (cid:16) µ ( γ + n −
1) (1 − µ ) − A ( i )1 ,n ( x ) − B ( i )1 ,n ( x ) (cid:17) (cid:16) f ( i ) n ( x ) (cid:17) k (cid:16) f ( i ) n ( x ) − (cid:17) k . Therefore M i,λn ( x ) = − (cid:16) µ ( γ + n −
1) (1 − µ ) − A ( i )1 ,n ( x ) − B ( i )1 ,n ( x ) (cid:17) × ( γ ) n − (cid:18) µµ − (cid:19) n − X ≤ k ≤ n ( − n ) k ( − x ) k (cid:16) f ( i ) n ( x ) (cid:17) k ( γ ) k (cid:16) f ( i ) n ( x ) − (cid:17) k (cid:0) − µ − (cid:1) k k ! , which coincides with (28). This completes the proof. In this section, we will obtain a second order linear difference equation that the sequence ofmonic Meixner-Sobolev type orthogonal polynomials {M i,λn } n ≥ satisfies. In order to do that,we will find the ladder (creation and annihilation) operators , using the connection formula (23),the three term recurrence relation (7) satisfied by { M γ,µn } n ≥ and the structure relation (8).From (23) and recurrence relation (7) we deduce the following resut M i,λn − ( x ) = A ( i )2 ,n ( x ) M γ,µn ( x ) + B ( i )2 ,n ( x ) M γ,µn − ( x ) , (30)where A ( i )2 ,n ( x ) = − B ( i )1 ,n − ( x ) β γ,µn − , and B ( i )2 ,n ( x ) = A ( i )1 ,n − ( x ) + A ( i )2 ,n ( x ) (cid:0) α γ,µn − − x (cid:1) . LINEAR DIFFERENCE EQUATION OF SECOND ORDER D i operator to (23) and using (18) we have D i M i,λn ( x ) = M γ,µn ( x ) D i A ( i )1 ,n ( x ) + A ( i )1 ,n ( x ± D i M γ,µn ( x ) M γ,µn − ( x ) D i B ( i )1 ,n ( x ) + B ( i )1 ,n ( x ± D i M γ,µn − ( x ) . Then, multiplying the previous expression by ( x + γδ i, ) and using the structure relation (8) aswell as the recurrence relation (7) we deduce( x + γδ i, ) D i M i,λn ( x ) = C ( i )1 ,n ( x ) M γ,µn ( x ) + D ( i )1 ,n ( x ) M γ,µn − ( x ) , (31)and ( x + γδ i, ) D i M i,λn − ( x ) = C ( i )2 ,n ( x ) M γ,µn ( x ) + D ( i )2 ,n ( x ) M γ,µn − ( x ) , (32)respectively, where C ( i )1 ,n ( x ) = ( x + γδ i, ) D i A ( i )1 ,n ( x ) + nA ( i )1 ,n ( x ± − ( n −
1) ( n + γ − µ δ i, B ( i )1 ,n ( x ± β γ,µn − (1 − µ ) ,D ( i )1 ,n ( x ) = ( x + γδ i, ) D i B ( i )1 ,n ( x ) + ( n − B ( i )1 ,n ( x ± n −
1) ( n + γ − (cid:0) x − α γ,µn − (cid:1) µ δ i, B ( i )1 ,n ( x ± β γ,µn − (1 − µ ) + n ( n + γ − µ δ i, A ( i )1 ,n ( x ± − µ , and C ( i )2 ,n ( x ) = − D ( i )1 ,n − ( x ) β γ,µn − ,D ( i )2 ,n ( x ) = C ( i )1 ,n − ( x ) + C ( i )2 ,n ( x ) (cid:0) α γ,µn − − x (cid:1) . Moreover, from (23)-(30) we have M γ,µn ( x ) = B ( i )2 ,n ( x ) M i,λn ( x ) − B ( i )1 ,n ( x ) M i,λn − ( x )Θ n ( x ; i ) , and M γ,µn − ( x ) = A ( i )1 ,n ( x ) M i,λn − ( x ) − A ( i )2 ,n ( x ) M i,λn ( x )Θ n ( x ; i ) , where Θ n ( x ; i ) = det A ( i )1 ,n ( x ) B ( i )1 ,n ( x ) A ( i )2 ,n ( x ) B ( i )2 ,n ( x ) ! . Thus, replacing the above in (31)-(32) we conclude˜Θ n ( x ; i ) D i M i,λn ( x ) + Λ (1)2 ,n ( x ; i ) M i,λn ( x ) = Λ (1)1 ,n ( x ; i ) M i,λn − ( x ) . and ˜Θ n ( x ; i ) D i M i,λn − ( x ) + Λ (2)1 ,n ( x ; i ) M i,λn − ( x ) = Λ (2)2 ,n ( x ; i ) M i,λn ( x ) , LINEAR DIFFERENCE EQUATION OF SECOND ORDER n ( x ; i ) = ( x + γδ i, ) Θ n ( x ; i ) , (33)and Λ ( k ) j,n ( x ; i ) = ( − k det C ( i ) k,n ( x ) A ( i ) j,n ( x ) D ( i ) k,n ( x ) B ( i ) j,n ( x ) ! , j = 1 , , k = 1 , . (34) Proposition 4
Let n M i,λn o n ≥ be the sequence of monic Meixner-Sobolev orthogonal polyno-mials defined by ( ) and let I be the identity operator. Then, the ladder (destruction andcreation) operators a , a † are defined by a = ˜Θ n ( x ; i ) D i + Λ (1)2 ,n ( x ; i ) I, a † = ˜Θ n ( x ; i ) D i + Λ (2)1 ,n ( x ; i ) I, which verify a (cid:16) M i,λn ( x ) (cid:17) = Λ (1)1 ,n ( x ; i ) M i,λn − ( x ) , (35) a † (cid:16) M i,λn − ( x ) (cid:17) = Λ (2)2 ,n ( x ; i ) M i,λn ( x ) , where ˜Θ n ( x ; i ) and Λ ( k ) j,n ( x ; i ) with i, j, k = 1 , are given in ( ) - ( ) . Theorem 3
Let n M i,λn o n ≥ be the sequence of monic polynomials orthogonal with respect tothe inner product ( ) . Then, the following statement holds. For all n ≥ F n ( x ; i ) D i M i,λn ( x ) + G n ( x ; i ) D i M i,λn ( x ) + H n ( x ; i ) M λn ( x ) ( x ) = 0 , (36) where F n ( x ; i ) = ˜Θ n ( x ; i ) ˜Θ n ( x ± i )Λ (1)1 ,n ( x ± i ) , G n ( x ; i ) = ˜Θ n ( x ; i ) D i ˜Θ n ( x ; i )Λ (1)1 ,n ( x ± i ) − ˜Θ n ( x ; i ) D i Λ (1)1 ,n ( x ; i )Λ (1)1 ,n ( x ; i ) Λ (1)1 ,n ( x ± i )+ ˜Θ n ( x ; i ) Λ (1)2 ,n ( x ± i )Λ (1)1 ,n ( x ± i ) + ˜Θ n ( x ; i ) Λ (2)1 ,n ( x ; i )Λ (1)1 ,n ( x ; i ) . and H n ( x ; i ) = ˜Θ n ( x ; i ) D i Λ (1)2 ,n ( x ; i )Λ (1)1 ,n ( x ± i ) − ˜Θ n ( x ; i ) Λ (1)2 ,n ( x ; i ) D i Λ (1)1 ,n ( x ; i )Λ (1)1 ,n ( x ; i ) Λ (1)1 ,n ( x ± i )+ Λ (2)1 ,n ( x ; i ) Λ (1)2 ,n ( x ; i )Λ (1)1 ,n ( x ; i ) − Λ (2)2 ,n ( x ; i ) . where ˜Θ n ( x ; i ) and Λ ( k ) j,n ( x ; i ) with i, j, k = 1 , are given in ( ) - ( ) . LINEAR DIFFERENCE EQUATION OF SECOND ORDER Proof.
From (35) we have 1Λ (1)1 ,n ( x ; i ) a (cid:16) M i,λn ( x ) (cid:17) = M i,λn − ( x ) . Next, applying the operator a † to both members of the previous expression, we get a † " (1)1 ,n ( x ; i ) a (cid:16) M i,λn ( x ) (cid:17) = Λ (2)2 ,n ( x ; i ) M i,λn ( x ) . Thus ˜Θ n ( x ; i ) D i " (1)1 ,n ( x ; i ) a (cid:16) M i,λn ( x ) (cid:17) + Λ (2)1 ,n ( x ; i )Λ (1)1 ,n ( x ; i ) a (cid:16) M i,λn ( x ) (cid:17) = Λ (2)2 ,n ( x ; i ) M i,λn ( x ) . On the other hand, we have D i " (1)1 ,n ( x ; i ) a (cid:16) M i,λn ( x ) (cid:17) = D i " (1)1 ,n ( x ; i ) (cid:16) ˜Θ n ( x ; i ) D i M i,λn ( x ) + Λ (1)2 ,n ( x ; i ) M i,λn ( x ) (cid:17) = D i ˜Θ n ( x ; i ) D i M i,λn ( x )Λ (1)1 ,n ( x ; i ) + D i Λ (1)2 ,n ( x ; i ) M i,λn ( x )Λ (1)1 ,n ( x ; i ) . Then, using D i (cid:26) f ( x ) g ( x ) (cid:27) = g ( x ) D i f ( x ) − f ( x ) D i g ( x ) g ( x ) g ( x ± , we deduce D i " (1)1 ,n ( x ; i ) a (cid:16) M i,λn ( x ) (cid:17) == Λ (1)1 ,n ( x ; i ) D i (cid:16) ˜Θ n ( x ; i ) D i M i,λn ( x ) (cid:17) Λ (1)1 ,n ( x ; i ) Λ (1)1 ,n ( x ± i ) − ˜Θ n ( x ; i ) D i M i,λn ( x ) D i Λ (1)1 ,n ( x ; i )Λ (1)1 ,n ( x ; i ) Λ (1)1 ,n ( x ± i )+ Λ (1)1 ,n ( x ; i ) D i (cid:16) Λ (1)2 ,n ( x ; i ) M i,λn ( x ) (cid:17) Λ (1)1 ,n ( x ; i ) Λ (1)1 ,n ( x ± i ) − Λ (1)2 ,n ( x ; i ) M i,λn ( x ) D i Λ (1)1 ,n ( x ; i )Λ (1)1 ,n ( x ; i ) Λ (1)1 ,n ( x ± i ) . THE (2 J + 3) -TERM RECURRENCE RELATION n ( x ; i ) ˜Θ n ( x ± i )Λ (1)1 ,n ( x ± i ) D i M i,λn ( x ) + " ˜Θ n ( x ; i ) D i ˜Θ n ( x ; i )Λ (1)1 ,n ( x ± i ) − ˜Θ n ( x ; i ) D i Λ (1)1 ,n ( x ; i )Λ (1)1 ,n ( x ; i ) Λ (1)1 ,n ( x ± i )+ ˜Θ n ( x ; i ) Λ (1)2 ,n ( x ± i )Λ (1)1 ,n ( x ± i ) + ˜Θ n ( x ; i ) Λ (2)1 ,n ( x ; i )Λ (1)1 ,n ( x ; i ) D i M i,λn ( x )+ " ˜Θ n ( x ; i ) D i Λ (1)2 ,n ( x ; i )Λ (1)1 ,n ( x ± i ) − ˜Θ n ( x ; i ) Λ (1)2 ,n ( x ; i ) D i Λ (1)1 ,n ( x ; i )Λ (1)1 ,n ( x ; i ) Λ (1)1 ,n ( x ± i )+ Λ (2)1 ,n ( x ; i ) Λ (1)2 ,n ( x ; i )Λ (1)1 ,n ( x ; i ) − Λ (2)2 ,n ( x ; i ) M i,λn ( x ) = 0 , which coincide with (36). (2 j + 3) -term recurrence relation In this section we find the (2 j + 3)-term recurrence relation that the sequence of monic Meixner-Sobolev type orthogonal polynomials (28) satisfies. For this purpose, we will use the remark-able fact, which is a straightforward consequence of (4), that the multiplication operator by h x − α i j +1 i is a symmetric operator with respect to such a discrete Sobolev inner product. In-deed, for any p, q ∈ P we have D h x − α i j +1 i p ( x ) , q ( x ) E λ = D p ( x ) , h x − α i j +1 i q ( x ) E λ = D h x − α i j +1 i p ( x ) , q ( x ) E (37)= D p ( x ) , h x − α i j +1 i q ( x ) E . Notice that, having the expressions (23), (24) and (25) into account, we deduce the followingresult.
Lemma 1
Let n M i,λn o n ≥ be the sequence of monic Meixner-Sobolev type orthogonal polyno-mials defined by ( ) . Then, the following holds. h x − α i j +1 i M i,λn ( x ) = A n ( x ; i ) M γ,µn ( x ) + B n ( x ; i ) M γ,µn − ( x ) , (38) where A n ( x ; i ) = h x − α i j +1 i + X ≤ k ≤ j a ( k ) i,n h x − α i ki , and B n ( x ; i ) = X ≤ k ≤ j b ( k ) i,n h x − α i ki , with a ( k ) i,n and b ( k ) i,n given in ( ) and ( ) , respectively. THE (2 J + 3) -TERM RECURRENCE RELATION Theorem 4
Let n M i,λn o n ≥ be the sequence of monic Meixner-Sobolev type orthogonal polyno-mials defined by ( ) . Then, for every λ ∈ R + , j ∈ N , α ≤ , γ > and < µ < the norm ofthese polynomials, orthogonal with respect to ( ) is (cid:13)(cid:13)(cid:13) M i,λn (cid:13)(cid:13)(cid:13) λ = k M γ,µn k + b ( j ) i,n (cid:13)(cid:13) M γ,µn − (cid:13)(cid:13) , (39) where b ( j ) i,n is the polynomial coefficient defined by ( ) . Proof.
Clearly (cid:13)(cid:13)(cid:13) M i,λn (cid:13)(cid:13)(cid:13) λ = D M i,λn ( x ) , h x − α i j +1 i π n − j − ( x ) E λ , for every monic polynomials π n − j − of degree n − j −
1. From (37) we have D M i,λn ( x ) , h x − α i j +1 i π n − j − ( x ) E λ = D h x − α i j +1 i M i,λn ( x ) , π n − j − ( x ) E λ = D h x − α i j +1 i M i,λn ( x ) , π n − j − ( x ) E . Next we use the connection formula (38).Taking into account that A n ( x ; i ) is a monic polynomialof degree exactly j + 1 and B n ( x ; i ) is a polynomial of degree exactly j with the leandingcoefficient b ( j ) i,n we deduce (cid:13)(cid:13)(cid:13) M i,λn (cid:13)(cid:13)(cid:13) λ = hA n ( x ; i ) M γ,µn ( x ) , π n − j − ( x ) i + (cid:10) B n ( x ; i ) M γ,µn − ( x ) , π n − j − ( x ) (cid:11) , = h M γ,µn ( x ) , x n i + b ( j ) i,n (cid:10) M γ,µn − ( x ) , x n − (cid:11) , which coincides with (39). Theorem 5 ( (2 j + 3) -term recurrence relation) For every λ ∈ R + , j ∈ N , α ≤ , γ > and < µ < , the monic Meixner-Sobolev type polynomials (cid:8) M λn (cid:9) n ≥ , orthogonal with respectto ( ) satisfy the following (2 j + 3) -term recurrence relation h x − α i j +1 i M i,λn ( x ) = M i,λn + j +1 ( x ) + X n − j − ≤ k ≤ n + j c ( i ) n,k M i,λk ( x ) , where c ( i ) n,n + j = D h x − α i j +1 i M γ,µn ( x ) , M i,λn + j ( x ) E λ (cid:13)(cid:13)(cid:13) M i,λn + j (cid:13)(cid:13)(cid:13) λ + a ( j ) i,n , (40) MEHLER-HEINE TYPE FORMULA c ( i ) n,k = D h x − α i j +1 i M γ,µn ( x ) , M i,λk ( x ) E λ (cid:13)(cid:13)(cid:13) M i,λk (cid:13)(cid:13)(cid:13) λ + X k − n +1 ≤ l ≤ j a ( l ) i,n D h x − α i li M γ,µn ( x ) , M i,λk ( x ) E λ (cid:13)(cid:13)(cid:13) M i,λk (cid:13)(cid:13)(cid:13) λ + X k − n +2 ≤ l ≤ j b ( l ) i,n D h x − α i li M γ,µn − ( x ) , M i,λk ( x ) E λ (cid:13)(cid:13)(cid:13) M i,λk (cid:13)(cid:13)(cid:13) λ + a ( k − n ) i,n + b ( k − n +1) i,n , k = n − , . . . , n + j − , a ( − i,n = 0 , (41) c ( i ) n,k = D h x − α i j +1 i M i,λn ( x ) , M i,λk ( x ) E λ (cid:13)(cid:13)(cid:13) M i,λk (cid:13)(cid:13)(cid:13) λ , k = n − j + 1 , . . . , n − ,c ( i ) n,n − j = D M i,λn ( x ) , h x − α i j +1 i M γ,µn − j ( x ) E λ (cid:13)(cid:13)(cid:13) M i,λn − j (cid:13)(cid:13)(cid:13) λ + a ( j ) i,n (cid:13)(cid:13)(cid:13) M i,λn (cid:13)(cid:13)(cid:13) λ (cid:13)(cid:13)(cid:13) M i,λn − j (cid:13)(cid:13)(cid:13) λ , (42) and c ( i ) n,n − j − = (cid:13)(cid:13)(cid:13) M i,λn (cid:13)(cid:13)(cid:13) λ (cid:13)(cid:13)(cid:13) M i,λn − j − (cid:13)(cid:13)(cid:13) λ . (43) Proof.
Let consider the Fourier expansion of h x − α i j +1 i M i,λn ( x ) in terms of n M i,λn o n ≥ h x − α i j +1 i M i,λn ( x ) = M i,λn + j +1 ( x ) + X ≤ k ≤ n + j c n,k M i,λk ( x ) , Thus c ( i ) n,k = D h x − α i j +1 i M i,λn ( x ) , M i,λk ( x ) E λ (cid:13)(cid:13)(cid:13) M i,λk (cid:13)(cid:13)(cid:13) λ , k = 0 , . . . , n + j. Evidently, from the properties of orthogonality of M i,λn , we deduce that c ( i ) n,k = 0 for k =0 , . . . , n − j −
2. In order to compute (40)-(43) it is sufficient to use (37) and (38)-(27) as wellas the orthogonality conditions of (4).
The main result of this section will be to establish Mehler–Heine type formula of { Q ,λn } n ≥ forthe case where α = 0. Let us see the following result. MEHLER-HEINE TYPE FORMULA Lemma 2
For γ > , < µ < , ≤ k ≤ j and α = 0 the following limits lim n →∞ ( γ ) n [ n ] j [ n − k µ n ( n − , (44) and lim n →∞ ( γ ) n [ n ] j [ n ] k ( n + γ − µ n +1 ( n − , (45) hold. Proof.
In fact, firstly let us prove (44). Taking µ = p − with p > z ) = lim n →∞ ( n − n z ( z ) n , we deduce lim n →∞ ( γ ) n [ n ] j [ n − k µ n ( n − γ ) lim n →∞ N j,k ( n ; γ ) p n = 1Γ ( γ ) lim n →∞ n γ + k + j p n . (46)Since N j,k ( n ; γ ) = n γ [ n ] j [ n − k = n γ +1 ( n − · · · ( n − k ) ( n − k − · · · ( n − j − n γ +2 k + j − k − (cid:18) − n (cid:19) · · · (cid:18) − kn (cid:19) (cid:18) − k + 1 n (cid:19) · · · (cid:18) − j + 1 n (cid:19) ∼ n γ + k + j . Therefore, applying to (46) L’Hospital’s rule several times we obtain the desired result. In orderto prove (45) one proceeds analogously
Lemma 3
For γ > , < µ < , ≤ k ≤ j and α = 0 the following limits lim n →∞ A (1)1 ,n ( x ) = 1 and lim n →∞ B (1)1 ,n ( x ) = 0 . (47) hold. Proof.
For such purpose it is enough to checklim n →∞ a ( k )1 ,n = lim n →∞ b ( k )1 ,n = 0 . From (9), (10) and (11) we have a ( k )1 ,n = λj ! (1 − µ ) γ − ( γ ) j (cid:16) λ K ( j,j ) n − (0 , (cid:17) ( γ ) n [ n ] j [ n − k µ n ( n − γ ) k (cid:18) µ − µ (cid:19) j + k k ! . and b ( k )1 ,n = λj ! (1 − µ ) γ − ( γ ) j (cid:16) λ K ( j,j ) n − (0 , (cid:17) ( γ ) n [ n ] j [ n ] k ( n + γ − µ n +1 ( n − γ ) k (cid:18) µ − µ (cid:19) j + k k ! . Then, having into account the Theorem 1, the Corollary 1 and the previous Lemma we deduce(47).
MEHLER-HEINE TYPE FORMULA - - - Figure 1: Limit function in (48) for n = 50, (red color) left member and (green color) rightmember. Data: γ = 7, µ = 1 / λ = 10 − and j = 177. - - - Figure 2: Limit function in (48) for n = 70, (red color) left member and (green color) rightmember. Data: γ = 7, µ = 1 / λ = 10 − and j = 177. Theorem 6
Let be γ > , < µ < , ≤ k ≤ j and α = 0 . Then, we have lim n →∞ ( µ − n Q ,λn ( x )Γ ( n − x ) = 1(1 − µ ) γ + x Γ ( − x ) , (48) uniformly on compact subsets of the complex plane. Proof.
In fact, multiplying (23) by the factor ( µ − n / Γ ( n − x ) we have( µ − n Q ,λn ( x )Γ ( n − x ) = A (1)1 ,n ( x ) ( µ − n M γ,µn ( x )Γ ( n − x ) + B (1)1 ,n ( x ) ( µ − n M γ,µn − ( x )Γ ( n − x ) . Then, applying the previous Lemma as well as the (12) we arrived to the desired result.Finally, we show some graphical experiments of the limit function in (48) for several valuesof n using Mathematica software, see Figures 1–4. EFERENCES - - - Figure 3: Limit function in (48) for n = 100, (red color) left member and (green color) rightmember. Data: γ = 7, µ = 1 / λ = 10 − and j = 177. - - - Figure 4: Limit function in (48) for n = 150, (red color) left member and (green color) rightmember. Data: γ = 7, µ = 1 / λ = 10 − and j = 177. References [1] R. ´Alvarez Nodarse. Polinomios hipergeom´etricos y q -polinomios. Monograf´ıas del Sem-inario Matem´atico Garc´ıa de Galdeano, Zaragoza: Presas Universitarias de Zaragoza ,341(26), 2003.[2] R. ´Alvarez Nodarse and F. Marcell´an. Difference equation for modifications of meixnerpolynomials.
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