Generalized Little q-jacobi Polynomials as Eigensolutions of Higher-Order q-difference Operators

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1 Geeralized Little q-jacobi Polyomials as Eigesolutios of Higher-Order q-differece Operators Luc Viet Alexei Zhedaov CRM-2583 December 1998 Cetre de recherches mathématiques, Uiversité de Motréal, C.P. 6128, succ. Cetre-Ville, Motréal QC H3C 3J7, Caada. The work of L.V. is supported i part through fuds provided by NSERC (Caada) ad FCAR (Quebec), viet@crm.umotreal.ca. Doetsk Istitute for Physics ad Techology, Doetsk , Ukraie. The work of A.Zh. was supported i part through fuds provided by SCST (Ukraie) Project #2.4/197, INTAS grat ad project supported by RFBR (Russia). A.Zh. thaks the Cetre de recherches mathématiques of the Uiversité de Motréal for its hospitality, zhedaov@crm.umotreal.ca.

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3 Abstract We cosider the polyomials p (x; a, b; M) obtaied from the little q-jacobi polyomials p (x; a, b) by isertig a discrete mass M at x = 0 i the orthogoality measure. We show that for a = q j, j = 0, 1, 2,... the polyomials p (x; a, b; M) are eigesolutios of a liear q-differece operator of order 2j + 4 with polyomial coefficiets. This provides a q-aalog of results recetly obtaied for the Krall polyomials. Keywords: Krall s polyomials, little q-jacobi polyomials. Mathematics Subject Classificatios: 33D45. Résumé Nous cosidéros les polyômes p (x; a, b; M) obteus des petits q-polyômes de Jacobi p (x; a, b) e isérat ue masse discrète M à x = 0 das la mesure. Nous motros que pour a = q j, j = 0, 1, 2,... les polyômes p (x; a, b; M) sot solutios propres d u opérateur liéaire aux q-différeces d ordre 2j + 4 avec coefficiets polyomiaux. Nous offros aisi u q-aalogue des résultats obteus récemmet pour les polyômes de Krall.

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5 1 q-derivative operators ad their represetatio coefficiets Let T be the q-shift operator that acts o fuctios accordig to whith 0 < q < 1 a real umber. Obviously Itroduce the q-derivative operator (see, e.g. [2]) It is called q-derivative because its actio o moomials is with T F (x) = F (qx), (1.1) T F (x) = F (q x), = 0, ±1, ±2,.... (1.2) D q F (x) = (x(1 q)) 1 (1 T ). (1.3) D q x = [] x 1, (1.4) [] = (q 1)/(q 1) (1.5) the so-called q-umber. Moreover lim q 1 D q = D, where D is the ordiary derivative operator with respect to x: DF (x) = F (x). Cosider operators of the form L q = 2N k=0 a k (x) D k q, (1.6) where N is a fixed positive iteger ad a k (x) are polyomials i x of degrees ot exceedig k: a k (x) = k α ks x s, k = 0, 1,..., 2N. (1.7) s=0 Let us itroduce also the related operators L q = T N L q = 2N k=0 a k (q N x) T N D k q. (1.8) The operator L q is a liear combiatio of the operators T 2N, T 2N 1,..., T, T 0 = I, while the operator L q is a liear combiatio of the operators T N, T N 1,..., T 1 N, T N. For q 1 both operators L q ad L q become 2N-order differetial operators with polyomial coefficiets: lim L q = lim L q = q 1 q 1 2N k=0 a k (x) D k. (1.9) The operator L q will prove more practical i searchig for orthogoal polyomials P (x) satisfyig eigevalue equatios of the kid L q P (x) = λ P (x). (1.10) 1

6 It is kow that for N = 1, the little q-jacobi polyomials satisfy a equatio of the form (1.10) with N = 1 [4]. We wish to determie other systems of orthogoal polyomials satisfyig equatio (1.10) with N > 1. To this ed, we shall exted to q-differece operators, the method proposed i [7]. The mai idea of the method is the followig. Cosider the actio of the operator L q upo the moomials x. From (1.6) ad (1.7) we get where ad π s (q ) = α s0 + L q x = s=0 A (s) x s, (1.11) A (s) = q N(s ) [][ 1]... [ s + 1] π s (q ), (1.12) 2N s i=1 α s+i,i [ s][ s 1]... [ s i + 1] (1.13) are polyomials i z = q of degrees ot exceedig 2N s. It is clear that, moreover, A (s) = 0, s > 2N. (1.14) The coefficiets A (s) completely characterize the operator L q. We will call A (s) coefficiets of the operator L q. the represetatio Propositio 1.1 Assume that there are coefficiets A (s) expressible as i (1.12), where π s (q ) are arbitrary polyomials i q of degrees ot exceedig 2N s. Assume also that A (s) = 0, s > 2N (i.e. π s (q ) = 0 for s > 2N). The there exists a uique operator L q of the form (1.8) such that are its represetatio coefficiets. A (s) Proof. Ay polyomial π s (q ) of degree ot exceedig 2N s ca be preseted i form (1.13) with some coefficiets α ik. These coefficiets are determied usig Newto s iterpolatio formula α s+i,i = (q 1)i q si+i(i 1)/2 D i [i]! qπ s (x) i = 0, 1,..., 2N s, (1.15) x=q s, where [i]! = [1][2]... [i] is the q-factorial. Clearly, the coefficiets α ik are determied uiquely by (1.15) from the give polyomials π s (q ). Hece the operator L q is defied uiquely. 2 Basic relatios for polyomials satisfyig eigevalue equatio I this sectio we cosider the basic relatios betwee the represetatio coefficiets A (s) expasio coefficets of polyomials P (x) satisfyig eigevalue equatios. Assume that where B (s) are moic, i.e. that P (x) = s=0 ad the B (s) x s, (2.1) are expasio coefficiets. I what follows we will assume that the polyomials P (x) B (0) = 1. (2.2) 2

7 Substitutig (2.1) ito the eigevalue equatio (1.10) we arrive at the followig set of algebraic relatios s i=0 B (s i) These will be cetral i our aalysis. For s = 0, (2.3) gives A (i) s+i = λ B (s), s = 0, 1, 2,...,. (2.3) λ = A (0). (2.4) Thus from (1.12) we fid that the eigevalues λ have the followig expressio λ = q N π 0 (q ), (2.5) where π 0 (q ) is a polyomial i q of degree ot exceedig 2N. Similarly, for s = 1, (2.3) yields A (1) = Ω B (1), (2.6) where Ω = λ λ 1. From this relatio we fid that B (1) = [] Relatio (2.3) ca be rewritte i the form (λ λ s ) B (s) π 1 (q ) q N π 0 (q ) π 0 (q 1 ). (2.7) = B (s 1) A (1) s+1 + B (s 2) A (2) s A (s). (2.8) >From this relatio we ca coclude, by iductio, that the coefficiets B (s) are ratioal fuctios i q, amely that B (s) = [][ 1]... [ s + 1] Q 1,s(q ) Q 2,s (q ), (2.9) where Q 1,s (q ) is a polyomial of degree ot exceedig 2Ns s, whereas the degree of the polyomial Q 2,s (q ) does ot exceed 2Ns. The problem cosidered up to this poit of fidig the expasio coefficiets B (s) of the polyomials P (x) whe the represetatios coefficiets A (s) are give, always has a uique solutio. Propositio 2.1 Assume that the represetatio coefficiets A (s) of the operator L q satisfy the requiremet A (0) A (0) m, m. (2.10) The there exists a uique set of moic polyomials P (x), = 0, 1,... satisfyig equatio (1.10). Proof. We fid λ m from (2.4); i view of coditio (2.10), a uique B (1) is foud from (2.6). Assumig that all B (1), B (2),..., B (s 1) have thus bee foud recursively, B (s) is the determied i uambigous way owig (2.10). The polyomials P (x) are ot orthogoal i geeral. The requiremet that they form a orthogoal set implies strog additioal restrictios upo the coefficets B (s) ad A (s). Ideed, orthogoal polyomials satisfy three-term recurrece relatio of the form [1] P +1 (x) + b P (x) + u P 1 (x) = xp (x), (2.11) 3

8 with b ad u referred to as the recurrece parameters. From (2.11) ad (2.1) we get the set of relatios B (s+1) +1 B (s+1) + u B (s 1) 1 + b B (s) = 0, s = 0, 1,...,, (2.12) where it is assumed that B ( 1) = B (+1) Takig ito accout that the coefficiets B (s) propositio. = 0. Puttig s = 0 ad s = 1 i (2.12), we fid b = B (1) B (1) +1, u = B (2) B (2) +2 b B (1). (2.13) are ratioal fuctios i q we arrive at the followig Propositio 2.2 If the orthogoal polyomials P (x) are eigefuctios of the operator (1.8), their recurrece coefficiets b, u are ratioal fuctios of the argumet q. The problem of recostructig the represetatio coefficiets A (s) whe the expasio coefficiets B (s) are give, is more difficult. The coefficiets B (s) must of course be ratioal fuctios i q, sice otherwise the problem has o solutios. Assume therefore that B (s) = [][ 1]... [ s + 1] Q 1,s(q ) Q 2,s (q ), (2.14) where the degree of the polyomial Q 1,s (q ) does ot exceed 2Ms s whereas the degree of the polyomial Q 2,s (q ) does ot exceed 2Ms for some positive iteger M N. Cosider relatio (2.6) writte i the form A (1) Ω = [] Q 1,1(q ) Q 2,1 (q ). (2.15) Sice both A (1) ad Ω should be polyomials i q of degrees ot exceedig 2N, we have A (1) = q N(1 ) ρ 1 (q ) [] Q 1,1 (q ), Ω = q N(1 ) ρ 1 (q ) Q 1,2 (q ), (2.16) where ρ 1 (q ) is some polyomial of degree ot exceedig 2N 2M. The relatio (2.3), for s = 2, ca the be rewritte i the form A (2) = (λ λ 2 ) B (2) A (1) 1 B (1) = (Ω + Ω 1 ) B (2) A (1) 1 B (1). (2.17) The represetatio coefficiet A (2) is thus determied uiquely if A (1) ad Ω are kow. Assume that the coefficiets A (2), A (3),..., A (k 1) have bee determied iteratig this process. With s = k, (2.3) ca ow be rewritte i the form A (k) = (λ λ k )B (k) Takig ito accout the fact that B (k 1) A (1) k+1 B(k 2) A (2) λ λ k = Ω + Ω Ω k+1, k+2 B(1) A (k 1) 1. (2.18) we see that A (k) is completely determied from the coefficiets Ω, A (1),..., A (k 1). For the A (s) thus obtaied to actually be represetatio coefficiets of a operator L q, they ecessary eed to satisfy i additio, the coditios of Propositio (1.1). Whe this is so, the correspodig polyomials are eigefuctios of the operator L q. 4

9 3 Little q-jacobi polyomials The moic little q-jacobi polyomials [4] are defied as p (x; a, b) = ( 1) q( 1)/2 (aq; q) (abq +1 ; q) 2φ 1 ( ) q, abq +1 aq qx, (3.1) where (a; q) = (1 a)(1 aq)... (1 aq 1 ) is the q-shifted factorial ad 2 φ 1 deotes the q- hypergeometric fuctio (see, e.g. [2]). The orthogoality relatio is w k p (q k ; a, b) p m (q k ; a, b) = h δ m, (3.2) k=0 where h are appropriate ormalizatio costats, ad the ormalized weight fuctio is w k = (aq; q) (abq 2 ; q) (bq; q) k (aq) k (q; q) k. (3.3) It is assumed that 0 < aq < 1, b < q 1. The expasio coefficiets of the little q-jacobi polyomials are B (s) = b s (q, a 1 q ; q) s. (3.4) (q, a 1 b 1 q 2 ; q) s It is easily verified that equatios (2.8) have the followig solutios A (0) = λ = [](q 1 abq 2 ), A (1) = [](aq q 1 ), A (s) = 0, s 2. (3.5) Hece, the little q-jacobi polyomials satisfy a secod-order q-differece equatio as is well kow [4]. We also eed the value of the fuctio of secod kid, Q (z), at z = 0 (the poit z = 0 is a accumulatio poit of the orthogoality measure): Q (0; a, b) = k=0 P (q k ; a, b) w k q k. (3.6) This sum ca be evaluated usig the q-biomial theorem ad the q-saalschütz formula (see, e.g [2]): Takig ito accout that Q (0; a, b) = ( 1) +1 a ( 1)/2 1 abq q 1 a we ote that if a = q j, j = 1, 2, 3,..., P (0; a, b) = ( 1) q ( 1)/2 (q; q) (bq; q) (abq; q) (abq +1 ; q). (3.7) (aq; q) (abq +1 ; q), (3.8) Φ = Q (0; q j, b) + βp (0; q j, b) = ( 1) q ( 1)/2 (q j+1 ; q) (bq +j+1 ; q) ( β q j ) (1 bq j+1 ) (bq; q) j (q; q) j. (3.9) (1 q j ) (q +1 ; q) j (bq +1 ; q) j 5

10 4 Trasformed q-jacobi polyomials Let P (x) be arbitrary orthogoal polyomials with measure localized o the iterval [a, b]. The correspodig weight fuctio w(x) is assumed to be ormalized to 1, i.e. Itroduce the fuctios of secod kid b a Q (z) = w(x) dx = 1. b a P (x) w(x) dx. (4.1) z x Let c be a poit beyod the orthogoality iterval [a, b] such that Q (c) exists. Cosider the polyomials where G(c){P (x)} = P (x) Φ Φ 1 P 1 (x), = 1, 2,..., P0 (x) = 1, (4.2) Φ = Q (c) + β P (c). (4.3) The otatio G(c){P (x)} stads for the Geroimus trasformatio of the polyomials P (x) at the poit x = c (for details see, e.g. [6]). The weight fuctio w(x) of the polyomials G(c){P (x)} is ( ) w(x) w(x) = κ β δ(x c), (4.4) x c where κ is a appropriate ormalizatio costat. The Geroimus trasformatio thus iserts a cocetrated mass at the poit x = c. The value of this mass depeds o the parameter β. Retur to the case of the little q-jacobi polyomials with a = q j, j = 1, 2, 3,... Perform the Geroimus trasformatio (4.2) with Φ give by (3.9). (I this case c = 0.) The weight fuctio w(x) for the polyomials G(c){P (x)} is ( ) w(x) = κ w k δ(x q k ) β δ(x), (4.5) where k=0 w k = (qj+1 ; q) (bq j+2 ; q) The weight fuctio (4.5) ca be rewritte i the form where M = β (bq; q) k q jk (q; q) k. (4.6) w(x) = κ 1 (w(x; j 1) + M δ(x)), (4.7) 1 qj 1 bq j+1, κ 1 = κ 1 bqj+1 1 q j, ad w(x; j 1) is the weight fuctio correspodig to the q-little Jacobi polyomials with the parameter a = q j replaced with a = q j 1, i.e. w(x; j 1) = w k (j 1)δ(x q k ), (4.8) k=0 6

11 ad w k (j 1) = (qj ; q) (bq; q) k q jk. (4.9) (bq j+1 ; q) (q; q) k Thus the weight fuctio w(x; j) for the polyomials G(0){P (x)} is obtaied from the weight fuctio w(x; j 1) through the additio of a arbitrary mass M at the poit x = 0. I the expasio G(0){P (x)} = the coefficiets B (s) are foud from (3.1) ad (3.9) s=0 B (s) x s, (4.10) B (s) = b s (q ; q) s (q j ; q) s (q; q) s (b 1 q j 2 ; q) s ( 1 q +j s (1 bq )(1 q s ) (1 q +j )(1 bq j+2 s ) ) Y j (), (4.11) Y j ( 1) with Y j () = βq j (q +1 ; q) j (bq +1 ; q) j (q; q) j 1 (bq; q) j+1. (4.12) 5 Costructio of the coefficiets A (s). I this sectio we costruct the coefficiets A (s) for the q-differece operator L q that has the polyomials P (x) as eigefuctios. We start from the relatio A (1) = Ω B (1), (5.1) where Choose Ω proportioal to the deomiator of B (1) : Ω = λ λ 1 = A (0) A (0) 1. (5.2) Ω = bq +j 1 (q 1)(1 b 1 q 1 j 2 ) ( βq j( 1) (q ; q) j (bq ; q) j (bq; q) j+1 (q; q) j 1 ). (5.3) The from (5.1) we get for A (1) the expressio A (1) = (1 q ) ( βq j(1 ) (q +1 ; q) j (bq ; q) j (1 q 1 ) (q; q) j 1 (bq; q) j+1 (1 q +j 1 ) ). (5.4) Usig (5.2) it is ot difficult to show that A (0) = λ = β (q 1)q (j+1) 1 (q ; q) j+1 (bq ; q) j+1 1 q j 1 (Note that A (0) 0 = 0). 7 (q 1)(1 bq +j )(bq; q) j+1 (q; q) j 1. (5.5)

12 Usig ow the explicit expressios for B (1,2) ad A (0,1), we fid A (2) = β(q 1)q (2 )(j+1) 1 (1 q j )(q 2 ; q) j+3 (bq ; q) j 1 (q; q) 2. (5.6) Repeatig this procedure we arrive at the expressio A (s) = β(q 1)q (s )(j+1) 1 (q j ; q) s 1 (q s ; q) s+j+1 (bq ; q) j s+1 (q; q) s + ξ δ s,1 + η δ s,0, (5.7) where ξ = (q 1)(q; q) j 1 (bq; q) j+1 (1 q j+ 1 ), η = (1 q )(1 bq +j )(bq; q) j+1 (q; q) j 1, ad s = 0, 1, 2,.... Propositio 5.1 The coefficiets A (s) give by (5.7) satisfy the basic relatios s i=0 B (s i) A (i) s+i = A(0) B (s). (5.8) Proof. Usig the explicit expressios for B (s) ad A (s) we ca rewrite the lhs of (5.8) i the form s i=0 B (s i) A (i) s+i = η sb (s) + ξ s+1 B (s 1) + κ (S 1 + ν S 2 ), (5.9) where κ = β(q 1)b s q (j+1)(s ) 1 (q ; q) s (q j ; q) s (q s ; q) j+1 (bq s ; q) j+1, (q; q) s (b 1 q j 2 ; q) s (1 q j 1 ) q +j (1 bq )(1 q s )Y j () ν = (1 q +j )(1 bq j+2 s )Y j ( 1), ad S 1, S 2 are the sums S 1 = S 2 = s i=0 s i=0 q i (q s ; q) i (bq j+2+1 s ; q) i (q j 1 ; q) i (q; q) i (q +1 s ; q) i (bq s ; q) i, q i (q 1 s ; q) i (bq j+2 s ; q) i (q j 1 ; q) i (q; q) i (q +1 s ; q) i (bq s ; q) i. These sums ca be evaluated usig the q-aalog of the Saalschütz formula [2]: S 1 = q (j+1)s (b 1 q j, q 1 j ; q) s (b 1 q 1, q ; q) s, S 2 = q (j+1)(s 1) (b 1 q 1 j, q j ; q) s 1 (b 1 q 2, q 1 ; q) s 1. 8

13 Relatio (5.8) ow becomes (η s λ )B (s) + ξ s+1 B (s 1) + κ (S 1 + ν S 2 ) = 0 (5.10) ad is see to be idetically satisfied. This proves the propositio. Note that A (s) = 0, if s j + 2. (5.11) Moreover, for s < j + 2 the coefficiets A (s) Q 2j+2 (q ; s) is a polyomial i q of degree 2j + 2. Hece we have have the form A (s) = q (j+1) Q 2j+2 (q ; s), where Propositio 5.2 The polyomials G(0){P (x; q j, b)} are the eigefuctios of a q-differece operator L q of order 2N = 2j + 2. We kow that the polyomials G(0){p (x; q j, b)} coicide with the polyomials p (x; q j 1, b; M) obtaied from the little q-jacobi polyomials by addig to the orthogoality measure a mass M at x = 0. We thus have equivaletly the followig Propositio 5.3 The polyomials p (x; q j, b; M) obtaied from the little q-jacobi polyomials by isertig a discrete mass at x = 0 i the orthogoality measure are the eigefuctios of a q- differece operator of order 2N = 2j The case of little q-laguerre polyomials The moic little q-laguerre polyomials [4] p (x; a) = ( 1) q ( 1)/2 (aq; q) 2 φ 1 ( q, 0 aq ) qx, (6.1) are obtaied from the little q-jacobi polyomials by settig b = 0. Hece, these polyomials also satisfy a q-differece equatio. Cosider the polyomials G(0){p (x; q j )}, obtaied from the little q-laguerre polyomials by the Geroimus trasformatio at x = 0. All formulas for these polyomials are obtaied from those for little q-jacobi polyomials by puttig b = 0. I particular, their coefficiets A (s) are easily obtaied from (5.7). We thus have Propositio 6.1 The polyomials G(0){p (x; q j )} are the eigefuctios of a q-differece operator of order 2N = 2j + 2. I this case, the polyomials G(0){p (x; q j )} coicide with polyomials p (x; q j 1 ; M) obtaied from the little q-laguerre polyomials by addig to the orthogoality measure a mass M at x = 0. Hece Propositio 6.2 The polyomials p (x; q j ; M) are the eigefuctios of a q-differece operator of order 2N = 2j + 4. Whe q 1 we get Koorwider s geeralized Laguerre polyomials [5] L (j;m) (x) whose measure differs from that of the ordiary Laguerre polyomials L (j) (x) by isertig a cocetrated mass M at the edpoit x = 0 of the orthogoality iterval (0, ). These polyomials are kow to satisfy a differetial equatio of order 2j + 4 [3]. 9

14 Refereces [1] T. Chihara, A Itroductio to Orthogoal Polyomials, Gordo ad Breach, NY, [2] G. Gasper ad M. Rahma, Basic hypergeometric series, Cambridge Uiversity Press, Cambridge, [3] J. Koekoek ad R. Koekoek, O a differetial equatio for Koorwider s geeralized Laguerre polyomials, Proc. Amer. Math. Soc. 112 (1991), [4] R. Koekoek ad R. F. Swarttouw 1994 The Askey scheme of hypergeometric orthogoal polyomials ad its q-aalogue, Report 94-05, Faculty of Techical Mathematics ad Iformatics, Delft Uiversity of techology. [5] T. H. Koorwider, Orthogoal polyomials with weight fuctio (1 x) α (1 + x) β + Mδ(x + 1) + Nδ(x 1) Ca.Math.Bull. 27, (1984), [6] A. Zhedaov, Ratioal spectral trasformatios ad orthogoal polyomials, J. Comput. ad Appl.Math. 85 (1997), [7] A. Zhedaov, A method of costructig Krall s polyomials, Preprit (1998). 10