A q-extension of the generalized Hermite polynomials with the continuous orthogonality property on R

In e na ional Jou nal o Pu e and Applied Ma hema ics ————————————————————————– Volume 10 No. 3 2004, 331-342 Aq-EXTENSION OF THE GENERALIZED HERMITE POLYNOMIALS WITH THE CONTINUOUS ORTHOGONALITY PROPERTY ON R R. ´ Al a ez-Noda se1, M.K. A akishiye a2, N.M. A akishiye 3§ 1Depa amen o de An´alisis Ma em´a ico Uni e sidad de Se illa Apa ado Pos al 1160, E-41080 Se illa, SPAIN and 1Ins i u o Ca los I de F´ısica Te´o ica y Compu acional Uni e sidad de G anada, E-18071 G anada, SPAIN e-mail: [email p o ec ed] 2Facul ad de Ciencias UAEM – Uni e sidad Au ´onoma del Es ado de Mo edos Apa ado Pos al 396-3, C.P. 62250, Cue na aca, Mo elos, MEXICO e-mail: m[email p o ec ed] 3Ins i u o de Ma em´a icas UNAM – Uni e sidad Nacional Au ´onoma de Mexico Apa ado Pos al 273-3, C.P. 62210, Cue na aca San a Fe 45, Col. Ma a illas, Mo elos, M´ EXICO e-mail: na ig@ma cue .unam.mx Abs ac : In his pape we s udy in de ail a q-ex ension o he gene alized He mi e polynomials o Szeg˝o. A con inuous o hogonali y p ope y on Rwi h espec o he posi i e weigh unc ion is es ablished, a q-di e ence equa ion and a h ee- e m ecu ence ela ion a e de i ed o his amily o q-polynomials. AMS Subjec Classi ica ion: 26C05, 33D45, 39A13 Key Wo ds: gene alized He mi e polynomials, con inuous o hogonali y, q- di e ence equa ion, h ee- e m ecu ence ela ion Recei ed: No embe 12, 2003 c 2004, Academic Publica ions L d. §Co espondence au ho 332 R. ´ Al a ez-Noda se, M.K. A akishiye a, N.M. A akishiye 1. In oduc ion The gene alized He mi e polynomials we e in oduced by Szeg˝o [12] as H(µ) 2n(x) := (−1)n22nn!L(µ−1/2) n(x2), H(µ) 2n+1(x) := (−1)n22n+1 n!x L(µ+1/2) n(x2), (1.1) whe e µ > −1/2, L(α) n(x) a e he Lague e polynomials, L(α) n(z) := (α+ 1)n n!1F1−n α+ 1  z =(α+ 1)n n! n X k=0 (−n)k (α+ 1)k zk k!,(1.2) and (a)n= Γ(a+n)/Γ(a), n= 0,1,2,..., is he shi ed ac o ial. Obse e ha he ze o alue o he pa ame e µin (1.1) co esponds o he o dina y He mi e polynomials Hn(x), i.e., H(0) n(x) = Hn(x). The gene alized He mi e polynomials (1.1) a e o hogonal wi h espec o he weigh unc ion |x|2µe−x2,x∈R, i.e., Z∞ −∞ H(µ) n(x)H(µ) m(x)|x|2µe−x2dx = 22nhn 2i! Γ n+ 1 2+µ+1 2δnm,(1.3) whe e [x] deno es he g ea es in ege no exceeding x. They sa is y a h ee- e m ecu ence ela ion 2xH(µ) n(x) = H(µ) n+1(x) + 2(n+ 2µ θn)H(µ) n−1(x), n ≥0,(1.4) and a second-o de di e en ial equa ion xd2 dx2+ 2(µ−x2)d dx + 2nx −2µ θnx−1H(µ) n(x) = 0, n ≥0,(1.5) wi h θn:= n−2[n/2] (see Szeg˝o [12], Chiha a [4]). A de ailed discussion o o he p ope ies o H(µ) n(x) can be ound in Ma ke [9], Rosenblum [11]. Aq-EXTENSION OF THE GENERALIZED... 333 The eason o in e es in s udying he gene alized He mi e polynomials (1.1) is wo old. Pu e ma hema ically hey a e o in e es as an explici exam- ple o he comple e o hono mal se in L2 µ(R), he Hilbe space o Lebesgue measu able unc ions (x), x∈R, wi h || ||µ:= Z∞ −∞ | |2|x|2µdx1/2 <∞.(1.6) Hence one can build he Bose-like oscilla o calculus in e ms o hese poly- nomials, which gene alizes he well-known calculus, based on he quan um- mechanical ha monic oscilla o in physics (see, o example, Rosenblum [11]). So we y o make one s ep u he by conside ing a gene aliza ion o he clas- sical He mi e polynomials Hn(x) wi h wo addi ional pa ame e s, µand q. The aim o his pape is o in es iga e in de ail a q-ex ension o he gen- e alized He mi e polynomials (1.1) wi h he con inuous o hogonali y p ope y on R( he case o disc e e o hogonali y equi es a di e en echnique, see, o example, Be g e al [3]). In Sec ion 2 we in oduce his amily {H(µ) n(x;q)}in e ms o he q-Lague e polynomials and ind a ele an q-di e ence equa ion o i . In Sec ion 3 he con inuous o hogonali y p ope y o {H(µ) n(x;q)}wi h espec o he posi i e weigh unc ion on Ris explici ly o mula ed. Sec ion 4 is de o ed o he de i a ion o a h ee- e m ecu ence ela ion o his amily o q-polynomials. 2. Gene alized He mi e Polynomials I is known om Hahn [7], Ex on [5], and Moak [10] ha he q-Lague e poly- nomials L(α) n(x;q) a e explici ly gi en as L(α) n(x;q) := (qα+1;q)n (q;q)n1φ1 q−n qα+1  q, −qn+α+1 x! =1 (q;q)n2φ1 q−n,−x 0 q, qn+α+1!, (2.1) 334 R. ´ Al a ez-Noda se, M.K. A akishiye a, N.M. A akishiye whe e (a;q)0= 1 and (a;q)n=Qn−1 j=0 (1 −aqj), n= 1,2,..., is he q-shi ed ac o ial, and φpq−n, a2,··· , a b1, b2,··· , bp q , z = n X k=0 (q−n;q)k(a2;q)k···(a ;q)k (b1;q)k(b2;q)k···(bp;q)k zk (q;q)kh(−1)kqk(k−1)/2ip− +1 (2.2) is he basic hype geome ic polynomial o deg ee nin he a iable z( h oughou his pape , we will employ he s anda d no a ions o he q-special unc ions heo y, see Gaspe e al [6] o And ews e al [2]). The q-Lague e polynomials (2.1) sa is y wo kinds o o hogonali y ela ions, an absolu ely con inuous one and a disc e e one. The o me o hogonali y ela ion, in which we a e in e es ed in he p esen pape , is gi en by Z∞ 0 xα Eq(x)L(α) m(x;q)L(α) n(x;q)dx =d−1 n(α)δmn , α > −1,(2.3) whe e Eq(x) is he Jackson q-exponen ial unc ion, Eq(z) := ∞ X n=0 qn(n−1)/2 (q;q)n zn= (−z;q)∞,(2.4) and he no maliza ion cons an dn(α) is equal o dn(α) = 1 πsin π(α+ 1) qn(q;q)n (qα+1;q)n (q;q)∞ (q−α;q)∞ .(2.5) The q-Lague e polynomials (2.1) a e de ined in such a way ha in he limi as q→1 hey educe o he o dina y Lague e polynomials L(α) n(x), i.e., lim q→1L(α) n((1 −q)x;q) = L(α) n(x).(2.6) We can now de ine, in comple e analogy wi h he ela ionship (1.1), a q- ex ension o he gene alized He mi e polynomials H(µ) n(x) o he o m H(µ) 2n(x;q) := (−1)n(q;q)nL(µ−1/2) n(x2;q), H(µ) 2n+1(x;q) := (−1)n(q;q)nx L(µ+1/2) n(x2;q), (2.7) Aq-EXTENSION OF THE GENERALIZED... 335 which a e o hogonal on he eal line R. Indeed, since lim q→1 (qa;q)n (1 −q)n= (a)n,(2.8) wi h he aid o (2.6) one eadily e i ies ha lim q→1(1 −q)−n/2H(µ) n(p1−q x;q) = 2−nH(µ) n(x).(2.9) Obse e also ha he ze o alue o he pa ame e µin (2.7) co esponds o polynomials Hn(x;q)≡ H(0) n(x;q). The sequence {Hn(x;q)}can be exp essed ei he in e ms o he q-Lague e polynomials L(α) n(x;q), α=±1/2 (as i ob i- ous om de ini ion (2.7) i sel ), o h ough he disc e e q-He mi e polynomials ˜ hn(x;q) o ype II: Hn(x;q2) = qn(n−1)/2˜ hn(x;q).(2.10) A de ailed discussion o he p ope ies o he polynomials Hn(x;q) can be ound in ou p e ious pape ´ Al a ez-Noda se e al [1] on his subjec . Aq-di e ence equa ion o he in oduced polynomials H(µ) n(x;q) is, in ac , an easy consequence o he known q-di e ence equa ion qα(1 + x)L(α) n(q x;q) + L(α) n(q−1x;q) = [1 + qα(1 + qnx)] L(α) n(x;q) (2.11) o he q-Lague e polynomials (see, o example, o mula (3.21.6) in Koekoek e al [8]). Indeed, om his q-di e ence equa ion and de ini ion (2.7) i ollows immedia ely ha qµ−1/2(1 + x2)H(µ) n(q1/2x;q) + H(µ) n(q−1/2x;q) =hq−θn/2+qµ+(θn−1)/2(1 + q[n/2] x2)iH(µ) n(x;q), (2.12) whe e, as be o e, θn=n−2[n/2]. Taking in o accoun ha he dila ions x→ q±1xa e ep esen ed by he ope a o s q±xd dx , ha is, q±xd dx (x) = (q±1x), one now eadily e i ies ha he q-di e ence equa ion (2.12) coincides wi h he second-o de di e en ial equa ion (1.5) in he limi as q→1. 336 R. ´ Al a ez-Noda se, M.K. A akishiye a, N.M. A akishiye 3. O hogonali y Rela ion We begin his sec ion wi h he ollowing heo em: Theo em 1. The sequence o he q-polynomials {H(µ) n(x;q)}, which a e de ined by he ela ions (2.7), sa is ies he o hogonali y ela ion ∞ Z −∞ H(µ) m(x;q)H(µ) n(x;q)|x|2µdx Eq(x2) =π cos πµ (q1/2−µ;q)∞ (q;q)∞ q−n 2−µθn(q;q)[n 2](qµ+1/2;q)[n+1 2]δmn , (3.1) on he whole eal line Rwi h espec o he con inuous posi i e weigh unc ion w(x) = 1/Eq(x2). P oo . Since he weigh unc ion in (3.1) is an e en unc ion o he in- dependen a iable xand H(µ) n(−x;q) = (−1)nH(µ) n(x;q) by he de ini ion (2.7), he q-polynomials o an e en deg ee H(µ) 2m(x;q) and o an odd deg ee H(µ) 2n+1(x;q), m, n = 0,1,2,..., a e e iden ly o hogonal o each o he . Conse- quen ly, i su ices o p o e only hose cases in (3.1), when deg ees o polyno- mials mand na e ei he simul aneously e en o odd. Le us conside i s he o me case. F om (2.7) and (2.3) i ollows ha ∞ Z −∞ H(µ) 2m(x;q)H(µ) 2n(x;q)|x|2µdx Eq(x2) = (−1)m+n(q;q)m(q;q)n ∞ Z −∞ L(µ−1/2) m(x2;q)L(µ−1/2) n(x2;q)|x|2µdx Eq(x2) = 2(−1)m+n(q;q)m(q;q)n ∞ Z 0 L(µ−1/2) m(x2;q)L(µ−1/2) n(x2;q)x2µdx Eq(x2) = (−1)m+n(q;q)m(q;q)n ∞ Z 0 L(µ−1/2) m(y;q)L(µ−1/2) n(y;q)yµ−1/2dy Eq(y) = (q;q)2 nd−1 n(µ−1/2) δmn , Aq-EXTENSION OF THE GENERALIZED... 337 whe e he no maliza ion cons an dn(α) is de ined in (2.5). Thus ∞ Z −∞ H(µ) 2m(x;q)H(µ) 2n(x;q)|x|2µdx Eq(x2) =π cos πµ (q1/2−µ;q)∞ (q;q)∞ q−n(q;q)n(qµ+1/2;q)nδmn .(3.2) Likewise, one inds ha in he la e case ∞ Z −∞ H(µ) 2m+1(x;q)H(µ) 2n+1(x;q)|x|2µdx Eq(x2) = 2(−1)m+n(q;q)m(q;q)n ∞ Z 0 L(µ+1/2) m(x2;q)L(µ+1/2) n(x2;q)x2(µ+1) dx Eq(x2) = (−1)m+n(q;q)m(q;q)n ∞ Z 0 L(µ+1/2) m(y;q)L(µ+1/2) n(y;q)yµ+1/2dy Eq(y) = (q;q)2 nd−1 n(µ+ 1/2) δmn. Consequen ly, ∞ Z −∞ H(µ) 2m+1(x;q)H(µ) 2n+1(x;q)|x|2µdx Eq(x2) =π cos πµ (q1/2−µ;q)∞ (q;q)∞ q−n−µ−1/2(q;q)n(qµ+1/2;q)n+1 δmn . (3.3) Pu ing (3.2) and (3.3) oge he esul s in he o hogonali y ela ion (3.1). The posi i i y o Jackson q-exponen ial unc ion Eq(x2) o x∈Rand q∈(0,1) is ob ious om i s de ini ion (2.4): o i is ep esen ed as an in ini e sum o posi i e e ms (o an in ini e p oduc o posi i e ac o s). This comple es he p oo . To conclude his sec ion, we no e he ob ious ac ha in he limi as q→1 he (3.1) educes o he o hogonali y ela ion (1.3) o he gene alized He mi e polynomials (1.1). This ollows immedia ely om he limi ela ions (2.8) and (2.9), upon using he ac ha lim q→1Eq((1 −q)z) = ez.(3.4) 338 R. ´ Al a ez-Noda se, M.K. A akishiye a, N.M. A akishiye Also, in he e en he pa ame e µis ze o, hen he (3.1) coincides wi h he o hogonali y ela ion o he polynomials (2.10), de i ed in ´ Al a ez-Noda se e al [1]. 4. Recu ence Rela ion In his sec ion we de i e a h ee- e m ecu ence ela ion o he q-ex ension o he gene alized He mi e polynomials (2.7). Since an a bi a y amily o o hogonal polynomials pn(x) sa is ies a ecu ence ela ion o he o m (see Chiha a [4, p.19]) (anx+bn)pn(x) = pn+1(x) + cnpn−1(x), n ≥0,(4.1) one needs o ind coe icien s an,bn, and cn, which co espond o he case unde discussion. Be o e s a ing his de i a ion we no e ha in wha ollows i p o es con- enien o use he ollowing o m L(α) n(x;q) = (qα+1;q)n (q;q)n n X k=0 qk(k+α) (qα+1;q)kn kq (−x)k(4.2) o he q-Lague e polynomials L(α) n(x;q), which comes om he i s line in de ini ion (2.1), upon using he ela ion (q−n;q)k (q;q)k = (−1)kqk(k−1)/2−nk n kq .(4.3) Le us i s conside he case when nin (4.1) is e en. Then om (2.7) and (4.2) we ind ha H(µ) 2n+1(x;q) + c2n(q)H(µ) 2n−1(x;q) = (−1)nx(qµ+3/2;q)n n X k=0 qk(k+µ+1/2) (qµ+3/2;q)kn kq (−x2)k + (−1)n−1c2n(q)x(qµ+3/2;q)n−1 n−1 X k=0 qk(k+µ+1/2) (qµ+3/2;q)kn−1 kq (−x2)k. (4.4) The nex s ep is o employ he ela ion (1 −qα) (qα+1;q)n= (1 −qn+α) (qα;q)n(4.5) Aq-EXTENSION OF THE GENERALIZED... 339 in o de o ew i e he quo ien (qµ+3/2;q)n/(qµ+3/2;q)k om he i s e m in he igh side o (4.4) as (qµ+3/2;q)n (qµ+3/2;q)k =1−qn+µ+1/2 1−qk+µ+1/2 (qµ+1/2;q)n (qµ+1/2;q)k .(4.6) In he second e m in he igh side o (4.4) one can use he e iden ela ion (qµ+3/2;q)n−1= (qµ+1/2;q)n/(1 −qµ+1/2) and he same o mula (4.5) o he ac o (qµ+3/2;q)k. We ecall also he p ope y o he q-binomial coe icien n−1 kq =1−qn−k 1−qnn kq .(4.7) Pu ing his all oge he , we ob ain H(µ) 2n+1(x;q) + c2n(q)H(µ) 2n−1(x;q) = (−1)nx(qµ+1/2;q)n× n X k=0 qk(k+µ+1/2) (qµ+1/2;q)kn kq (−x2)k 1−qk+µ+1/21−qn+µ+1/2−c2n(q)1−qn−k 1−qn. (4.8) The igh -hand side o (4.8) should ma ch wi h H(µ) 2n(x;q) = (−1)n(qµ+1/2;q)n n X k=0 qk(k+µ−1/2) (qµ+1/2;q)kn kq (−x2)k,(4.9) mul iplied by a2n(q)x+b2n(q). This means ha he coe icien c2n(q) can be ound om he equa ion 1−qn+µ+1/2−c2n(q)1−qn−k 1−qn=dn(q)q−k(1 −qk+µ+1/2),(4.10) whe e dn(q) is some k-independen ac o . I is no di icul o e i y ha he only solu ion o he equa ion (4.10) is c2n(q) = 1 −qnand dn(q) = qn. Thus H(µ) 2n+1(x;q) + (1 −qn)H(µ) 2n−1(x;q) = qnxH(µ) 2n(x;q).(4.11) Simila ly, in he case o an odd n om (4.8) we ha e H(µ) 2n+2(x;q) + c2n+1(q)H(µ) 2n(x;q) = (−1)n+1 (qµ+1/2;q)n+1 n+1 X k=0 qk(k+µ−1/2) (qµ+1/2;q)kn+ 1 kq (−x2)k + (−1)nc2n+1(q) (qµ+1/2;q)n n X k=0 qk(k+µ−1/2) (qµ+1/2;q)kn kq (−x2)k.(4.12)