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)