Mellin transforms for some families of q-polynomials
Abstract
By using Ramanujan's q-extension of the Euler integral representation for the gamma function, we derive the Mellin integral transforms for the families of the discrete q-Hermite II, the Al-Salam–Carlitz II, the big q-Laguerre, the big q-Legendre, the big q-Jacobi and the q-Hahn polynomials.
Full text
MELLIN TRANSFORMS FOR SOME
FAMILIES OF
q
-POLYNOMIALS
Rena o
Al a ez-No da se
a
M. K. A akishiye a
b
N. M. A akishiye
a
Depa amen o de Analisis Ma ema io, Uni e sidad. de Se il la, Apdo. 1160,
E-41080 Se il la, and Ins i u o Ca los I de Fsia Teo ia y Compu aional,
Uni e sidad de G anada, E-18071, G anada, Spain. E-mail: anus.es
b
Faul ad de Cienias, UAEM, Apa ado Pos al 396-3, CP 62250, Cue na aa,
Mo elos, Mexio. E-mail: mesumase m. .uaem.mx
Ins i u o de Ma ema ias, UNAM, Apa ado Pos al 273-3, C.P. 62210
Cue na aa, Mo elos, Mexio. E-mail: na igma ue .unam.mx
Abs a
By using Ramanujan's
q
-ex ension o he Eule in eg al ep esen a ion o he
gamma un ion, we de i e he Mellin in eg al ans o ms o he amilies o he dis-
e e
q
-He mi e I I, he Al-Salam{Ca li z I I, he big
q
-Lague e, he big
q
-Legend e,
he big
q
-Jaobi and he
q
-Hahn p olynomials.
Key wo ds:
Mellin in eg al ans o ms,
q
-p olynomials.
1 In o du ion
Mellin in eg al ans o ms o some amilies o basi hyp e geome i p oly-
nomials om he Askey sheme [15℄ we e onside ed in [7℄. De i a ion o
1
This esea h has b een supp o ed in pa by he Minis e io de Cienias y Te-
nologa o Spain unde he g an BFM-2000-0206-C04-02, he Jun a de Andalua
unde g an FQM-262, he Eu op ean p o je INTAS 2000-272, and by he Mexian
UNAM-DGAPA p o je IN112300. Two o us (MKA and NMA) a e mos g a e ul o
he Faul ad de Ma ema ias, Uni e sidad de Se illa o he hospi ali y ex ended o
hem du ing hei isi o Se illa in May-June 2001, when his wo k was omple ed.
P ep in submi ed o Else ie P ep in 13 O ob e 2003
hese Mellin ans o m pai s is essen ially based on he use o Ramanujan's
q
-
ex ension [17,4,5℄ o he Eule in eg al ep esen a ion o he gamma un ion
(
x
) (1
x
)
q
(1
x
)
=
1
Z
0
x
1
d
E
q
((1
q
)
)
;
<
x >
0
;
(1)
whe e
q
(
z
) is he
q
-gamma un ion
q
(
x
) :=
(
q
;
q
)
1
(
q
x
;
q
)
1
(1
q
)
1
x
;
0
< q <
1
;
and
E
q
(
z
) is Jakson's
q
-exp onen ial un ion
E
q
(
z
) :=
1
X
n
=0
q
(
n
2
)
(
q
;
q
)
n
z
n
= (
z
;
q
)
1
;
(2)
whe e
n
2
=
n
(
n
1)
=
2. We employ he s anda d no a ion o he
q
-sp eial
un ions heo y, see e.g. [12℄ o [2℄. In pa iula , he
q
-shi ed a o ials a e
gi en by
(
a
;
q
)
0
= 1
;
(
a
;
q
)
n
=
n
1
Y
k
=0
(1
aq
k
)
; n
= 1
;
2
;:::;
(
a
;
q
)
1
=
1
Y
k
=0
(1
aq
k
)
;
(3)
and we will use he no a ion
p
0
B
a
1
; : : : ; a
b
1
; : : : ; b
p
q ; z
1
C
A
:=
1
X
k
=0
(
a
1
;
q
)
k
(
a
;
q
)
k
(
b
1
;
q
)
k
(
b
p
;
q
)
k
z
k
(
q
;
q
)
k
(
1)
k
q
(
n
2
)
p
+1
(4)
o he basi hyp e geome i se ies.
I is well known ha Ramanujan e alua ed a numb e o in eg als ha ex end
he lassial b e a in eg al o Eule (see [17℄, [13℄, [4℄-[6℄). These in eg als ha e
asso ia ed o hogonal p olynomials ha ha e played a signian ole in he
de elopmen o he
q
-sp eial un ions heo y. Ramanujan's now lassial
q
-
ex ension o he b e a in eg al o Eule is
1
Z
0
x
E
q
(
q
b
+
x
)
E
q
(
x
)
dx
=
(
) (1
)
q
(
b
)
q
(1
)
q
(
b
+
)
:
(5)
2
The o mula (1) is an easy onsequene o (5) and he limi ela ion
lim
b
!1
q
(
b
)
q
(
b
+
)
= (1
q
)
;
when he hange o a iables
x
= (1
q
)
is made.
I was shown in [7℄ ha by using a
q
-analogue o Eule 's ee ion o mula
q
(
x
)
q
(1
x
) =
iq
1
=
8
(1
q
) (
q
;
q
)
3
1
q
x=
2
1
(ln
q
ix=
2
; q
1
=
2
)
;
1
(
z ; q
) is he he a- un ion o Jaobi, one an ep esen (1) in he o m
q
(
x
)
q
(
x
) =
1
q
ln
q
1
q
x
(
x
1)
=
2
1
Z
0
x
1
d
E
q
((1
q
)
)
;
<
x >
0
;
(6)
whe e
q
(
x
) is some p e io di a o , i.e.
q
(
x
+
n
) =
q
(
x
) o any non-nega i e
in ege
n
(an explii o m o
q
(
x
) an b e ound in [7℄). I should b e em-
phasized ha his o mula is simply Jaobi's iple p o du iden i y o he
he a- un ion
1
(
z ; q
), ew i en in e ms o he
q
-gamma un ion
q
(
z
). The
obse a ion ha Jaobi's iple p o du iden i y is a
q
-analogue o Eule 's
ee ion o mula o he gamma un ion (
z
) was known o G. And ews
and R. Askey sine he 1975-1976 aademi yea ". Un o una ely, hey ne e
published any hing ab ou his a . Besides, R. Askey b elie es ha his ob-
se a ion is due o Geo ge And ews" (see he e y end o [3℄), al hough G.
And ews ommen s ha Dik mo des ly a ibu es i o me; his is mo e a
measu e o his gene osi y han his au a e memo y" (e-mail ommunia ion,
No emb e 16, 2001). Anyway, he idea o ega ding Jaobi's iple p o du
iden i y as a
q
-ex ension o Eule 's ee ion o mula is a leas 25 yea s old.
Two o us (MKA and NMA) eg e ha we we e no al eady awa e o his
a a he ime o he w i ing o [7℄ (whih was he s a ing p oin o using
his idea in he de i a ion o Mellin in eg al ans o ms o some amilies o
q
-p olynomials).
In iew o he p e io dii y o
q
(
x
), one an hen de i e om (6) a Mellin
in eg al ans o m o he p o du
p
n
(
;
q
)
E
1
q
((1
q
)
), whe e
p
n
(
z
;
q
) is
some p olynomial in
z
o deg ee
n
. In his way he Mellin in eg al ans o ms
we e ob ained o all hose amilies o
q
-p olynomials om he Askey sheme,
in whih indep enden a iable is he a gumen o an app op ia e e mina ing
basi hyp e geome i se ies. They onsis o he S iel jes{Wige , he Roge s{
Szego, he
q
-Lague e, he Wall, he al e na i e
q
-Cha lie , and he li le
q
-
Jaobi p olynomials.
In his pap e we wish o apply he ehnique o [7℄ o he s udy o hose am-
ilies o
q
-p olynomials om he Askey sheme, whih on ain he indep enden
3
a iable
x
in one o he pa ame e s o he o esp onding basi hyp e geome i
se ies. The simples example o his yp e is he dis e e
q
-He mi e I I p olyno-
mials
~
h
n
(
x
;
q
) :=
i
n
q
(
n
2
)
2
0
0
B
q
n
; ix
|
q ;
q
n
1
C
A
=
i
n
q
(
n
2
)
n
X
k
=0
(
q
n
;
q
)
k
(
ix
;
q
)
k
(
q
;
q
)
k
q
nk
(
k
2
)
;
he e a e also he Al-Salam{Ca li z I I, he big
q
-Lague e, he big
q
-Legend e,
he big
q
-Jaobi, and he
q
-Hahn p olynomials (see [15℄).
Ou mo i a ion o he s udy o
q
-analogues o Mellin in eg al ans o ms
omes om ma hema ial physis. I is well known ha in non ela i is i
quan um mehanis he o o dina e and momen um ealiza ions a e in e e-
la ed by he Fou ie in eg al ans o m. Bu in some ela i is i app oahes
o quan um mehanis he passage om he momen um o he ongu a ion
ealiza ion is aomplished by he Mellin in eg al ans o m. Fo ins ane, a
ela i is i quasip o en ial [14,16℄ mo del o he linea ha moni osilla o , s ud-
ied in de ail in [8,9,11℄, is go e ned by a die ene Hamil onian in he ong-
u a ion
x
- ealiza ion. The passage om he ongu a ion o he momen um
ealiza ion is equi alen o he Mellin in eg al ans o m in he ligh - on a i-
able
p
+
=
p
0
+
p
, a he han he Fou ie ans o m as in he non ela i is i
ase. The e o e we b elie e ha ou ehnique an b e applied o ons u ing
a ious
q
-ex ensions o suh quan um-mehanial mo dels, based on die ene
equa ions.
2 The Mellin ans o m o a pa iula amily o
q
-p olynomials
I is well known ha Eule 's in eg al ep esen a ion
(
x
) =
1
Z
0
x
1
e
d ;
<
x >
0
;
(7)
o he gamma un ion (
x
) is an ins ane o he Mellin in eg al ans o m
g
(
z
) =
M
(
);
z
g
=
1
Z
0
(
)
z
1
d
4
o he exp onen ial un ion
(
) =
e
. Simila ly, (1) o (6) is he Mellin
ans o m o he un ion
E
1
q
((1
q
)
):
g
q
(
x
) :=
M
E
1
q
((1
q
)
);
x
g
=
ln
q
1
1
q
q
(
x
)
q
x
(
x
1)
=
2
q
(
x
)
;
<
x >
0
:
Le
p
n
(
x
;
q
) =
n
X
k
=0
a
nk
(
q
) (
x
;
q
)
k
(8)
b e a
q
-p olynomial o deg ee
n
in
x
(wi h o eÆien s
a
nk
(
q
), whih may dep end,
in addi ion o
q
, on some o he pa ame e s ). Then
M
p
n
(
;
q
)
E
1
q
((1
q
)
);
x
g
=
n
X
k
=0
a
nk
(
q
)
M
(
;
q
)
k
E
1
q
((1
q
)
);
x
g
;
whe e
is a ons an . Sine he
q
-shi ed a o ial
(
z
;
q
)
k
=
k
X
j
=0
"
k
j
#
q
q
(
j
2
)
(
z
)
j
;
"
k
j
#
q
:=
(
q
;
q
)
k
(
q
;
q
)
k
j
(
q
;
q
)
j
;
is a p olynomial o deg ee
k
in
z
, we ha e
M
p
n
(
;
q
)
E
1
q
((1
q
)
);
x
g
=
n
X
k
=0
a
nk
(
q
)
k
X
j
=0
"
k
j
#
q
q
(
j
2
)
(
)
j
M
j
E
1
q
((1
q
)
);
x
g
=
n
X
k
=0
a
nk
(
q
)
k
X
j
=0
"
k
j
#
q
q
(
j
2
)
(
)
j
g
q
(
x
+
j
)
:
(9)
Bu
g
q
(
x
+
j
) =
ln
q
1
1
q
q
(
x
)
q
(
x
+
j
)(
x
+
j
1)
=
2
q
(
x
+
j
)
=
q
j x
(
j
2
)
(
q
x
;
q
)
j
(1
q
)
j
g
q
(
x
)
;
(10)
b eause
q
(
x
+
j
) =
(
q
x
;
q
)
j
(1
q
)
j
q
(
x
)
5
by he deni ion o he
q
-gamma un ion
q
(
x
). Subs i u ing (10) in o he
igh -hand side o (9), one hus ob ains ha
M
p
n
(
;
q
)
E
1
q
((1
q
)
);
x
g
=
g
q
(
x
)
n
X
k
=0
a
nk
(
q
)
k
X
j
=0
"
k
j
#
q
(
q
x
;
q
)
j
q
x
1
q
!
j
:
(11)
Finally, he
q
-binomial o eÆien
h
k
j
i
q
an b e w i en as
"
k
j
#
q
= (
1)
j
q
k j
(
j
2
)
(
q
k
;
q
)
j
(
q
;
q
)
j
:
(12)
The e o e he sum o e he index
j
in he igh -hand side o (11) ep esen s a
e mina ing basi hyp e geome i se ies
2
0
and we ob ain he desi ed esul
M
p
n
(
;
q
)
E
1
q
((1
q
)
);
x
g
=
g
q
(
x
)
n
X
k
=0
a
nk
(
q
)
2
0
0
B
q
k
; q
x
|
q ;
q
k
x
1
q
1
C
A
:
(13)
This o mula gi es an explii o m o he Mellin in eg al ans o m o he
un ion
E
1
q
((1
q
)
), mul iplied by a p olynomial
p
n
(
;
q
) o he yp e (8)
(wi h an a bi a y ons an
). Obse e ha he igh -hand side o (13) is a
p olynomial o deg ee
n
in he a iable
q
x
, imes he un ion
g
q
(
x
). Also, he
e mina ing basi hyp e geome i se ies
2
0
in (13) an b e w i en as
2
0
0
B
q
k
; q
x
|
q ;
q
k
x
1
q
1
C
A
=
C
k
q
x
;
1
q
q
k
;
q
1
!
;
whe e he
q
1
-Cha lie p olynomials
C
n
(
z
;
a
;
q
1
) a e dened ( . [15℄, p. 112)
as
C
n
(
z
;
a
;
q
1
) :=
2
0
0
B
q
n
; z
1
|
q ;
z =a
1
C
A
:
(14)
Fo a pa iula hoie o he ons an
, he sum o e
j
in (11) is simplied;
in o he wo ds, he e mina ing basi hyp e geome i se ies
2
0
in (13) edues
6
o a monomial in he a iable
q
x
. The p oin is ha
2
0
0
B
q
n
; z
1
|
q ; z q
n
1
C
A
=
z
n
:
(15)
To e i y (15), simply e e se he o de o summa ion in he deni ion o
2
0
and use he limi ase o he
q
-Chu{Vande monde sum
2
1
0
B
q
n
; b
q ; q
1
C
A
=
(
=b
;
q
)
n
(
;
q
)
n
b
n
wi h he anishing pa ame e
b
. We no e ha he ela ion (15) an b e ex-
p essed in e ms o he
q
1
-Cha lie p olynomials (14) as
C
n
(
z
;
q
n
;
q
1
) =
z
n
:
F om (15) i ollows ha i one ho oses
=
q
1, hen he Mellin in eg al
ans o m (13) edues o
M
p
n
((
q
1)
;
q
)
E
1
q
((1
q
)
);
x
g
=
g
q
(
x
)
n
X
k
=0
a
nk
(
q
)
q
k x
:
(16)
No ie ha suh simplia ion o (13) in he ase when
=
q
1 is a onse-
quene o he ollowing p op e y
E
q
(
z
) = (
z
;
q
)
k
E
q
(
q
k
z
) (17)
o Jakson's
q
-exp onen ial un ion (2). Indeed, by he deni ion (8),
M
p
n
((
q
1)
;
q
)
E
1
q
((1
q
)
);
x
g
=
n
X
k
=0
a
nk
(
q
)
1
Z
0
((
q
1)
;
q
)
k
E
q
((1
q
)
)
x
1
d
=
n
X
k
=0
a
nk
(
q
)
1
Z
0
x
1
d
E
q
((1
q
)
q
k
)
;
(18)
whe e we ha e employed he p op e y (17) wi h
z
= (1
q
)
. The hange o
he a iable
!
q
k
in (18) leads immedia ely o he Mellin ans o m (16).
Now i emains only o onside on e e examples o he a o emen ioned
amilies o
q
-p olynomials om he Askey sheme.
7
3 Con e e examples
1.
We s a wi h he Al-Salam{Ca li z I I p olynomials om he
q
-Askey sheme
(see [15℄, p. 114)
V
(
a
)
n
(
x
;
q
) := (
a
)
n
q
(
n
2
)
2
0
0
B
q
n
; x
|
q ;
q
n
a
1
C
A
= (
a
)
n
q
(
n
2
)
n
X
k
=0
(
q
n
;
q
)
k
(
x
;
q
)
k
(
q
;
q
)
k
q
k
[
n
(
k
1)
=
2℄
(
a
1
)
k
;
(19)
whih o upy he seond (i.e. nex - o-lowes ) le el in he Askey sheme o
basi hyp e geome i p olynomials wi h he dis e e o hogonali y p op e y
(see [15℄, p. 62). F om (19) i ollows ha he o eÆien s
a
nk
(
q
) in (8) in his
pa iula ase a e equal o
a
nk
(
q
) = (
a
)
n
q
(
n
2
)
a
n
k
n
k
q
;
(20)
whe e we ha e used he ela ion (12). Subs i u ing (20) in o (16), we hus
ob ain a Mellin in eg al ans o m
M
V
(
a
)
n
((
q
1)
;
q
)
E
1
q
((1
q
)
);
x
g
= (
a
)
n
q
(
n
2
)
H
n
(
q
x
=a
;
q
)
g
q
(
x
)
;
(21)
whe e he Roge s{Szego p olynomials
H
n
(
z
;
q
) a e he
q
-analogue o He mi e
p olynomials on he uni i le (see [18,1,10℄), dened as
H
n
(
z
;
q
) :=
n
X
k
=0
n
k
q
z
k
=
2
0
0
B
q
n
;
0
|
q ; z q
n
1
C
A
:
(22)
No ie ha he sp eial ase o he Al-Salam{Ca li z I I p olynomials (19) wi h
a
=
1 is known as he dis e e
q
-He mi e I I p olynomials
e
h
n
(
x
;
q
) (see [15℄,
p. 119). The e o e, om (21) one ob ains a Mellin ans o m
M
e
h
n
(
i
(1
q
)
x
;
q
)
E
1
q
((1
q
)
);
x
g
=
i
n
q
(
n
2
)
H
n
(
q
x
;
q
)
g
q
(
x
)
;
whih in e ela es he dis e e
q
-He mi e I I and he Roge s{Szego p olynomials.
F om (22) i is e iden ha
lim
q
!
1
H
n
(
z
;
q
) = (
z
+ 1)
n
:
8
The Mellin ans o m (21) in he limi as he pa ame e
q
!
1
hus oinides
wi h Eule 's in eg al ep esen a ion o he gamma un ion (7), b o h sides o
whih a e mul iplied by he ons an a o (
1)
n
(1 +
a
)
n
.
2.
A he hi d le el o he
q
-Askey sheme wi h he dis e e o hogonali y
he e is only one amily o
q
-p olynomials o he yp e (8), namely he big
q
-
Lague e p olynomials (see [15℄, p. 91)
P
n
(
x
;
a; b
;
q
) :=
3
2
0
B
q
n
;
0
; x
aq ; bq
q ; q
1
C
A
=
n
X
k
=0
(
q
n
;
q
)
k
(
x
;
q
)
k
q
k
(
aq
;
q
)
k
(
bq
;
q
)
k
(
q
;
q
)
k
:
(23)
The o eÆien s
a
nk
(
q
) in his ase a e equal o
a
nk
(
q
) =
(
q
n
;
q
)
k
q
k
(
aq
;
q
)
k
(
bq
;
q
)
k
(
q
;
q
)
k
:
(24)
Subs i u ing (24) in o (16), one ob ains he ollowing Mellin in eg al ans o m
M
P
n
((
q
1)
;
a; b
;
q
)
E
1
q
((1
q
)
);
x
g
=
3
2
0
B
q
n
;
0
;
0
aq ; bq
q ; q
1
x
1
C
A
g
q
(
x
) (25)
o he big
q
-Lague e p olynomials
P
n
(
x
;
a; b
;
q
).
F om he deni ion (23) i is lea ha
lim
q
!
1
P
n
((
q
1)
;
a; b
;
q
) = [1
(1
a
)
1
(1
b
)
1
℄
n
:
The e o e he Mellin ans o m (25) in he limi as
q
!
1
oinides wi h (7),
mul iplied by he ons an a o [1
(1
a
)
1
(1
b
)
1
℄
n
.
3.
The big
q
-Jaobi p olynomials (see [15℄, p. 73)
P
n
(
x
;
a; b;
;
q
) :=
3
2
0
B
q
n
; abq
n
+1
; x
aq ; q
q ; q
1
C
A
(26)
o upy he ou h le el in he Askey sheme o
q
-p olynomials wi h he dis e e
o hogonali y. Taking in o aoun ha he o eÆien s
a
nk
(
q
) in his ase a e
equal o
a
nk
(
q
) =
(
q
n
;
q
)
k
(
abq
n
+1
;
q
)
k
q
k
(
aq
;
q
)
k
(
q
;
q
)
k
(
q
;
q
)
k
;
9