TJMM
6 (2014), No. 1, 59-68
ON THE NATURAL q 2 -ANALOGUE OF THE GENERALIZED
GEGENBAUER FORM
´
I. BEN SALAH AND L. KHERIJI
Abstract. The aim of this paper is to highlight a q 2 -analogue of the generalized
Gegenbauer polynomials orthogonal with respect to the form G(α, β, q 2 ). Integral
representation and discrete measure of G(α, β, q 2 ) are given for some values of parameters.
1. Introduction
The generalized Gegenbauer orthogonal polynomials is the one of monic orthogonal
polynomials sequences which appear in many applications like the weighted Lp mean
convergence of Hermite-Fej´er interpolation, the Clifford analysis and the Lie algebra A2
[5, 14, 15]. The generalized Gegenbauer orthogonal polynomials is in connection with the
Dunkl-classical character [3].
Denoting by {Sn }n≥0 the (MOPS) of the generalized Gegenbauer polynomials and let
G(α, β) be its corresponding regular form. The (MOPS) {Sn }n≥0 satisfies the three-term
recurrence relation (see (8) below) [4]
βn = 0,
(n + β + 1)(n + α + β + 1)
γ2n+1 =
,
(1)
(2n + α + β + 1)(2n + α + β + 2) n ≥ 0,
(n
+
1)(n
+
α
+
1)
,
γ2n+2 =
(2n + α + β + 2)(2n + α + β + 3)
with the positive-definite case occurring for α > −1, β > −1.
In [1], the authors proved that the generalized Gegenbauer form G(α, β) is D-semiclassical
of class one satisfying the functional equation
n
o
D x(x2 − 1)G(α, β) + − 2(α + β + 2)x2 + 2(β + 1) G(α, β) = 0
(2)
for α 6= −n − 1, β 6= −n − 1, β 6= − 21 , α + β 6= −n − 1, n ≥ 0 from which they recovered
an integral representation and derived the moments of G(α, β) for all f ∈ P, <α > −1,
<β > −1
Z +1
Γ(α + β + 2)
hG(α, β), f i =
(1 − x2 )α |x|2β+1 f (x)dx,
(3)
Γ(α + 1)Γ(β + 1) −1
Γ(α + β + 2)Γ(n + β + 1)
, (G(α, β))2n+1 = 0, n ≥ 0.
(4)
Γ(β + 1)Γ(n + α + β + 2)
For other characterizations of the generalized Gegenbauer polynomials as a consequence
of its D-semiclassical character see [2]. To enrich the quantum calculus it is interesting to
build some q-analogous of the generalized Gegenbauer polynomials. In fact, the problem
(G(α, β))2n =
2010 Mathematics Subject Classification. 33C45, 42C05.
Key words and phrases. q-difference operator, Hq -semiclassical form, q-distributional equation, moments, discrete measure, integral representation.
59
´
I. BEN SALAH AND L. KHERIJI
60
of defining q-analogous of symmetrical (MOPS) has been the interest of some authors
from different point of views [4, 6, 7, 13].
In [7], the classification of the symmetric Hq -semiclassical orthogonal q-polynomials of
class one is given where Hq is the q-difference operator. Among the obtained canonical
situations we get the natural q 2 -analogue of the generalized Gegenbauer polynomials
2
{Sn (., q 2 )}n≥0 orthogonal with respect to the form G(α, β, q 2 ) for α + β 6= 3−2q
q 2 −1 , α + β 6=
−[n]q2 − 2, β 6= −[n]q2 − 1, α + β + 2 − (β + 1)q 2n + [n]q2 6= 0, n ≥ 0 and having the
recurrence coefficients
βn = 0,
(α + β + 2 + [n − 1]q2 )(β + 1 + [n]q2 )
γ
2n
,
2n+1 = q
n ≥ 0,
(5)
(α + β + 2 + [2n − 1]q2 )(α + β + 2 + [2n]q2 )
2n
α + β + 2 − (β + 1)q + [n]q2
2n
,
γ2n+2 = q [n + 1]q2
(α + β + 2 + [2n]q2 )(α + β + 2 + [2n + 1]q2 )
where
[n]q :=
qn − 1
,
q−1
q 6= 1,
n ≥ 0.
(6)
Also in that work it is showed that the form G(α, β, q 2 ) is Hq -semiclassical of class one
2
2n
2
2
for α + β 6= 3−2q
+ [n]q2 6= 0,
q 2 −1 , α + β 6= −[n]q − 2, β 6= −[n]q − 1, α + β + 2 − (β + 1)q
1
n ≥ 0, β 6= q(q+1) − 1 satisfying the q-distributional equation
2
2
2
Hq x(x − 1)G(α, β, q ) − (q + 1) (α + β + 2)x − (β + 1) G(α, β, q 2 ) = 0.
(7)
When q → 1 in (5) and (7) we recover (1)-(2) since [n]q tends to n and Hq tends to D.
So the aim of our contribution is to highlight the moments, integral representation and
discrete measure for G(α, β, q 2 ) when it is possible.
2. Preliminaries
Let P be the vector space of polynomials with coefficients in C and let P 0 be its dual.
We denote by hu, f i the effect of a form u ∈ P 0 (linear functional) on f ∈ P. In particular,
we denote by (u)n := hu, xn i, n ≥ 0 the moments of u. Let {Pn }n≥0 be a sequence of
monic polynomials with deg Pn = n, n ≥ 0. The sequence {Pn }n≥0 is called orthogonal
(MOPS) if we can associate with it a form u ((u)0 = 1) and a sequence of numbers
{rn }n≥0 (rn 6= 0, n ≥ 0) such that
hu, Pm Pn i = rn δn,m ,
n, m ≥ 0
and the form u is then said regular. The (MOPS) {Pn }n≥0 fulfils the three-term recurrence
relation
P0 (x) = 1, P1 (x) = x − β0 ,
(8)
Pn+2 (x) = (x − βn+1 )Pn+1 (x) − γn+1 Pn (x), n ≥ 0,
where
rn+1
hu, xPn2 i
; γn+1 =
6= 0, n ≥ 0.
rn
rn
The regular form u is positive definite if and only if ∀n ≥ 0, βn ∈ R, γn+1 > 0. Also, its
corresponding (MOPS) {Pn }n≥0 is symmetric if and only if βn = 0, n ≥ 0 or equivalently
(u)2n+1 = 0, n ≥ 0.
βn =
ON THE NATURAL q 2 -ANALOGUE OF THE GENERALIZED GEGENBAUER FORM
61
Let us introduce some useful operations in P 0 . For any form u, any a ∈ C − {0}, any
c ∈ C and any q 6= 1, we let Du = u0 , ha u, (x − c)−1 u and Hq u, be the forms defined by
duality [11, 12]
hu0 , f i := −hu, f 0 i, hha u, f i := hu, ha f i, h(x − c)−1 u, f i := hu, θc f i,
and
hHq u, f i := −hu, Hq f i,
for all f ∈ P where
(ha f )(x) = f (ax),
f (x) − f (c)
,
x−c
(θc f )(x) =
e := C −
We will usually suppose that q ∈ C
{0}
(Hq f )(x) =
S
S
f (qx) − f (x)
[8].
(q − 1)x
!!
{z ∈ C, z n = 1}
. When
n≥0
q → 1, we again meet the derivative D.
A form u is called Hq -semiclassical when it is regular and there exist two polynomials
Φ and Ψ, Φ monic, deg Φ = t ≥ 0, deg Ψ = p ≥ 1 such that
Hq (Φu) + Ψu = 0
(9)
the corresponding orthogonal sequence {Pn }n≥0 is called Hq -semiclassical [9]. The Hq semiclassical form u is said to be of class s = max(p − 1, t − 2) ≥ 0 if and only if [10]
Y
{|q(hq Ψ)(c) + (Hq Φ)(c)| + |hu, q(θcq Ψ) + (θcq ◦ θc Φ)i|} > 0,
(10)
c∈ZΦ
where ZΦ is the set of zeros of Φ.
Remark 1. When q → 1 in (9)-(10) we meet the D-semiclassical character [11, 12].
Regarding integral representations through weight-functions for a Hq -semiclassical
form u satisfying (9), we look for a function U such that
Z +∞
hu, f i =
U (x)f (x)dx, f ∈ P,
(11)
−∞
where we suppose that U is regular as far as necessary. On account of (9), we get [9]
Z +∞
−1
q
Hq−1 (ΦU ) (x) + Ψ(x)U (x) f (x)dx = 0, f ∈ P,
−∞
with the additional condition [9]
Z
lim
→+0
1
U (x) − U (−x)
dx
x
exists or is continuous at the origin. Therefore
q −1 Hq−1 (ΦU ) (x) + Ψ(x)U (x) = λg(x),
(12)
(13)
where λ ∈ C and g is a locally integrable function with rapid decay representing the null
form. For instance
(
0,
x ≤ 0,
1
g(x) =
1
−x 4
sin x 4 , x > 0,
e
was given by Stieltjes [16]. When λ = 0, the equation (13) becomes
Φ(q −1 x)U (q −1 x) = {Φ(x) + (q − 1)xΨ(x)} U (x),
´
I. BEN SALAH AND L. KHERIJI
62
so that, if q > 1, we have
Φ(x) + (q − 1)xΨ(x)
U (x), x ∈ R,
(14)
Φ(q −1 x)
and if 0 < q < 1, with x → qx, we have
Φ(x)
U (qx) =
U (x), x ∈ R.
(15)
Φ(qx) + (q − 1)qxΨ(qx)
Lastly, let us recall the following standard expressions needed to the q-calculus in the
sequel [6, 9]
n
Y
(a; q)0 := 1; (a; q)n :=
(1 − aq k−1 ), n ≥ 1,
(16)
U (q −1 x) =
k=1
+∞
Y
(1 − aq k ), |q| < 1,
(17)
(a; q)∞
, 0 < q < 1.
(aq n ; q)∞
(18)
(a; q)∞ :=
k=0
(a; q)n =
1
(a; q)n = (−1)n an (a−1 ; q −1 )n q 2 n(n−1) , n ≥ 0,
the q-binomial theorem
+∞
X
(a; q)k
k=0
(q; q)k
zk =
(19)
(az; q)∞
, |z| < 1, |q| < 1,
(z; q)∞
(20)
the q-analogue of the exponential function
+∞ 1 k(k−1)
X
q2
k=0
(q; q)k
z k = (−z; q)∞ , |q| < 1.
(21)
3. Moments, discrete measure and integral representation of G(α, β, q 2 )
Firstly, let us state this technical lemma needed to the sequel and is easy to establish:
Lemma 1. Let
ξµ (q) = 1 + (µ + 1)(1 − q 2 ),
and
r
q(µ,ω) =
1+
q > 0, µ > −1,
(22)
ω
, µ > −1, ω > −µ − 1.
µ+1
(23)
We have
ξµ (q) = 1 ⇐⇒ q = 1, ξµ (q) = 0 ⇐⇒ q = q(µ,1) , ξµ (q) = −1 ⇐⇒ q = q(µ,2) ,
ξµ (q) < −1 ⇐⇒ q ∈]q(µ,2) , +∞[, −1 < ξµ (q) < 0 ⇐⇒ q ∈]q(µ,1) , q(µ,2) [,
(24)
0 < ξµ (q) < 1 ⇐⇒ q ∈]1, q(µ,1) [, ξµ (q) > 1 ⇐⇒ q ∈]0, 1[.
Secondly, from (5) and according to the lemma 1, the natural q 2 -analogue of the
generalized Gegenbauer orthogonal polynomials is positive definite for 0 < q < 1, α > −1,
β > −1 or 1 < q < q(β,1) , α > −1, β > −1.
2
Thirdly, from the Hq -semiclassical of class one conditions α + β 6= 3−2q
q 2 −1 , α + β 6=
1
−[n]q2 − 2, β 6= −[n]q2 − 1, α + β + 2 − (β + 1)q 2n + [n]q2 6= 0, n ≥ 0, β 6= q(q+1)
−1
2
concerning the form G(α, β, q ) and by virtue of the lemma 1 another time we get
ξα+β+1 (q) 6= 0, ξβ (q) 6= q −1 ,
(25)
ξα+β+1 (q) 6= q 2n , ξβ (q) 6= q 2n , ξα+β+1 (q) 6= q 2n ξβ (q), n ≥ 0.
ON THE NATURAL q 2 -ANALOGUE OF THE GENERALIZED GEGENBAUER FORM
63
Now, we are able to highlight discrete measure and integral representations of G(α, β, q 2 )
in the positive definite case and for some values of parameters.
Proposition 1. The form G(α, β, q 2 ) has the following properties.
(1) The moments of G(α, β, q 2 ) are
(G(α, β, q 2 ))2n+1 = 0,
n ≥ 0,
Qn
(G(α, β, q 2 ))0 = 1, (G(α, β, q 2 ))2n
2k−2
− ξβ (q)
k=1 q
,
= Qn
2k−2 − ξ
α+β+1 (q))
k=1 (q
n ≥ 1.
(26)
(2) For all α > −1, β > −1 and 0 < q < 1, the form G(α, β, q 2 ) has the discrete
measure
!
+∞
(ξβ (q))−1 ; q 2 ∞ X
2
r
r
G(α, β, q ) =
∆k δ
+δ
(27)
ξ (q)
ξ (q)
((ξα+β+1 (q))−1 ; q 2 )∞
q k ξ β (q)
−q k ξ β (q)
α+β+1
α+β+1
k=0
where
l
2
k
ql
ξβ (q)
(ξβ (q))−k X
, k ≥ 0.
−
∆k =
2
(q 2 ; q 2 )l (q 2 ; q 2 )k−l
qξα+β+1 (q)
(28)
l=0
(3) For all α > −1, β > −1 and 1 < q < q(β,1) , the form G(α, β, q 2 ) has the discrete
measure
+∞
ξβ (q); q −2 ∞ X
2
(29)
G(α, β, q ) =
Λk δ−q−k + δq−k
−2
(ξα+β+1 (q); q )∞
k=0
where
Λk =
l
2
k
(ξβ (q))k X
q −l
qξα+β+1 (q)
−
, k ≥ 0.
2
(q −2 ; q −2 )l (q −2 ; q −2 )k−l
ξβ (q)
(30)
l=0
Proof. For (1), equivalently with (7), we have
hHq x(x2 − 1)G(α, β, q 2 ) − (q + 1) (α + β + 2)x2 − (β + 1) G(α, β, q 2 ), xn i = 0, n ≥ 0.
Consequently, according to the symmetric character of this form and the definition in
(22), this yields the recurrence relation
G(α, β, q 2 ) 0 = 1; G(α, β, q2 ) 1 = 0,
(q n − ξα+β+1 (q)) G(α, β, q 2 ) n+2 = (q n − ξβ (q)) G(α, β, q 2 ) n , n ≥ 0.
Thus the desired result (26) since the properties in (25).
To establish (27) and (29), by virtue of (24)-(25) and (16)-(19) we may write the
moment of index even in (26) as follows: for all n ≥ 0
n
((ξβ (q))−1 ;q2 )∞ ((ξα+β+1 (q))−1 q2n ;q2 )∞
ξβ (q)
, 0 < q < 1,
ξα+β+1 (q)
((ξα+β+1 (q))−1 ;q 2 )∞
((ξβ (q))−1 q 2n ;q 2 )∞
2
G(α, β, q ) 2n =
−2n −2
−2
ξ
(q);q
ξ
(q)q
;q
)∞
)∞ ( α+β+1
(β
,
q > 1.
(ξα+β+1 (q);q −2 )∞
(ξβ (q)q −2n ;q −2 )∞
(31)
But, by the q-binomial theorem (20), the q-analogue of the exponential function (21),
the two latest properties in (24) and since
∀n ≥ 0, ∀q ∈]0, 1[, 0 < q 2n (ξβ (q))−1 < 1; ∀n ≥ 0, ∀q ∈]1, q(β,1) [, 0 < q −2n ξβ (q) < 1,
´
I. BEN SALAH AND L. KHERIJI
64
the equality in (31) yields to
2
2n
chG(α, β, q ), x i =
ξβ (q)
ξα+β+1 (q)
n
(ξβ (q))−1 ; q 2 ∞
((ξα+β+1 (q))−1 ; q 2 )∞
+∞
+∞
X
(ξβ (q))−k q 2nk X (−1)k q k(k−1) (ξα+β+1 (q))−k q 2nk
, 0 < q < 1, n ≥ 0,
×
(q 2 ; q 2 )k
(q 2 ; q 2 )k
(32)
k=0
k=0
and
ξβ (q); q −2 ∞
hG(α, β, q ), x i =
(ξα+β+1 (q); q −2 )∞
2
2n
+∞
+∞
X
(−1)k q −k(k−1) (ξα+β+1 (q))k q −2nk X (ξβ (q))k q −2nk
×
, 1 < q < q(β,1) , n ≥ 0.
(q −2 ; q −2 )k
(q −2 ; q −2 )k
k=0
(33)
k=0
Using the Cauchy product between the two power series in (32) since and those in (33),
according to the definitions in (28) and (30) we get successively for all n ≥ 0
s
!2n
+∞
(ξβ (q))−1 ; q 2 ∞ X
ξβ (q)
k
2
2n
∆k q
, 0 < q < 1,
hG(α, β, q ), x i = 2
((ξα+β+1 (q))−1 ; q 2 )∞
ξα+β+1 (q)
k=0
+∞
X
ξβ (q); q
∞
Λk (q −k )2n , 1 < q < q(β,1) .
(ξα+β+1 (q); q −2 )∞
−2
hG(α, β, q 2 ), x2n i = 2
k=0
2
By the fact that the form G(α, β, q ) is symmetric we obtain the desired results (27)
and (29). Thus, the points (2)-(3) are proved.
Proposition 2. The form G(α, β, q 2 ) has the following integral representations.
(1) For −1 < α < 0, β > −1, 0 < q < 1 and for all f ∈ P
2
q ξα+β+1 (q) 2 2
+1
Z
x
;
q
ln ξ (q)
ξβ (q)
ln |x| − lnβq −1
2
∞ hG(α, β, q ), f i = K1 |x|
sin 2π ln q f (x)dx, (34)
(x2 ; q 2 )∞
−1
where
K1−1
Z
+1
=2
x
ln ξ (q)
− lnβq −1
q 2 ξα+β+1 (q) 2 2
x ;q
ξβ (q)
(x2 ; q 2 )∞
0
∞
sin 2π ln |x| dx.
ln q (35)
(2) For α ≥ 0, β > −1, q(α+β+1,−α) < q < 1 and for all f ∈ P
r
+ q2 ξ
ξβ (q)
α+β+1 (q)
Z
hG(α, β, q 2 ), f i = K2
ln ξ (q)
− lnβq −1
|x|
r
−
q 2 ξα+β+1 (q) 2 2
x ;q
ξβ (q)
(x2 ; q 2 )∞
∞
f (x)dx,
(36)
ξβ (q)
q 2 ξα+β+1 (q)
where
r
+ q2 ξ
K2−1 = 2
ξβ (q)
Zα+β+1 (q)
x
0
ln ξ (q)
− lnβq −1
q 2 ξα+β+1 (q) 2 2
x ;q
ξβ (q)
(x2 ; q 2 )∞
∞
dx.
(37)
ON THE NATURAL q 2 -ANALOGUE OF THE GENERALIZED GEGENBAUER FORM
(3) For α ≥ 0, β > −1, 1 < q < q(α+β+1,1) and for all f ∈ P
Z q
ln ξβ (q)
(q −2 x2 ; q −2 )∞
2
f (x)dx,
hG(α, β, q ), f i = K3
|x|− ln q −1 ξα+β+1 (q) 2 −2
−q
ξβ (q) x ; q
65
(38)
∞
where
K3−1 = 2
q
Z
ln ξβ (q)
−1
ln q
x−
0
(q −2 x2 ; q −2 )∞
dx.
ξα+β+1 (q) 2 −2
ξβ (q) x ; q
(39)
∞
(4) For −1 < α < 0, β > −1, 1 < q < q(α+β+1,1) and for all f ∈ P
hG(α, β, q 2 ), f i = K4
s
ξβ (q)
ξα+β+1 (q)
Z
×
|x|
s
−
ln ξ (q)
− lnβq −1
ξβ (q)
ξα+β+1 (q)
(q
−2 2
x ;q
−2
)∞
ξα+β+1 (q) 2 −2
ξβ (q) x ; q
∞
ξ
(q) ln α+β+1
x
(40)
ξβ (q)
sin 2π
f (x)dx,
−1
ln q
where
s
K4−1 = 2
ξβ (q)
ξα+β+1 (q)
Z
x−
ln ξβ (q)
−1
ln q
0
(q
−2 2
x ;q
−2
)∞
ξα+β+1 (q) 2 −2
ξβ (q) x ; q
∞
ξα+β+1 (q)
x
ln
ξβ (q)
sin 2π
dx. (41)
ln q −1
Proof. To establish the integral representations in (1)-(4) and by virtue of (11), we look
for a function U representing G(α, β, q 2 ). It is seen from the q-distributional equation (7)
that
Φ(x) = x(x2 − 1); Ψ(x) = −(q + 1) (α + β + 2)x2 − (β + 1) .
(42)
For (1)-(2), according to (11), (42) and (22), the q-difference equation (15) becomes
U (qx) = (qξβ (q))−1
1 − x2
1−
q2 ξ
α+β+1 (q)
ξβ (q)
x2
U (x).
(43)
But, taking α > −1, β > −1, 0 < q < 1, and using (23)-(24) it is quite straightforward
to get the following equivalences
0<
ξβ (q)
q2 ξ
α+β+1 (q)
< 1 ⇐⇒ q > q(α+β+1,−α) ,
(44)
0 < q(α+β+1,−α) < 1 ⇐⇒ α ≥ 0,
(45)
q(α+β+1,−α) > 1 ⇐⇒ α < 0.
(46)
and
Consequently, if −1 < α < 0, β > −1, 0 < q < 1 we seek U as
2
q ξα+β+1 (q) 2 2
x
;
q
ξβ (q)
∞
V (x)
, |x| < 1,
U (x) =
2 ; q2 )
(x
∞
0
, |x| ≥ 1.
Replacing in (43) this leads to V (qx) = (qξβ (q))−1 V (x), therefore
V (x) = |x|−
ln ξβ (q)
−1
ln q
W (x)
(47)
´
I. BEN SALAH AND L. KHERIJI
66
with W (qx) = W (x). Taking into account (47) we choose
ln |x| .
W (x) = K1 sin 2π
ln q Thus, for 0 < |x| <
1
2
we have
ln ξ (q)
− lnβq −1
0 ≤ U (x) ≤ K1 |x|
q 2 ξα+β+1 (q) 2 2
x ;q
ξβ (q)
(x2 ; q 2 )∞
∞
K1
∼
x→0
|x|
ln ξβ (q)
+1
ln q
,
ln ξβ (q)
+ 1 < 1,
ln q
and
2πK1
U (x) ∼
|x|→1
| ln q|
q 2 ξα+β+1 (q) 2
;q
ξβ (q)
Q+∞
k=1 (1
−
∞
q 2k )
| ln |x||
−→
1 − x2 |x|→1
πK1
q 2 ξα+β+1 (q) 2
;q
ξβ (q)
| ln q|
Q+∞
k=1 (1
∞
− q 2k )
.
It follows the result in (34) with (35) since the first condition in (12) is valid.
Also, if α ≥ 0, β > −1, q(α+β+1,−α) < q < 1 we seek U as
2
s
q ξα+β+1 (q) 2 2
x
;
q
ξ
(q)
ξβ (q)
β
∞
,
, |x| ≤
V (x)
2
2
2
(x ; q )∞
q ξα+β+1 (q)
U (x) =
s
ξβ (q)
0
, |x| >
.
q 2 ξα+β+1 (q)
Replacing in (43) this leads to V (qx) = (qξβ (q))−1 V (x), therefore
V (x) = K2 |x|−
ln ξβ (q)
−1
ln q
.
It follows the result in (36) with (37) since the first condition in (12) is valid.
From the hypothesis of (3)-(4), we have α > −1, β > −1, 1 < q < q(α+β+1,1) . By
virtue of (11), (42) and (22), the q-difference equation (14) becomes
U (q
−1
ξα+β+1 (q) 2
ξβ (q) x
U (x).
1 − q −2 x2
1−
x) = qξβ (q)
According to (24) and (45)-(46) we have
0 < ξα+β+1 (q) < ξβ (q) < 1, 1 < q < min q(α+β+1,1) , q(β,1) = q(α+β+1,1) ,
ξβ (q)
> q 2 ⇐⇒ q > q(α+β+1,−α) .
ξα+β+1 (q)
Consequently, if α ≥ 0, β > −1, 1 < q < q(α+β+1,1) we seek U as
q −2 x2 ; q −2 ∞
V (x) , |x| ≤ q,
ξα+β+1 (q) 2 −2
U (x) =
x
;
q
ξβ (q)
∞
0
, |x| > q.
Replacing in (48) this leads to V (qx) = qξβ (q)V (x), therefore
V (x) = K3 |x|−
ln ξβ (q)
−1
ln q
.
It follows the result in (38) with (39) since the first condition in (12) is valid.
(48)
ON THE NATURAL q 2 -ANALOGUE OF THE GENERALIZED GEGENBAUER FORM
67
Moreover, if −1 < α < 0, β > −1, 1 < q < min(q(α+β+1,−α) , q(α+β+1,1) ) = q(α+β+1,1)
we seek U as
s
−2 2 −2
q
x
;
q
ξβ (q)
∞
, |x| <
,
V (x)
ξα+β+1 (q) 2 −2
ξ
α+β+1 (q)
x ;q
ξ
(q)
β
U (x) =
(49)
∞
s
ξ
(q)
β
0
, |x| ≥
.
ξα+β+1 (q)
Replacing in (48) this leads to V (qx) = qξβ (q)V (x), therefore
V (x) = |x|−
ln ξβ (q)
−1
ln q
W (x),
with W (q −1 x) = W (x). According to (49), one may choose
ξ
(q) x
ln α+β+1
ξβ (q)
.
W (x) = K4 sin 2π
ln q −1
It follows the result in (40) with (41) since the first condition in (12) is valid and by a
similar reasoning likewise in (1).
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[13] Mejri, M., q-Extension of generalized Gegenbauer polynomials, J. Difference Equ. Appl. 16 (12)
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[15] De Shi, Y.G., Mean convergence of Hermite-Feg´
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68
´
I. BEN SALAH AND L. KHERIJI
´ des Sciences de Monastir
Faculte
´partement de Mathe
´matiques
De
5019, Monastir, Tunisia.
E-mail address: [email protected]
´
´paratoire aux Etudes
´nieurs El Manar
Institut Pre
d’Inge
Campus universitaire El Manar
B.P.244 El Manar II - 2092 Tunis, Tunisia.
E-mail address: [email protected]
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