IJMMS 2003:67, 4241–4248
PII. S016117120320613X
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
THE ROOTS OF THE THIRD JACKSON
q-BESSEL FUNCTION
L. D. ABREU, J. BUSTOZ, and J. L. CARDOSO
Received 16 July 2002
We derive analytic bounds for the zeros of the third Jackson q-Bessel function
(3)
Jν (z; q).
2000 Mathematics Subject Classification: 33D15, 33D67.
1. Introduction. There are three known q-analogs of classical Bessel functions [6, 5] that are due to Jackson [7]. Following the notation of Ismail [6, 5],
(k)
these are designated by Jν (z; q), k = 1, 2, 3.
The parameter q is taken to satisfy 0 < q < 1. The third Jackson q-Bessel
(3)
function Jν (z; q) is defined as
Jν(3) (z; q)
ν+1 0
q
;q ∞ ν
2
:=
z 1 Φ1
; q, qz .
qν+1
(q; q)∞
(1.1)
This function is also known as the Hahn-Exton q-Bessel function [8, 9]. The
notation 1 Φ1 in (1.1) is the standard in use for q-hypergeometric series [4]. The
(3)
function Jν (z; q) satisfies a linear q-difference equation and it is known that
(3)
Jν (z; q) has an infinite number of simple real zeros [8]. In this paper, we will
give lower and upper bounds for these zeros. The roots of these functions are
of interest for several reasons. Firstly, it is intrinsically interesting to provide
information about the roots of a function such as (1.1), which is an entire func(2)
(3)
tion of order zero. Also, the roots of Jν (z; q) and Jν (z; q) figure prominently
in expansions in terms of “q-Fourier series” [2, 3]. Lastly, if we denote the roots
(3)
(3)
of Jν (z; q) by jn,ν , then the mass points of the orthogonality measure for a q(3)
analog of Lommel polynomials are located at the points 1/jn,ν . Furthermore,
although the function defined in (1.1) is of a simpler character than the remaining Jackson q-Bessel functions, it is hoped that the results given here for
(3)
(k)
Jν (z; q) may be extended in the future to Jν (z; q), k = 1, 2.
(3)
2. The roots of Jν (z; q). We prove two lemmas stating the existence of an
(3)
odd number of roots in a certain interval and then we prove that Jν (z; q) has
only one root in such an interval.
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L. D. ABREU ET AL.
First we apply the following transformation to (1.1):
0
0
; q, z = (z; q)∞1 Φ1
; q, c .
c
z
(2.1)
2 0
qz ; q ∞ ν
ν+1
=
z 1 Φ1
; q, q
.
qz2
(q; q)∞
(2.2)
(c; q)∞1 Φ1
This produces
Jν(3) (z; q)
This last relation in a series form gives
2 ∞
qz ; q
zν
(−1)k 2 ∞
qk(k+2ν+1)/2
(q; q)∞ k=0
qz ; q k (q; q)k
k+1 2 ∞
z ; q ∞ k(k+2ν+1)/2
zν
k q
=
(−1)
q
.
(q; q)∞ k=0
(q; q)k
Jν(3) (z; q) =
(2.3)
This representation will be critical in the proof of the next two lemmas.
(3)
Lemma 2.1. If qν+1 < (1−q)2 , then sgn[Jν (q−m/2 ; q)] = (−1)m , m = 1, 2, . . . .
Proof. Set z = q−m/2 in (2.3) to obtain
−m+k+1 ∞
; q ∞ k(k+2ν+1)/2
(q; q)∞ (3) −m/2 k q
q
J
;
q
=
(−1)
q
.
ν
−mν/2
q
(q; q)k
k=0
(2.4)
Now observing that (q−m+k+1 ; q)∞ = 0 if k < m, the series on the right-hand
side of this last equality can be written as
∞
(−1)k
k=m
−m+k+1 q
; q ∞ k(k+2ν+1)/2
q
.
(q; q)k
(2.5)
Setting j = k − m in this last series yields
∞
(q; q)m (3) −m/2 Jν q
; q = (−1)m
(−1)j Aj ,
−mν/2
q
j=0
(2.6)
where
Aj =
j+1 q ; q ∞ (j+m)(j+m+2ν+1)/2
q
.
(q; q)j+m
(2.7)
Now we prove that Aj+1 < Aj . A calculation shows that Aj+1 < Aj is equivalent to qm+j+ν+1 < (1 − qm+j+1 )(1 − qj+1 ). But the left-hand side of this inequality is decreasing in m and j, while the right-hand side is increasing in m
and j. So we only need to verify the case j = m = 0, that is, qν+1 < (1−q)2 , but
THE ROOTS OF THE THIRD JACKSON q-BESSEL FUNCTION
4243
this is the hypothesis of the lemma. Clearly, since Aj+1 < Aj , then
sgn(−1)m
∞
(−1)j Aj = (−1)m .
(2.8)
j=0
Lemma 2.1 states that there exist an odd number of roots in the interval
(q−m/2+1/2 , q−m/2 ). The next lemma refines this statement.
Lemma 2.2. Let qν+1 < (1 − q)2 and define
(ν)
(q)
αm
log 1 − qm+ν / 1 − qm
.
=
log q
(2.9)
Then
(ν)
sgn Jν(3) q−m/2+αm (q)/2 ; q = (−1)m−1 ,
m = 1, 2, . . . .
(2.10)
(ν)
Proof. First observe that the function αm (q) is well defined because if
< (1 − q)2 , then qν+1 < (1 − q) and so, for positive integer m, qm+ν <
q
(ν)
(1 − qm ), that is, 1 − qm+ν /(1 − qm ) > 0. Being defined, it is clear that αm (q)
m+ν
m
is positive because 1 − q
/(1 − q ) < 1 holds for any q ∈ (0, 1).
Observe also that
ν+1
qm+ν
(ν)
(q) < 1 ⇐⇒ log 1 −
αm
> log q ⇐⇒ qm+ν < (1 − q) 1 − qm ,
1 − qm
(2.11)
(ν)
which is true if qν+1 < (1 − q)2 . So, we have 0 < αm (q) < 1.
Now, set z = q
(ν)
−m/2+αm (q)/2
(q; q)∞
(ν)
q−mν/2+ναm (q)/2
=
m−2
(−1)k
in (2.3). The substitution gives
(ν)
Jν(3) q−m/2+αm (q)/2 ; q
(ν)
q−m+αm (q)+k+1 ; q
(q; q)k
k=0
+
∞
k
(−1)
∞
qk(k+2ν+1)/2
(ν)
q−m+αm (q)+k+1 ; q
k=m−1
∞
(q; q)k
(2.12)
qk(k+2ν+1)/2 .
Denote the first sum above by S1 and the second by S2 . If 0 ≤ k ≤ m − 2,
(ν)
(ν)
then 0 < αm (q) < 1 implies that sgn(q−m+αm
sgn S1 = (−1)m−1 , m = 1, 2, . . . .
In S2 , set j = k − m + 1 to obtain
S2 = (−1)m−1
∞
(q)+k+1
(−1)j Aj ,
j=0
; q)∞ = (−1)m−k−1 . Thus
(2.13)
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L. D. ABREU ET AL.
where
(ν)
qαm (q)+j ; q
Aj =
(q; q)j+m−1
∞
q(j+m−1)(j+m+2ν)/2 .
(2.14)
(ν)
A calculation shows that Aj+1 < Aj is reduced to qj+m+ν < (1−qαm
(ν)
αm (q)
(q)+j
)(1−
q
), which holds because q
= (1−q
)(1−q ) and because the lefthand side of the last inequality is decreasing in j and the right-hand side is
increasing in j. The infinite series is thus positive and therefore
m+j
m+ν
m
sgn S2 = (−1)m−1 = sgn S1 .
(2.15)
(3)
From Lemmas 2.1 and 2.2, we know that Jν (z; q) has an odd number of
(ν)
−m/2+αm (q)
, q−m/2 ). The next theorem proves that there
roots in the interval (q
is exactly one root in each such interval and that there are no other roots.
(ν)
Theorem 2.3. If qν+1 < (1 − q)2 and if wk (q) are the positive roots of
(3)
(ν)
Jν (z; q), ordered increasingly in k, then wk (q) = q−k/2+k (ν) , with 0 < k (ν) <
(ν)
αk (q), k = 1, 2, . . . .
(3)
Proof. From the preceding lemmas, we know that Jν (z; q) has roots of
(ν)
(ν)
the form wk = q−k/2+k , with 0 < k < αk (q).
To simplify the notation, we set
(q; q)∞
F (z) = ν+1 ν Jν(3) (z; q).
; q ∞z
q
(2.16)
We prove that the only positive roots of F (z) inside the disk |z| < q−m/2 are
(ν)
wk (q), k = 1, 2, . . . , m.
Suppose there are other roots ±λk , k = 1, 2, . . . , Pm ; λk > 0. By Jensen’s theorem [1], we can write
1
2π
2π
0
=2
log F q−m/2 eiθ dθ
m
log
k=1
=2
m
log q−m/2+k/2−k + 2
k=1
=
Pm
q−m/2
q−m/2
+2
log
wk
λk
k=1
Pm
k=1
log
q−m/2
λk
Pm
−m + m
q−m/2
k + 2
log
.
log q − 2 log q
2
λk
k=1
k=1
2
m
(2.17)
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THE ROOTS OF THE THIRD JACKSON q-BESSEL FUNCTION
On the other hand, by the definition of the q-Bessel function, we have
∞
qk(k+1)/2−mk
F q−m/2 eiθ =
(−1)k ν+1 e2ikθ
; q k (q; q)k
q
k=0
2
q(−m +m)/2
= (−1)m ν+1 e2imθ
; q m (q; q)m
q
×
(2.18)
∞
qk(k+1)/2
(−1)k m+ν+1 m+1 e2ikθ .
;q k q
;q k
q
k=−m
Then we have
−m2 + m
log q − log qν+1 ; q m − log(q; q)m
log F q−m/2 eiθ =
2
∞
qk(k+1)/2
k
2ikθ
e
+ log (−1)
m+ν+1
m+1
;q k q
;q k
q
k=−m
(2.19)
so that
1
m→∞ 2π
2π
lim
0
log F q−m/2 eiθ dθ
−m2 + m
log q − log qν+1 ; q ∞ − log(q; q)∞
(2.20)
m→∞
2
∞
1 2π
qk(k+1)/2
k
2ikθ + lim
log (−1)
e
dθ.
m+ν+1
m+1
m→∞ 2π 0
;q k q
;q k
q
k=−m
= lim
Now observe that
∞
∞
qk(k+1)/2
(−1)k m+ν+1 m+1 e2ikθ =
(−1)k qk(k+1)/2 e2ikθ .
m→∞
q
q
;
q
;
q
k
k
k=−m
k=−∞
lim
(2.21)
The above limit is uniform in θ. By the Jacobi triple product identity [4],
∞
k=−∞
− q1/2 e2iθ
k 1/2 k2 = q; qe2iθ ; e−2iθ ; q ∞ .
q
(2.22)
4246
L. D. ABREU ET AL.
Using the uniform convergence to interchange the limit and the integral,
1
lim
m→∞ 2π
=
2π
0
1
2π
∞
qk(k+1)/2
k
2ikθ
dθ
e
log (−1)
m+ν+1
m+1
q
;q k q
;q k
k=−m
2π
0
log q; qe2iθ ; e−2iθ ; q ∞ dθ
1 2π
log qe2iθ ; q ∞ dθ
2π 0
1 2π
+
log e−2iθ ; q ∞ dθ
2π 0
∞
1 2π
log 1 − qj+1 e2iθ ∞ dθ
= log(q; q)∞ +
2π
0
j=0
= log(q; q)∞ +
+
(2.23)
∞
1 2π
log 1 − qj e−2iθ ∞ dθ
2π
0
j=0
= log(q; q)∞ .
The integrals in the third equality above vanish because of the mean value
theorem for harmonic functions.
We have thus concluded that
1
m→∞ 2π
2π
lim
0
log F q−m/2 eiθ dθ
−m2 + m
log q − log qν+1 ; q ∞ .
= lim
m→∞
2
(2.24)
From (2.17), we can write

lim 2
m→∞



m
q−m/2 

log
k  = − log qν+1 ; q ∞ .
− lim 2 log q
m→∞
λk
k=1
k=1
Pm
(2.25)
(ν)
But, as can be seen by the Taylor expansion of αk (q), k = O(qk ), and this
∞
implies k=1 k < ∞.
Also
Pm
k=1
log
q−m/2
q−m/2
> log
→ ∞ as m → ∞.
λk
λ1
(2.26)
So the identity (2.25) can only hold if the first sum is empty, and the only
(ν)
roots are thus ± wk (q), k = 1, 2, . . . .
THE ROOTS OF THE THIRD JACKSON q-BESSEL FUNCTION
Remark 2.4. It follows from (2.25) that
∞
k=0 k
4247
= log(qν+1 ; q)∞ /2 log q.
Remark 2.5. Observe that for fixed j, we can always choose m sufficiently
large so that
qm+j+ν+1 < 1 − qm+j+1 1 − qj+1 ,
ν
qj+m+ν < 1 − qαm (q)+j 1 − qm+j .
(2.27)
Thus, we have the asymptotic behaviour
wk q−m/2
when m → ∞
(2.28)
without the restriction qν+1 < (1 − q)2 . In [5], Ismail conjectured that
(ν)
(1) limm→∞ qm/2 wm (q) = 1,
(ν)
(ν)
(2) limm→∞ wm+1 (q2 )/wm (q2 ) = 1/q.
The asymptotic relation (2.28) establishes these conjectures.
Remark 2.6. Lemmas 2.1 and 2.2 and Theorem 2.3 state that the roots
(ν)
wk (q) satisfy the inequalities
(ν)
q−m/2+αm
(q)
< wk (q) < q−m/2 .
(ν)
(2.29)
These bounds are quite accurate. This is evident if we estimate the length of
the interval containing the roots. A somewhat tedious calculation with Taylor
series shows that
(ν)
q−m/2 − q−m/2+αm
(q)
= qm/2+ν O(1).
(2.30)
Clearly, for fixed q satisfying the conditions of the theorem, the bounds become increasingly accurate as either k or ν increases.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
L. V. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, New York, 1979.
J. Bustoz and J. L. Cardoso, Basic analog of Fourier series on a q-linear grid, J.
Approx. Theory 112 (2001), no. 1, 134–157.
J. Bustoz and S. K. Suslov, Basic analog of Fourier series on a q-quadratic grid,
Methods Appl. Anal. 5 (1998), no. 1, 1–38.
G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications, vol. 35, Cambridge University Press, Cambridge, 1990.
M. E. H. Ismail, Some properties of Jacksons third q-Bessel function, to appear.
, The zeros of basic Bessel functions, the functions Jν+ax (x), and associated
orthogonal polynomials, J. Math. Anal. Appl. 86 (1982), no. 1, 1–19.
F. H. Jackson, On generalized functions of Legendre and Bessel, Trans. Roy. Soc.
Edin. 41 (1904), 1–28.
H. T. Koelink and R. F. Swarttouw, On the zeros of the Hahn-Exton q-Bessel function
and associated q-Lommel polynomials, J. Math. Anal. Appl. 186 (1994), no. 3,
690–710.
4248
[9]
L. D. ABREU ET AL.
R. F. Swarttouw, The Hahn-Exton q-Bessel function, Ph.D. thesis, Technische Universiteit Delft, Delft, the Netherlands, 1992.
L. D. Abreu: Department of Mathematics, University of Coimbra, Coimbra 3000, Portugal
E-mail address: daniel@mat.uc.pt
J. Bustoz: Department of Mathematics, Arizona State University, Tempe, AZ 852871804, USA
E-mail address: bustoz@asu.edu
J. L. Cardoso: Department of Mathematics, University of Tras-os-Montes and Alto
Douro, Vila Real 5000-410, Portugal
E-mail address: jluis@utad.pt