A Connection Formula of the Hahn-Exton q-Bessel Function
We show a connection formula of the Hahn-Exton q-Bessel function around the origin and the infinity. We introduce the q-Borel transformation and the q-Laplace transformation following C. Zhang to obtain the connection formula. We consider the limit p→1⁻ of the connection formula.
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Інститут математики НАН України
2011
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| Цитувати: | A Connection Formula of the Hahn-Exton q-Bessel Function / T. Morita // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-1480802019-02-17T01:26:29Z A Connection Formula of the Hahn-Exton q-Bessel Function Morita, T. We show a connection formula of the Hahn-Exton q-Bessel function around the origin and the infinity. We introduce the q-Borel transformation and the q-Laplace transformation following C. Zhang to obtain the connection formula. We consider the limit p→1⁻ of the connection formula. 2011 Article A Connection Formula of the Hahn-Exton q-Bessel Function / T. Morita // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 9 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D15; 34M40; 39A13 DOI: http://dx.doi.org/10.3842/SIGMA.2011.115 http://dspace.nbuv.gov.ua/handle/123456789/148080 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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| description |
We show a connection formula of the Hahn-Exton q-Bessel function around the origin and the infinity. We introduce the q-Borel transformation and the q-Laplace transformation following C. Zhang to obtain the connection formula. We consider the limit p→1⁻ of the connection formula. |
| format |
Article |
| author |
Morita, T. |
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Morita, T. A Connection Formula of the Hahn-Exton q-Bessel Function Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Morita, T. |
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Morita, T. |
| title |
A Connection Formula of the Hahn-Exton q-Bessel Function |
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A Connection Formula of the Hahn-Exton q-Bessel Function |
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A Connection Formula of the Hahn-Exton q-Bessel Function |
| title_fullStr |
A Connection Formula of the Hahn-Exton q-Bessel Function |
| title_full_unstemmed |
A Connection Formula of the Hahn-Exton q-Bessel Function |
| title_sort |
connection formula of the hahn-exton q-bessel function |
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Інститут математики НАН України |
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2011 |
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http://dspace.nbuv.gov.ua/handle/123456789/148080 |
| citation_txt |
A Connection Formula of the Hahn-Exton q-Bessel Function / T. Morita // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 9 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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AT moritat aconnectionformulaofthehahnextonqbesselfunction AT moritat connectionformulaofthehahnextonqbesselfunction |
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2025-07-12T18:10:56Z |
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2025-07-12T18:10:56Z |
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1837465714416418816 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 115, 11 pages
A Connection Formula
of the Hahn–Exton q-Bessel Function
Takeshi MORITA
Graduate School of Information Science and Technology, Osaka University,
1-1 Machikaneyama-machi, Toyonaka, 560-0043, Japan
E-mail: t-morita@cr.math.sci.osaka-u.ac.jp
Received May 11, 2011, in final form December 14, 2011; Published online December 16, 2011
http://dx.doi.org/10.3842/SIGMA.2011.115
Abstract. We show a connection formula of the Hahn–Exton q-Bessel function around
the origin and the infinity. We introduce the q-Borel transformation and the q-Laplace
transformation following C. Zhang to obtain the connection formula. We consider the limit
p→ 1− of the connection formula.
Key words: Hahn–Exton q-Bessel function; q-Borel transformation; connection problems
2010 Mathematics Subject Classification: 33D15; 34M40; 39A13
1 Introduction
In this paper, we show a connection formula of the Hahn–Exton q-Bessel function J
(3)
ν (x; q). At
first, we review the Bessel function and q-analogues of the Bessel function. The Bessel equation
d2u
dz2
+
1
z
du
dz
+
(
1− ν2
z2
)
u = 0
has a solution u(z) = Jν(z), J−ν(z). Here, the Bessel function Jν(z) is
Jν(z) =
1
Γ(ν + 1)
(z
2
)ν
0F1
(
−, ν + 1,−z
2
4
)
.
The degenerated confluent hypergeometric function 0F1(−, α, z) is defined by
0F1(−, α, z) =
∑
n≥0
1
(α)nn!
zn, (α)n = α{α+ 1} · · · {α+ (n− 1)}.
Both Jν(z) and J−ν(z) are linearly independent if ν 6∈ Z.
It is known that there exists three different q-analogues of the Bessel function.
J (1)
ν (x; q) :=
(qν+1; q)∞
(q; q)∞
(x
2
)ν
2ϕ1
(
0, 0; qν+1; q,−x
2
4
)
, |x| < 2,
J (2)
ν (x; q) :=
(qν+1; q)∞
(q; q)∞
(x
2
)ν
0ϕ1
(
−; qν+1; q,−q
ν−1x2
4
)
, x ∈ C,
J (3)
ν (x; q) :=
(qν+1; q)∞
(q; q)∞
xν1ϕ1
(
0; qν+1; q, qx2
)
, x ∈ C.
Here,
(a; q)n :=
{
1, n = 0,
(1− a)(1− aq) · · · (1− aqn−1), n ≥ 1,
mailto:t-morita@cr.math.sci.osaka-u.ac.jp
http://dx.doi.org/10.3842/SIGMA.2011.115
2 T. Morita
(a; q)∞ = lim
n→∞
(a; q)n
and
(a1, a2, . . . , am; q)∞ = (a1; q)∞(a2; q)∞ · · · (am; q)∞.
Moreover, the basic hypergeometric series rϕs is
rϕs(a1, . . . , ar; b1, . . . , bs; q, x) :=
∑
n≥0
(a1, . . . , ar; q)n
(b1, . . . , bs; q)n(q; q)n
[
(−1)nq
n(n−1)
2
]1+s−r
xn.
The first and the second one are called Jackson’s first and second q-Bessel function and the
third one is called the Hahn–Exton q-Bessel function. They satisfy the following q-difference
equations:
J (1)
ν : u(xq)−
(
qν/2 + q−ν/2
)
u(xq1/2) +
(
1 +
x2
4
)
u(x) = 0,
J (2)
ν :
(
1 +
qx2
4
)
u(xq)−
(
qν/2 + q−ν/2
)
u
(
xq1/2
)
+ u(x) = 0,
J (3)
ν : u(xq)−
{
(qν/2 + q−ν/2)− q−ν/2+1x2
}
u
(
xq1/2
)
+ u(x) = 0. (1)
The limits of these q-analogues of the Bessel function are the Bessel function when q → 1−:
lim
q→1−
J (k)
ν ((1− q)x; q) = Jν(x), k = 1, 2
and
lim
q→1−
J (3)
ν ((1− q)x; q) = Jν(2x).
The relation between J
(1)
ν (x; q) and J
(2)
ν (x; q) was found by Hahn [3] as follows:
J (2)
ν (x; q) =
(
−x
2
4
; q
)
∞
J (1)
ν (x; q). (2)
Connection problems of the q-difference equation between the origin and the infinity are
studied by G.D. Birkhoff [1]. We review connection formulae for several q-difference functions.
1. Watson’s formula. In 1910 [6], Watson showed the connection formula of the basic hyper-
geometric function 2ϕ1 as follows:
2ϕ1 (a, b; c; q;x) =
(b, c/a; q)∞(ax, q/ax; q)∞
(c, b/a; q)∞(x, q/x; q)∞
2ϕ1 (a, aq/c; aq/b; q; cq/abx)
+
(a, c/b; q)∞(bx, q/bx; q)∞
(c, a/b; q)∞(x, q/x; q)∞
2ϕ1 (b, bq/c; bq/a; q; cq/abx) .
2. Connection formula of J
(1)
ν (x; q). C. Zhang has given some connection formulae for the
solutions of the q-difference equations of confluent type [7, 8] and [9]. In [8], Zhang
has shown connection formulae for J
(1)
ν (x; q) and J
(2)
ν (x; q). The connection formula of
J
(1)
ν (x; q) is given by(
α√
px ; p
)
∞
θp
(
−α
x
) 2ϕ1
(
pν+
1
2 , p−ν+
1
2 ;−p; p, α
√
px
)
A Connection Formula of the Hahn–Exton q-Bessel Function 3
=
1
θp
(
−α
x
)
θp
(
−αq
ν
2
x
)
(q, q−ν ; q)∞
2ϕ1
(
0, 0; qν+1; q,−x
2
4
)
+
θp
(
−αq−
ν
2
x
)
(q, qν ; q)∞
2ϕ1
(
0, 0; q−ν+1; q,−x
2
4
) , (3)
where q = p2 and α2 = −4q3/2.
The connection formula of J
(2)
ν (x; q) is obtained by (3) and (2). But it is not known the con-
nection formula of the Hahn–Exton q-Bessel function.
The Hahn–Exton q-Bessel equations (1) has two analytic solutions u(x) = J
(3)
ν (x), J
(3)
−ν (xp−ν)
around x = 0 and has one analytic solution z (1/x) = 1
θp(−pν+2/x)
∑
n≥0
anx
−n, a0 = 1. We show
a connection formula of J
(3)
ν (x; q) in Section 2 as follows:
Theorem 1. For any x ∈ C∗ \ [pν+2; p],
z
(
1
x
)
=
1
(p−2ν , p; p)∞
θp
(
−p2ν+2
x
)
θp
(
−pν+2
x
) 1ϕ1
(
0, p1+2ν ; p, x
)
+
1
(p2ν , p; p)∞
θp
(
−p2
x
)
θp
(
−pν+2
x
) 1ϕ1
(
0, p1−2ν ; p, p−2νx
)
. (4)
Here, θp(·) is the theta function of Jacobi and [λ; q] is the q-spiral (see Section 2). We use the
q-Borel transformation and the q-Laplace transformation which is defined by C. Zhang in [8].
In Section 3, we consider the limit p → 1− of the connection formula. If we take a suitable
limit p→ 1− of (4), we obtain
H(2)
ν
(√
z
)
=
−ieνπi
sin νπ
{
Jν
(√
z
)
− e−νπiJ−ν
(√
z
)}
.
Here, H
(2)
ν (z) is the Hankel function of the second kind. Thus we obtain a connection formula
of the Bessel function as a limit p→ 1− of (4).
2 The connection formula
In this section, we give a connection formula of the Hahn–Exton q-Bessel function. We introduce
the p-Borel transformation and the p-Laplace transformation to obtain the connection formula
between the origin and the infinity. These transformations are useful to consider connection
problems. We assume that q ∈ C∗ satisfies 0 < |q| < 1 and q = p2. The q-difference operator σq
is given by σqf(x) = f(qx).
2.1 The theta function of Jacobi
Before we study connection problems, we review the theta function of Jacobi. The theta function
of Jacobi is given by the following series:
Definition 1. For any x ∈ C∗,
θq(x) = θ(x) :=
∑
n∈Z
q
n(n−1)
2 xn.
4 T. Morita
We denote by θq(x) or more shortly θ(x). The theta function satisfies Jacobi’s triple product
identity:
θ(x) =
(
q,−x,− q
x
; q
)
∞
.
The theta function satisfies the q-difference equation as follows
θ(qkx) = q−
k(k−1)
2 x−kθ(x), ∀x ∈ C∗.
The theta function has the inversion formula xθ(1/x) = θ(x). For all fixed λ ∈ C∗, we define
a q-spiral [λ; q] := λqZ = {λqk : k ∈ Z}. We remark that θ
(
λqk/x
)
= 0 if and only if x ∈ [−λ; q].
2.2 The Hahn–Exton q-Bessel function
The Hahn–Exton q-Bessel function is defined by
J (3)
ν (x; q) :=
(qν+1; q)∞
(q; q)∞
xν
∑
n≥0
(−1)nq
n(n−1)
2
(qν+1, q; q)n
(
qx2
)n
.
The function J
(3)
ν (x; q) satisfies the q-difference equation[
σ2p −
{
(pν + p−ν)− x2p2−ν
}
σp + 1
]
y(x) = 0. (5)
If we replace ν by −ν and x by xp−ν , we obtain J
(3)
−ν (xp−ν ; q) which is another solution of (5)
around the origin. This solution corresponds to the classical Neumann function Yν(x) [5]. We
consider the behavior of equation (5) around the infinity. We set 1/t, formally t2 7→ t and
z(t) = y(1/t). Then z(t) satisfies[
σ2p −
{
(pν + p−ν)− p−2−ν
t
}
σp + 1
]
z (t) = 0. (6)
We set E(t) = 1/θp(−pν+2t) and f(t) =
∑
n≥0
ant
n, a0 = 1. We assume that z(t) can be
described as
z(t) = E(t)f(t) =
1
θp(−pν+2t)
∑
n≥0
ant
n
.
Since E(t) satisfies the following q-difference equation
σpE(t) = −pν+2tE(t), σ2pE(t) = p2ν+5t2E(t),
we can check out that the function f(t) satisfies the equation{
p2ν+5t2σ2p + pν+2(pν + p−ν)tσp − σp + 1
}
f(t) = 0. (7)
2.3 The p-Borel transformation and the p-Laplace transformation
We define the p-Borel transformation and the p-Laplace transformation to solve the equation (7),
following Zhang [8].
A Connection Formula of the Hahn–Exton q-Bessel Function 5
Definition 2. For f(t) =
∑
n≥0
ant
n, the p-Borel transformation is defined by
g(τ) = (Bpf) (τ) :=
∑
n≥0
anp
−n(n−1)
2 τn,
and the p-Laplace transformation is given by
(Lpg) (t) :=
1
2πi
∫
|τ |=r
g(τ)θp
(
t
τ
)
dτ
τ
.
Here, r0 > 0 is enough small number.
The p-Borel transformation is considered as a formal inverse of the p-Laplace transformation.
Lemma 1. We assume that the function f can be p-Borel transformed to the analytic func-
tion g(τ) around τ = 0. Then,
Lp ◦ Bpf = f.
Proof. We can prove this lemma calculating residues of the p-Laplace transformation around
the origin. �
The p-Borel transformation has the following operational relation.
Lemma 2. For any l,m ∈ Z≥0,
Bp
(
tmσlp
)
= p−
m(m−1)
2 τmσl−mp Bp.
Applying the p-Borel transformation to the equation (7) and using Lemma 2, g(τ) satisfies
the first order difference equation
g(pτ) =
(
1 + p2ν+2τ
) (
1 + p2τ
)
g(τ).
Since g(0) = 1, we get an infinite product of g(τ):
g(τ) =
1
(−p2ν+2τ ; p)∞(−p2τ ; p)∞
.
Then g(τ) has single poles at{
−p−2ν−2−k,−p−2−k; k ∈ Z≥0
}
.
We set
0 < r < r0 := min
{
1
|p2ν+2|
,
1
|p2|
}
.
and choose the radius r > 0 such that 0 < r < r0. By Cauchy’s residue theorem, the p-Laplace
transform of g(τ) is
f(t) =
1
2πi
∫
|τ |=r
g(τ)θp
(
t
τ
)
dτ
τ
= −
∑
k≥0
Res
{
g(τ)θp
(
t
τ
)
1
τ
; τ=−p−2ν−2−k
}
−
∑
k≥0
Res
{
g(τ)θp
(
t
τ
)
1
τ
; τ=−p−2−k
}
,
where 0 < r < r0. To calculate the residue, we use the following lemma.
6 T. Morita
Lemma 3. For any k ∈ N, λ ∈ C∗, we have
1. Res
{
1
(τ/λ; p)∞
1
τ
: τ = λp−k
}
=
(−1)k+1p
k(k+1)
2
(p; p)k(p; p)∞
,
2.
1
(λp−k; p)∞
=
(−λ)−kp
k(k+1)
2
(λ; p)∞ (p/λ; p)k
, λ 6∈ pZ.
Summing up all of the residues, we obtain the convergent series f(t) as follows
f(t) =
θp
(
−p2ν+2t
)
(p−2ν , p; p)∞
1ϕ1
(
0, p1+2ν ; p, x
)
+
θp
(
−p2t
)
(p2ν , p; p)∞
1ϕ1
(
0, p1−2ν ; p, p−2νx
)
,
where xt = 1. Therefore, we acquire the connection formula for z(t) = E(t)f(t).
3 The limit of the connection formula
In this section, we show that the limit p → 1− of the connection formula gives a connection
formula of the Bessel function. At first, we assume that 0 < p < 1 and 0 <
√
p < 1. For the
Bessel function, we set the Hankel function of the first and the second kind H
(1)
ν (z) and H
(2)
ν (z).
Definition 3. The Hankel function of the first kind is given by
H(1)
ν (z) :=
Γ
(
1
2 − ν
)
πi
√
π
(z
2
)ν ∫ (1+)
1+∞i
eizt
(
t2 − 1
)ν− 1
2 dt, −π < arg z < 2π.
The Hankel function of the second kind is defined by
H(2)
ν (z) :=
Γ
(
1
2 − ν
)
πi
√
π
(z
2
)ν ∫ (−1−)
−1+∞i
eizt
(
t2 − 1
)ν− 1
2 dt, −2π < arg z < π.
The contour for H
(1)
ν (z) is a path starting from t = +1 +∞i, rounding the circle around
t = 1 counterclockwise, and going back to t = +1 +∞i. Moreover, the contour for H
(2)
ν (z) is
a path starting from t = −1 +∞i, rounding the circle around t = 1 clockwise, and going back
to t = −1 +∞i.
The Hankel functions can be written by Jν(z):
H(1)
ν (z) =
ie−νπi
sin νπ
{
Jν(z)− eνπiJ−ν(z)
}
, (8)
H(2)
ν (z) = − ieνπi
sin νπ
{
Jν(z)− e−νπiJ−ν(z)
}
. (9)
The Hankel functions have asymptotic expansions around z = 0 [4]:
H(1)
ν (z) ∼
(
2
πz
) 1
2
eiζ
∑
s≥0
is
As(ν)
zs
, −π + δ ≤ arg z ≤ 2π − δ,
H(2)
ν (z) ∼
(
2
πz
) 1
2
e−iζ
∑
s≥0
(−i)sAs(ν)
zs
, −2π + δ ≤ arg z ≤ π − δ,
as z →∞. Here, δ is an any small constant,
As(ν) =
(4ν2 − 12)(4ν2 − 32) · · ·
{
4ν2 − (2s− 1)2
}
s!8s
and
ζ = z − 1
2
νπ − 1
4
π.
In this sense, (8) and (9) considered as connection formula of the Bessel equation.
A Connection Formula of the Hahn–Exton q-Bessel Function 7
3.1 Limit of the connection formula
We rewrite the connection formula in Theorem 1 in order to take a limit p → 1−. We set new
functions hν(t; p) and J±ν (x; p). We set hν(t; p) := (p1/2, p1/2; p)∞z(t). For any x ∈ C∗ \ [−λ; p]
and λ ∈ C∗, J+
ν,λ(x; p) is
J+
ν,λ(x; p) :=
(pν+1; p)∞
(p; p)∞
θp
(
λpν
x
)
θp
(
λ
x
) 1ϕ1
(
0; p1+2ν ; p, x
)
.
Similarly, J−ν,λ(x; p) is
J−ν,λ(x; p) :=
(pν+1; p)∞
(p; p)∞
θp
(
λpν
x
)
θp
(
λ
x
) 1ϕ1
(
0; p1+2ν ; p, p2νx
)
.
We remark that the function θp(λp
ν/x)/θp(λ/x) satisfies the following q-difference equation
u(px) = pνu(x),
which is also satisfied by the function u(x) = xν . We remark that the pair (J+
ν,λ(x; p), J−−ν,λ(x; p))
gives a fundamental system of solutions of equation (6) if ν 6∈ Z. We set the function C+
ν (λ, t; p)
and C−ν (λ, t; p) as follow:
Definition 4. For any λ ∈ C∗, C+
ν (λ, t; p) is
C+
ν (λ, t; p) :=
(p
1
2 , p
1
2 ; p)∞
(pν+1, p−2ν ; p)∞
θp(−p2ν+2t)
θp(−pν+2t)
θp(λt)
θp(λpνt)
.
Similarly, the function C−ν (λ, t; p) is
C−ν (λ, t; p) :=
(p
1
2 , p
1
2 ; p)∞
(p−ν+1, p2ν ; p)∞
θp(−p2t)
θp(−pν+2t)
θp(λt)
θp(λp−νt)
.
Then, C+
ν (λ, t; p) and C−ν (λ, t; p) are single valued as a function of t. The function C+
ν (λ, t; p)
and C−ν (λ, t; p) are the p-elliptic functions. By using these new functions, our connection formula
is rewritten by
hν
(
1
x
; p
)
= C+
ν
(
λ,
1
x
; p
)
J+
ν (x; p) + C−ν
(
λ,
1
x
; p
)
J−−ν,λ(x; p).
Theorem 2. For any x ∈ C∗ \ (−∞, 0] where arg x ∈ (−π, π), we have
lim
p→1−
hν
(
1
(1− p)2x
; p
)
= −ie−νπiH(2)
2ν (2
√
x).
Here, H
(2)
2ν (·) is the Hankel function of the second kind.
The aim of this section is to give a proof of the theorem above.
By the definition, hν
(
1/{(1− p)2x}; p
)
can be described as follows
hν
(
1
(1− p)2x
; p
)
=
{
(p
1
2 , p
1
2 ; p)∞
(p−2ν , p; p)∞
(1− p)2ν
}θp
(
− p2ν+2
x(1−p)2
)
θp
(
− pν+2
x(1−p)2
)(1− p)−2ν
.
×
{
1ϕ1
(
0; p1+2ν ; p, (1− p)2x
)}
8 T. Morita
+
{
(p
1
2 , p
1
2 ; p)∞
(p2ν , p; p)∞
(1− p)−2ν
}θp
(
− p2
x(1−p)2
)
θp
(
− pν+2
x(1−p)2
)(1− p)2ν
×
{
1ϕ1
(
0; p1−2ν ; p, p−2ν(1− p)2x
)}
. (10)
We consider the limit of each part {·}.
Lemma 4. For any ν ∈ C∗ \ Z, we have
lim
p→1−
(p
1
2 , p
1
2 ; p)∞
(p−2ν , p; p)∞
(1− p)2ν = − 1
sin(2νπ)Γ(2ν + 1)
.
Proof. We can check out as follows
(p
1
2 , p
1
2 ; p)∞
(p−2ν , p; p)∞
(1− p)2ν =
(p;p)∞
(p−2ν ;p)∞
(1− p)1+2ν{
(p;p)∞
(p
1
2 ;p)∞
(1− p)
1
2
}{
(p;p)∞
(p
1
2 ;p)∞
(1− p)
1
2
} =
Γp(−2ν)
Γp
(
1
2
)
Γp
(
1
2
) .
Here, Γq(·) is Jackson’s q-gamma function which is defined by
Γq(x) :=
(q; q)∞
(qx; q)∞
(1− q)1−x, 0 < q < 1.
This function satisfies lim
q→1−
Γq(x) = Γ(x) [2]. Therefore,
lim
p→1−
(p
1
2 , p
1
2 ; p)∞
(p−2ν , p; p)∞
(1− p)2ν =
Γ(−2ν)
Γ
(
1
2
)
Γ
(
1
2
) .
By Euler’s reflection formula of the gamma function, we get
Γ(−2ν)
Γ
(
1
2
)
Γ
(
1
2
) = − 1
sin(2νπ)Γ(2ν + 1)
.
Therefore, we get the conclusion. �
If we replace ν by −ν, we get the limit
lim
p→1−
(p
1
2 , p
1
2 ; p)∞
(p2ν , p; p)∞
(1− p)−2ν =
1
sin(2νπ)Γ(1− 2ν)
.
In [8], the following proposition can be found:
Proposition 1. For any x ∈ C∗ (−π < arg x < π), we have
lim
p→1−
θp
(
pν1
(1−p2)x
)
θp
(
pν2
(1−p2)x
) (1− p2)ν2−ν1 = xν1−ν2 ,
and
lim
p→1−
θp
(
− pν1
(1−p2)x
)
θp
(
− pν2
(1−p2)x
) (1− p2)ν2−ν1 = (−x)ν1−ν2 .
A Connection Formula of the Hahn–Exton q-Bessel Function 9
Lemma 5. For any x ∈ C∗ (−π < arg x ≤ π) and fixed constant K, we have
θp(−
√
p)θp
(
−K
x
)
= θ√p
(√
K
x
)
θ√p
(
−
√
K
x
)
.
Proof. From Jacobi’s triple product identity and (a2; q2)n = (a,−a; q)n, we obtain
(
√
p;
√
p)∞
(−√p;√p)∞
θp
(
−K
x
)
= θ√p
(√
K
x
)
θ√p
(
−
√
K
x
)
.
We remark that (
√
p;
√
p)∞/(−
√
p;
√
p)∞ can be rewritten as follows [2]:
(
√
p;
√
p)∞
(−√p;√p)∞
=
∑
n∈Z
(−1)n(
√
p)n
2
= θp(−
√
p).
We obtain the conclusion. �
Therefore, we obtain the following relation.
Corollary 1. For any x ∈ C∗ (−π < arg x ≤ π), we have
θp
(
p2ν+2 −1
(1−p)2x
)
θp
(
pν+2 −1
(1−p)2x
) =
θ√p
(
pν+1 1
(1−p)
√
x
)
θ√p
(
pν+1 −1
(1−p)
√
x
)
θ√p
(
p
ν
2
+1 1
(1−p)
√
x
)
θ√p
(
p
ν
2
+1 −1
(1−p)
√
x
) (11)
and
θp
(
p2 −1
(1−p)2x
)
θp
(
pν+2 −1
(1−p)2x
) =
θ√p
(
p 1
(1−p)
√
x
)
θ√p
(
p −1
(1−p)
√
x
)
θ√p
(
p
ν
2
+1 1
(1−p)
√
x
)
θ√p
(
p
ν
2
+1 −1
(1−p)
√
x
) . (12)
We consider the limit p→ 1− (i.e.,
√
p→ 1−) of (11) and (12).
Lemma 6. For any x ∈ C∗ \ (−∞, 0] (−π < arg x ≤ π), we have
1. lim
p→1−
θp
(
− p2ν+2
x(1−p)2
)
θp
(
− pν+2
x(1−p)2
)(1− p)−2ν = eνπixν and
2. lim
p→1−
θp
(
− p2
x(1−p)2
)
θp
(
− pν+2
x(1−p)2
)(1− p)2ν = e−νπix−ν .
Proof. Combining Proposition 1 and Corollary 1, we consider the limit
√
p→ 1− as follows:
θp
(
p2ν+2 −1
(1−p)2x
)
θp
(
pν+2 −1
(1−p)2x
) (1− p)−2ν =
θ√p
(
pν+1 1
(1−p)
√
x
)
θ√p
(
pν+1 −1
(1−p)
√
x
)
θ√p
(
p
ν
2
+1 1
(1−p)
√
x
)
θ√p
(
p
ν
2
+1 −1
(1−p)
√
x
)(1− p)−2ν
=
θ
√
p
(
(
√
p)2ν+2 1
(1−(√p)2)
√
x
)
θ√p
(
(
√
p)ν+2 1
(1−{√p)2)
√
x
) {1− (
√
p)2
}−ν
×
θ
√
p
(
−(
√
p)2ν+2 1
(1−(√p)2)
√
x
)
θ√p
(
−(
√
p)ν+2 1
(1−{√p)2)
√
x
) {1− (
√
p)2
}−ν
→ (
√
x)ν · (−
√
x)ν = (−x)ν = eνπixν ,
√
p→ 1−.
Similarly, we can prove the latter one. We obtain the conclusion. �
10 T. Morita
We consider the last part.
Lemma 7. For any x ∈ C∗, we have
lim
p→1−
1ϕ1
(
0; p1+2ν ; p, (1− p)2x
)
= 0F1 (−, 1 + 2ν;−x)
and
lim
p→1−
1ϕ1
(
0; p1−2ν ; p, p−2ν(1− p)2x
)
= 0F1 (−, 1− 2ν;−x) .
Proof. We check each of the term of
1ϕ1
(
0; p1+2ν ; p, (1− p)2x
)
=
∑
n≥0
1
(p1+2ν , p; p)n
(−1)np
n(n−1)
2
{
(1− p)2x
}n
.
For any n ≥ 0,
1
(p1+2ν , p; p)n
(−1)np
n(n−1)
2
{
(1− p)2x
}n
=
(1− p)n(1− p)n
(p1+2ν ; p)n(p; p)n
p
n(n−1)
2 (−x)n → 1
(1 + 2ν)n · n!
(−x)n , p→ 1−.
Summing up all terms, we get∑
n≥0
1
(1 + 2ν)n · n!
(−x)n = 0F1 (−, 1 + 2ν;−x) .
Therefore, we obtain the conclusion. Similarly, we can prove the latter. �
We give the proof of Theorem 2.
Proof. Apply Lemma 4, Lemma 6 and Lemma 7 to (10), we obtain
hν
(
1
(1− p)2x
; p
)
→
{
− 1
sin(2νπ)Γ(1 + 2ν)
}
eνπixν0F1 (−, 1 + 2ν;−x)
+
{
1
sin(2νπ)Γ(1− 2ν)
}
e−νπix−ν0F1 (−, 1− 2ν;−x)
=
−eνπiJ2ν (2
√
x) + e−νπiJ−2ν (2
√
x)
sin(2νπ)
=
e−νπi
i
H
(2)
2ν
(
2
√
x
)
, p→ 1−.
Therefore, we acquire the conclusion. �
Acknowledgements
The author would like to express his deepest gratitude to Professor Yousuke Ohyama for many
valuable comments. The author also expresses his thanks to Professor Lucia Di Vizio for fruitful
discussions when she was invited to the University of Tokyo in the winter 2011. The author
would like to give thanks to the referee for some useful comments.
A Connection Formula of the Hahn–Exton q-Bessel Function 11
References
[1] Birkhoff G.D., The generalized Riemann problem for linear differential equations and the allied problems
for linear difference and q-difference equations, Proc. Amer. Acad. 49 (1913), 521–568.
[2] Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Appli-
cations, Vol. 96, Cambridge University Press, Cambridge, 2004.
[3] Hahn W., Beiträge zur Theorie der Heineschen Reihen. Die 24 Integrale der Hypergeometrischen q-
Differenzengleichung. Das q-Analogon der Laplace-Transformation, Math. Nachr. 2 (1949), 340–379.
[4] Olver F.W.J., Asymptotics and special functions, Computer Science and Applied Mathematics, Academic
Press, New York – London, 1974.
[5] Swarttouw R.F., Meijer H.G., A q-analogue of the Wronskian and a second solution of the Hahn–Exton
q-Bessel difference equation, Proc. Amer. Math. Soc. 129 (1994), 855–864.
[6] Watson G.N., The continuation of functions defined by generalized hypergeometric series, Trans. Camb.
Phil. Soc. 21 (1910), 281–299.
[7] Zhang C., Remarks on some basic hypergeometric series, in Theory and Applications of Special Functions,
Dev. Math., Vol. 13, Springer, New York, 2005, 479–491.
[8] Zhang C., Sur les fonctions q-Bessel de Jackson, J. Approx. Theory 122 (2003), 208–223.
[9] Zhang C., Une sommation discrè pour des équations aux q-différences linéaires et à coefficients, analytiques:
théorie générale et exemples, in Differential Equations and Stokes Phenomenon, World Sci. Publ., River
Edge, NJ, 2002, 309–329.
http://dx.doi.org/10.1017/CBO9780511526251
http://dx.doi.org/10.1002/mana.19490020604
http://dx.doi.org/10.2307/2160480
http://dx.doi.org/10.1007/0-387-24233-3_22
http://dx.doi.org/10.1016/S0021-9045(03)00073-X
http://dx.doi.org/10.1142/9789812776549_0012
1 Introduction
2 The connection formula
2.1 The theta function of Jacobi
2.2 The Hahn-Exton q-Bessel function
2.3 The p-Borel transformation and the p-Laplace transformation
3 The limit of the connection formula
3.1 Limit of the connection formula
References
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