A Connection Formula for the q-Confluent Hypergeometric Function
We show a connection formula for the q-confluent hypergeometric functions ₂φ₁(a,b;0;q,x). Combining our connection formula with Zhang's connection formula for ₂φ₀(a,b;−;q,x), we obtain the connection formula for the q-confluent hypergeometric equation in the matrix form. Also we obtain the conn...
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| Zitieren: | A Connection Formula for the q-Confluent Hypergeometric Function / T. Morita // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 10 назв. — англ. |
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| citation_txt | A Connection Formula for the q-Confluent Hypergeometric Function / T. Morita // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 10 назв. — англ. |
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| description | We show a connection formula for the q-confluent hypergeometric functions ₂φ₁(a,b;0;q,x). Combining our connection formula with Zhang's connection formula for ₂φ₀(a,b;−;q,x), we obtain the connection formula for the q-confluent hypergeometric equation in the matrix form. Also we obtain the connection formula of Kummer's confluent hypergeometric functions by taking the limit q→1− of our connection formula.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 050, 13 pages
A Connection Formula
for the q-Confluent Hypergeometric 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 October 09, 2012, in final form July 21, 2013; Published online July 26, 2013
http://dx.doi.org/10.3842/SIGMA.2013.050
Abstract. We show a connection formula for the q-confluent hypergeometric functions
2ϕ1(a, b; 0; q, x). Combining our connection formula with Zhang’s connection formula for
2ϕ0(a, b;−; q, x), we obtain the connection formula for the q-confluent hypergeometric equa-
tion in the matrix form. Also we obtain the connection formula of Kummer’s confluent
hypergeometric functions by taking the limit q → 1− of our connection formula.
Key words: q-Borel–Laplace transformation; q-difference equation; connection problem; q-
confluent hypergeometric function
2010 Mathematics Subject Classification: 33D15; 34M40; 39A13
1 Introduction
We show a new connection formula for two independent solutions to the q-confluent hypergeo-
metric equation
(1− abqx)u
(
q2x
)
− {1− (a+ b)qx}u(qx)− qxu(x) = 0. (1)
We use notations in accordance with [2]. Assume that q ∈ C∗ satisfies 0 < |q| < 1 and a/b is
not an integer power of q. The basic hypergeometric series rϕs is defined by
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,
where (a; q)n is the q-shifted factorial
(a; q)n :=
{
1, n = 0,
(1− a)(1− aq) · · ·
(
1− aqn−1
)
, n ≥ 1,
(a; q)∞ = lim
n→∞
(a; q)n,
and
(a1, a2, . . . , am; q)∞ = (a1; q)∞(a2; q)∞ · · · (am; q)∞.
Equation (1) has solutions
u1(x) = 2ϕ0(a, b;−; q, x), u2(x) =
(abx; q)∞
θ(−qx)
2ϕ1
(q
a
,
q
b
; 0; q, abx
)
mailto:t-morita@cr.math.sci.osaka-u.ac.jp
http://dx.doi.org/10.3842/SIGMA.2013.050
2 T. Morita
around the origin and has solutions
v1(x) = x−α2ϕ1
(
a, 0;
aq
b
; q,
q
abx
)
, v2(x) = x−β2ϕ1
(
b, 0;
bq
a
; q,
q
abx
)
around infinity. Here qα = a and qβ = b.
The connection formula for a linear q-difference equation of the second order is a linear
relation between u1(x), u2(x) and v1(x), v2(x):(
u1(x)
u2(x)
)
=
(
C11(x) C12(x)
C21(x) C22(x)
)(
v1(x)
v2(x)
)
,
where the connection coefficients Cjk(x) are q-periodic functions.
C. Zhang [10] proposed the connection formula for u1(x),
2f0(a, b;λ, q, x) =
(b; q)∞(
b
a ; q
)
∞
θ(aλ)
θ(λ)
θ
( qax
λ
)
θ
( qx
λ
) 2ϕ1
(
a, 0;
aq
b
; q,
q
abx
)
+
(a; q)∞(
a
b ; q
)
∞
θ(bλ)
θ(λ)
θ
(
qbx
λ
)
θ
( qx
λ
) 2ϕ1
(
b, 0;
bq
a
; q,
q
abx
)
, (2)
for x ∈ C∗ \ [−λ; q]. Here 2f0(a, b;λ, q, x) is the q-Borel–Laplace transform of the divergent series
2ϕ0(a, b;−; q, x), i.e.,
2f0(a, b;λ, q, x) := L+q,λ ◦ B
+
q 2ϕ0(a, b;−; q, x).
In this paper, we show the following new connection formula for u2(x):
2ϕ1(q/a, q/b; 0; q, abx) =
(q/a; q)∞
(b/a; q)∞
(aqx, 1/ax; q)∞
(abx; q)∞
2ϕ1(a, 0; aq/b; q, 1/abx)
+
(q/b; q)∞
(a/b; q)∞
(bqx, 1/bx; q)∞
(abx; q)∞
2ϕ1(b, 0; bq/a; q, 1/abx).
Since u1(x) is a divergent series, the q-Stokes phenomenon appears in Zhang’s connection
formula. But our formula gives the exact relation between the convergent series u2(x) around
the origin and the convergent series v1(x), v2(x) around infinity.
The theta function of Jacobi is given by the series
θq(x) :=
∑
n∈Z
q
n(n−1)
2 xn, ∀x ∈ C∗,
we denote θ(x) shortly. The theta function is written by the product form
θ(x) =
(
q,−x,− q
x
; q
)
∞
,
which is known as Jacobi’s triple product identity. For any k ∈ Z, the theta function also
satisfies the q-difference equation
θ
(
qkx
)
= q−
k(k−1)
2 x−kθ(x).
The theta function satisfies the inversion formula θ(x) = xθ(1/x). For all fixed λ ∈ C∗, we
define a q-spiral [λ; q] := λqZ = {λqk; k ∈ Z}. Note that θ(λqk/x) = 0 if and only if x ∈ [−λ; q].
A Connection Formula for the q-Confluent Hypergeometric Function 3
At first, we review the confluent hypergeometric equation (CHGE). In 1813 [3], C.F. Gauss
studied the hypergeometric series
2F1(α, β; γ; z) =
∑
n≥0
(α)n(β)n
(γ)nn!
zn, γ 6= 0,−1,−2, . . . ,
where (α)n = α{α+ 1} · · · {α+ (n− 1)}.
More generally, the generalized hypergeometric series is given by
rFs(α1, . . . , αr;β1, . . . , βs; z) =
∑
n≥0
(α1)n · · · (αr)n
(β1)n · · · (βs)nn!
zn.
The hypergeometric function 2F1(α, β; γ; z) satisfies the second-order differential equation
z(1− z)d
2u
dz2
+ {γ − (α+ β + 1)z} du
dz
− αβu = 0. (3)
Gauss gave the connection formula for the function 2F1(α, β; γ, z). We put z 7→ z/β, take the
limit β →∞ in equation (3), and obtain the confluent hypergeometric equation (CHGE)
z
d2u
dz2
+ (γ − z)du
dz
− αu = 0. (4)
Solutions of (4) around the origin are
û1(z) = 1F1(α; γ; z)
and
û2(z) = z1−γ1F1(α− γ + 1, 2− γ, z). (5)
Solutions around infinity are given by the divergent series
v̂1(z) = (−z)−α2F0(α, α− γ + 1;−; 1/z)
and
v̂2(z) = ezzα−γ2F0(1− α, γ − α;−; 1/z).
The asymptotic expansion of 1F1(α; γ; z) is given by
1F1(α; γ; z) ∼ Γ(γ)
Γ(γ − α)
(−z)−α2F0(α, α− γ + 1;−;−1/z)
+
Γ(γ)
Γ(α)
ezzα−γ2F0(1− α, γ − α;−; 1/z), (6)
where −π/2 < arg z < 3π/2. Note that the connection formula for the second solution around
infinity (5) can be derived from (6). In Section 2 we deal with another degeneration of equa-
tion (3) which is slightly different from the standard way.
It is known that there exists a q-analogue of 2F1(α, β; γ; z), which was introduced by E. Heine
in 1847 as
2ϕ1(a, b; c; q, x) :=
∑
n≥0
(a; q)n(b; q)n
(c; q)n(q; q)n
xn.
4 T. Morita
We assume that c is not integer powers of q. The function 2ϕ1(a, b; c; q, x) satisfies the second-
order q-difference equation (q-HGE)
x(c− abqx)D2
qu+
[
1− c
1− q
+
(1− a)(1− b)− (1− abq)
1− q
x
]
Dqu−
(1− a)(1− b)
(1− q)2
u = 0, (7)
where Dq is the q-derivative operator defined for fixed q by
Dqf(x) =
f(x)− f(qx)
(1− q)x
.
By replacing a, b, c by qα, qβ, qγ and then letting q → 1−, equation (7) tends to the hypergeo-
metric equation (3). The q-hypergeometric equation (7) can be rewritten as
(c− abqx)u
(
q2x
)
− {c+ q − (a+ b)qx}u(qx) + q(1− x)u(x) = 0. (8)
If we set x 7→ cx and c → ∞ in (8), we obtain the q-confluent hypergeometric equation (1).
Equation (1) is considered as a q-analogue of CHGE. Note that the first solution u1(x) is
a divergent series and u2(x) is a convergent series around the origin. Therefore we should study
the connection formula for u1(x) and u2(x) independently. We need different types of q-Borel–
Laplace transformations to obtain the connection formula for u1(x) and u2(x). This point is
essentially different from the differential equation case.
We study connection problems for linear q-difference equations with irregular singular points.
The irregularity of q-difference equations are studied using the Newton polygons by J.-P. Ramis,
J. Sauloy and C. Zhang [6]. For any q-difference operator P = σnq + a1(z)σ
n−1
q + · · · + an(z),
the Newton polygon is defined as the convex hull of {(i, j) ∈ Z2 | j ≥ v0(ai)}, provided that v0
are z-adic valuation in suitable fields. Graphically, the irregularity of q-difference equations and
q-difference modules are the height of the Newton polygon (from the bottom to the upper right
end).
Connection problems for linear q-difference equations [5] with regular singular points were
studied by G.D. Birkhoff [1]. Linear q-difference equations have formal power series solutions
xα
∑
n≥0
anx
n around the origin and xβ
∑
n≥0
bnx
−n around infinity for generic exponents. But
for connection problems for linear q-difference equations, we replace the function xκ with the
function θ(x)/θ(kx), where k = qκ, since these functions satisfy the same q-difference equation
σqf(x) = qκf(x). Then, the fundamental system of solutions is given by single valued functions
which have single poles at suitable q-spirals. Therefore, each connection coefficient has the
periods q and e2πi. The first example was given by G.N. Watson [7]. But a few examples of
irregular singular cases are known [4, 8, 9]. In this paper, we give a connection formula for the
q-confluent type function using the q-Borel–Laplace transformations.
In 1910, Watson [7] showed that the connection formula for the series 2ϕ1(a, b; c; q, x) has the
following form
2ϕ1 (a, b; c; q, x) =
(b, c/a; q)∞
(c, b/a; q)∞
θ(−ax)
θ(−x)
2ϕ1
(
a,
aq
c
;
aq
b
; q,
cq
abx
)
+
(a, c/b; q)∞
(c, a/b; q)∞
θ(−bx)
θ(−x)
2ϕ1
(
b,
bq
c
;
bq
a
; q,
cq
abx
)
. (9)
Note that we can not set a = 0 or b = 0 directly in this formula.
In 2002, C. Zhang [10] showed one of the connection formula for the q-CHGE (2). The
q-Borel–Laplace transformations were studied by C. Zhang in [10] (see Section 2 for more
details). When we study connection problems for q-difference equations, this resummation
A Connection Formula for the q-Confluent Hypergeometric Function 5
method becomes a powerful tool. Note that we can find a new parameter λ in the resumma-
tion 2f0(a, b;λ, q, x). Here λ is the direction of the summation. This parameter brings us new
viewpoints for the study of the q-Stokes phenomenon.
It is known that there exist two different types of the q-Borel–Laplace transformations.
The q-Borel–Laplace transformations of the first kind are defined in [10] and the q-Borel–
Laplace transformations of the second kind are studied in [9]. These q-Borel transformations
are formal inverse transformations of each of the q-Laplace transformations.
C. Zhang presented a connection formula for the series 2ϕ0(a, b;−; q, x) by the q-Borel–Laplace
transformations of the first kind. But the connection formula for the second solution of (1) is
not known. In this paper, we show the second connection formula for q-CHGE with the using
of the q-Borel transformation and the q-Laplace transformation of the second kind. Combining
with Zhang’s connection formula, we obtain the connection formula in the matrix form (see
Theorem 2). Using Watson’s formula (9) we also give another proof of the new connection
formula in Section 2.5.
In Section 3 we consider the limit q → 1− of our connection formula. If we take the limit
q → 1−, we formally obtain the connection formula for the confluent hypergeometric series 2F0.
2 The connection formula and the connection matrix
We review a q-confluent hypergeometric equation in Section 2.1. Then we show a connection
formula for the q-confluent hypergeometric function, which is different from Zhang’s formula.
2.1 Confluent hypergeometric equation
For the confluent hypergeometric equation (3), we take another degeneration. We put z 7→ zγ
and take the limit γ →∞. Then we obtain
z2
d2u
dz2
− {1− (α+ β + 1)z} du
dz
+ αβu = 0. (10)
Solutions to (10) around the origin are given by the divergent series
ũ1(z) = 2F0(α, β;−, z) and ũ2(z) = e
1
z (−z)1−α−β 2F0(1− α, 1− β;−, z).
Solutions around infinity are given by the convergent series
ṽ1(z) = (−z)α 1F1(α, 1 + α− β,−1/z) and ṽ2(z) = (−z)β 1F1(β, 1 + β − α,−1/z).
We consider a q-analogue of the confluent hypergeometric equation (8). The second-order
q-difference equation
x {abqx− (1− q)}D2
qu(x) +
{
1− (1− a)(1− b)− (1− abq)
1− q
x
}
Dqu(x)
+
(1− a)(1− b)
(1− q)2
u(x) = 0. (11)
can be rewritten as
(1− abqx)u
(
xq2
)
− {1− (a+ b)qx}u(xq)− qxu(x) = 0, (12)
which is called a q-confluent hypergeometric equation. When we take q → 1−, the limit of (11)
is the differential equation (8), provided that a = qα, b = qβ.
6 T. Morita
2.2 Local solutions to the q-confluent hypergeometric equation
Consider the connection problem of (12). At first we show local solutions for (12) around x = 0
and x =∞.
Lemma 1. Equation (12) has solutions
u1(x) = 2ϕ0(a, b;−; q, x), (13)
u2(x) =
(abx; q)∞
θ(−qx)
2ϕ1
(q
a
,
q
b
; 0; q, abx
)
(14)
around the origin and solutions
v1(x) = x−α2ϕ1
(
a, 0;
aq
b
; q,
q
abx
)
, (15)
v2(x) = x−β2ϕ1
(
b, 0;
bq
a
; q,
q
abx
)
, (16)
around inf inity, provided that a = qα and b = qβ.
Proof. We show a fundamental system of solutions of (12) around x = 0. If set u(x) =
∑
n≥0
anx
n,
a0 = 1, then we obtain
u1(x) = 2ϕ0(a, b;−; q, x).
We set E(x) = 1/θ(−qx) and f(x) =
∑
n≥0
anx
n, a0 = 1 to obtain another solution solution around
the origin. We assume that u(x) = E(x)f(x). Note that the function E(x) has the following
property
σqE(x) = −qxE(x), σ2qE(x) = q3x2E(x).
Therefore, we obtain the equation[
q3x(1− abqx)σ2q + q {1− (a+ b)qx}σq − q
]
f(x) = 0. (17)
Since the infinite product (abx; q)∞ satisfies the following q-difference relation
σq [(abx; q)∞] =
1
1− abx
(abx; q)∞,
we obtain the second solution. Therefore, solutions of equation (12) around the origin are
given by
u1(x) = 2ϕ0(a, b;−; q, x), u2(x) =
(abx; q)∞
θ(−qx)
2ϕ1
(q
a
,
q
b
; 0; q, abx
)
.
Around x =∞, we can easily determine local solutions by setting
v(x) =
θ(aµx)
θ(µx)
∑
n≥0
anx
−n, a0 = 1,
for any fixed µ ∈ C∗ and x ∈ C∗ \ [−µ; q]. �
Here u1(x) is a divergent series and u2(x), v1(x) and v2(x) are convergent series [2]. Therefore,
the q-Stokes phenomenon occurs for u1(x).
A Connection Formula for the q-Confluent Hypergeometric Function 7
Definition 1. For any f(x) =
∑
n≥0
anx
n, the q-Borel transformation B+q is
(
B+q f
)
(ξ) = ϕ(ξ) :=
∑
n≥0
anq
n(n−1)
2 ξn,
and the q-Laplace transformation L+q,λ is
(
L+q,λϕ
)
(x) :=
∑
n∈Z
ϕ(qnλ)
θ
(
qnλ
x
) .
C. Zhang determined a resummation of (13) by the q-Borel–Laplace transformations of the
first kind as follows
2f0(a, b;λ, q, x) := L+q,λ ◦ B
+
q 2ϕ0(a, b;−; q, x).
He also presented a connection formula (2) for 2f0(a, b;λ, q, x).
But the connection formula between (14) and (15), (16) is not known. In the next section,
we show the second connection formula by means of the q-Borel–Laplace transformations of the
second kind.
2.3 The second connection formula
We define the q-Borel transformation and the q-Laplace transformation of the second kind.
These transformations are introduced by C. Zhang to obtain the solution of equation (17).
Definition 2. For f(x) =
∑
n≥0
anx
n, the q-Borel transformation is defined by
g(ξ) =
(
B−q f
)
(ξ) :=
∑
n≥0
anq
−n(n−1)
2 ξn,
and the q-Laplace transformation is
(
L−q g
)
(x) :=
1
2πi
∫
|ξ|=r
g(ξ)θq
(
x
ξ
)
dξ
ξ
.
Here r > 0 is a sufficiently small number. The q-Borel transformation is considered as
a formal inverse of the q-Laplace transformation.
Lemma 2 ([9]). We assume that the function f can be q-Borel transformed to the analytic
function g(ξ) around ξ = 0. Then, we have
L−q ◦ B−q f = f.
Proof. We can prove this lemma calculating residues of the q-Laplace transformation around
the origin. �
The q-Borel transformation satisfies the following operational relation.
Lemma 3. For any l,m ∈ Z≥0,
B−q
(
xmσlq
)
= q−
m(m−1)
2 ξmσl−mq B−q .
8 T. Morita
We apply the q-Borel transformation to equation (17) and use Lemma 3. We use the no-
tation g(ξ) as the q-Borel transform of u2(x). We check out that g(ξ) satisfies the first-order
q-difference equation
g(qξ) =
(1 + aqξ)(1 + bqξ)
(1 + q2ξ)
g(ξ).
Since g(0) = a0 = 1, we have the infinite product of g(ξ) as follows
g(ξ) =
(−q2ξ; q)∞
(−qaξ; q)∞(−qbξ; q)∞
.
Note that g(ξ) has single poles at{
ξ ∈ C∗; ξ = − 1
aqk+1
,− 1
bqk+1
, k ∈ Z≥0
}
.
We set
r0 := max
{
1
|aq|
,
1
|bq|
}
and choose the radius r > 0 such that 0 < r < r0. By Cauchy’s residue theorem, the q-Laplace
transform of g(ξ) is
f(x) =
1
2πi
∫
|ξ|=r
g(ξ)θ
(
x
ξ
)
dξ
ξ
= −
∑
k≥0
Res
{
g(ξ)θ
(
x
ξ
)
1
ξ
; ξ = − 1
aqk+1
}
−
∑
k≥0
Res
{
g(ξ)θ
(
x
ξ
)
1
ξ
; ξ = − 1
bqk+1
}
,
where 0 < r < r0. Since there exists a positive constant CN (for any integer N) s.t.,
|g(ξ)| ≤ CNξ−N .
The following lemma plays a key role to calculate the residue.
Lemma 4. For any k ∈ N, λ ∈ C∗, we have:
1) Res
{
1
(ξ/λ; q)∞
1
ξ
: ξ = λq−k
}
=
(−1)k+1q
k(k+1)
2
(q; q)k(q; q)∞
,
2)
1
(λq−k; q)∞
=
(−λ)−kq
k(k+1)
2
(λ; q)∞ (q/λ; q)k
, λ 6∈ qZ.
Summing up all residues, we obtain f(x) as follows
f(x) =
( q
a ; q
)
∞(
b
a , q; q
)
∞
θ(−aqx)
θ(−qx)
2ϕ1
(
a, 0;
aq
b
; q,
q
abx
)
+
( q
b ; q
)
∞(
a
b , q; q
)
∞
θ(−bqx)
θ(−qx)
2ϕ1
(
b, 0;
bq
a
; q,
q
abx
)
.
Therefore, we obtain the following theorem.
Theorem 1. For any x 6∈ [1; q], we have
u2(x) =
(abx; q)∞
θ(−qx)
2ϕ1
(q
a
,
q
b
; 0; q, abx
)
=
( q
a ; q
)
∞(
b
a , q; q
)
∞
θ(−aqx)
θ(−qx)
2ϕ1
(
a, 0;
aq
b
; q,
q
abx
)
+
( q
b ; q
)
∞(
a
b , q; q
)
∞
θ(−bqx)
θ(−qx)
2ϕ1
(
b, 0;
bq
a
; q,
q
abx
)
. (18)
A Connection Formula for the q-Confluent Hypergeometric Function 9
2.4 The connection matrix
Combining Zhang’s connection formula and Theorem 1, we give the connection matrix for equa-
tion (1). At first, we define a new fundamental system of solutions around infinity. For any λ,
µ ∈ C∗, x ∈ C∗ \ [−µ; q], Sµ(a, b; q, x) is
Sµ(a, b; q, x) :=
θ(aµx)
θ(µx)
2ϕ1
(
a, 0;
aq
b
; q,
q
abx
)
.
The function θ(aµx)/θ(µx) satisfies the following q-difference equation
u(qx) =
1
a
u(x),
which is also satisfied by the function u(x) = x−α, a = qα. Note that the pair (Sµ(a, b; q, x) ,
Sµ(b, a; q, x)) gives a fundamental system of solutions of equation (1) if a/b 6∈ qZ. We define
q-elliptic functions Cλµ(a, b; q, x) and Cµ(a, b; q, x).
Definition 3. For any λ, µ ∈ C∗ we set functions Cλµ(a, b; q, x) and Cµ(a, b; q, x) as follows
Cλµ(a, b; q, x) :=
(b; q)∞(
b
a ; q
)
∞
θ(aλ)
θ(λ)
θ
( qax
λ
)
θ
( qx
λ
) θ(µx)
θ(aµx)
,
Cµ(a, b; q, x) :=
( q
a ; q
)
∞(
b
a , q; q∞
) θ(−aqx)
θ(−qx)
θ(µx)
θ(aµx)
.
Then Cλµ(a, b; q, x) and Cµ(a, b; q, x) are single valued as functions of x. They satisfy the
following relation
Cλµ(a, b; q, e2πix) = Cλµ(a, b; q, x), Cλµ(a, b; q, qx) = Cλµ(a, b; q, x)
and
Cµ
(
a, b; q, e2πix
)
= Cµ(a, b; q, x), Cµ(a, b; q, qx) = Cµ(a, b; q, x).
i.e. Cλµ(a, b; q, x) and Cµ(a, b; q, x) are q-elliptic functions. We set
2f1(a, b; q, x) := u2(x) =
(abx; q)∞
θ(−qx)
2ϕ1
(q
a
,
q
b
; 0; q, abx
)
.
Thus, we obtain the connection formula in the matrix form.
Theorem 2. For any λ, µ ∈ C∗, x ∈ C∗ \ [1; q] ∪ [−µ/a; q] ∪ [−µ/b; q] ∪ [−λ; q], we have(
2f0(a, b;λ, q, x)
2f1(a, b; q, x)
)
=
(
Cλµ(a, b; q, x) Cλµ(b, a; q, x)
Cµ(a, b; q, x) Cµ(b, a; q, x)
)(
Sµ(a, b; q, x)
Sµ(b, a; q, x)
)
.
2.5 Derivation from Watson’s formula
In this section, we give another proof of Theorem 1. Watson’s formula (9) is a connection formula
for the basic hypergeometric functions 2ϕ1(a, b; c; q, x). We derive the connection formula
(abx; q)∞
θ(−qx)
2ϕ1
(q
a
,
q
b
; 0; q, abx
)
=
(q/a; q)∞
(b/a, q; q)∞
θ(−aqx)
θ(−qx)
2ϕ1
(
a, 0;
aq
b
; q,
q
abx
)
+
(q/b; q)∞
(a/b, q; q)∞
θ(−bqx)
θ(−qx)
2ϕ1
(
b, 0;
bq
a
; q,
q
abx
)
from Watson’s formula (9). By take the limit c→ 0 of Watson’s formula, we obtain the following
proposition.
10 T. Morita
Proposition 1 ([10]). For any x ∈ C∗ \ qZ, we have
2ϕ1(a, b; 0; q, x) =
(b; q)∞
(b/a; q)∞
θ(−ax)
θ(−x)
1ϕ1
(
a;
aq
b
; q,
q2
bx
)
+
(a; q)∞
(a/b; q)∞
θ(−bx)
θ(−x)
1ϕ1
(
b;
bq
a
; q,
q2
ax
)
. (19)
In (19), we put a 7→ q/a, b 7→ q/b and x 7→ abx, we obtain the relation as follows
Corollary 1. For any x ∈ C∗ \ [ab; q], we have
2ϕ1
(q
a
,
q
b
; 0; q, abx
)
=
(q/b; q)∞
(a/b; q)∞
θ(−bqx)
θ(−abx)
1ϕ1
(
q
a
;
bq
a
; q,
q
ax
)
+
(q/a; q)∞
(b/a; q)∞
θ(−aqx)
θ(−abx)
1ϕ1
(q
b
;
aq
b
; q,
q
bx
)
. (20)
The function 1ϕ1(a; c; q, x) is related to 2ϕ1(a
′, 0; c′; q, x). Consider the relation between the
function 1ϕ1 and the function 2ϕ1.
Proposition 2. For any x ∈ C∗, we have
1ϕ1
(
a1; c1; q,
c1x
a1
)
= (x; q)∞2ϕ1
(
c1
a1
, 0; c1; q, x
)
, (21)
provided that c1/a1 6∈ qZ.
Proof. The function 1ϕ1(a1; c1; q, c1x/a1) satisfies the equation[
(c1 − c1qx)σ2q −
{
(c1 + q)− qc1
a1
x
}
σq + q
]
v(x) = 0.
We set v(x) = (x; q)∞ṽ(x), where ṽ(x) :=
∑
n≥0
ṽnx
n and ṽ0 := 1. Note that the function (x; q)∞
satisfies the first-order q-difference equation
σqf(x) =
1
1− x
f(x).
Then, we obtain the equation[
σ2q −
{(
1 +
q
c1
)
− qx
a1
}
σq +
q
c1
(1− x)
]
ṽ(x) = 0. (22)
Equation (22) has the solution
ṽ(x) = 2ϕ1
(
c1
a1
, 0; c1; q, x
)
.
Therefore, we obtain the conclusion. �
Corollary 2. In (21), we put a1 7→ q/a, c1 7→ bq/a and x 7→ q/abx. Then we obtain
1ϕ1
(
q
a
;
bq
a
; q,
q
ax
)
=
( q
abx
; q
)
∞
2ϕ1
(
b, 0;
bq
a
; q,
q
abx
)
. (23)
We also put a1 7→ q/b, c1 7→ aq/b and x 7→ q/abx. Then we obtain
1ϕ1
(q
b
;
aq
b
; q,
q
bx
)
=
( q
abx
; q
)
∞
2ϕ1
(
a, 0;
aq
b
; q,
q
abx
)
. (24)
A Connection Formula for the q-Confluent Hypergeometric Function 11
By relations (20), (23) and (24),
(abx; q)∞
θ(−qx)
2ϕ1
(q
a
,
q
b
; 0; q, abx
)
=
(q/b; q)∞
(a/b, q; q)∞
θ(−bqx)
θ(−abx)
(
q, abx, q
abx ; q
)
∞
θ(−qx)
2ϕ1
(
b, 0;
bq
a
; q,
q
abx
)
+
(q/a; q)∞
(b/a, q; q)∞
θ(−aqx)
θ(−abx)
(
q, abx, q
abx ; q
)
∞
θ(−qx)
2ϕ1
(
a, 0;
aq
b
; q,
q
abx
)
=
(q/b; q)∞
(a/b, q; q)∞
θ(−bqx)
θ(−qx)
2ϕ1
(
b, 0;
bq
a
; q,
q
abx
)
+
(q/a; q)∞
(b/a, q; q)∞
θ(−aqx)
θ(−qx)
2ϕ1
(
a, 0;
aq
b
; q,
q
abx
)
.
Therefore, we obtain the formula (18).
3 The limit q → 1− of the connection formula
In this section we show the limit q → 1− of our connection formula. In [10], C. Zhang proposed
the following limit.
Theorem 3 ([10]). For any α, β ∈ C∗(α−β 6∈ Z) and z in any compact domain of C∗ \ [−∞, 0],
we have
lim
q→1−
2f0
(
qα, qβ;λ, q,
z
(1− q)
)
=
Γ(β − α)
Γ(β)
z−α1F1
(
α, α− β + 1;
1
z
)
+
Γ(α− β)
Γ(α)
z−β1F1
(
β;β − α+ 1;
1
z
)
.
Our limit of the connection formula in Theorem 1 differs from the theorem above. By
Theorem 1, we have
u2(x) = Cµ(a, b; q, x)Sµ(a, b; q, x) + Cµ(b, a; q, x)Sµ(b, a; q, x) (25)
for any (x, q) ∈ C∗ × (0, 1].
The limit q → 1− of the left-hand side of (25) is formally given by e1/z(−z)1−α−β2F0(1 −
α, 1− β;−, z), provided that a = qα, b = qβ and x = z/(1− q). On the other hand, convergent
series 1F1(α;α+1−β; 1/z) and 1F1(β;β+1−α; 1/z) appear in the limit q → 1− of the right-hand
side of (25). The aim of this section is to prove the following theorem.
Theorem 4. The limit q → 1− of the new connection formula formally gives the following
asymptotic formula
e1/z(−z)1−α−β2F0(1− α, 1− β;−, z) =
Γ(β − α)
Γ(1− α)
(−z)−α1F1
(
α;α+ 1− β;
1
z
)
+
Γ(α− β)
Γ(1− β)
(−z)−β1F1
(
β;β + 1− α;
1
z
)
.
In [8], Zhang has shown a limit of theta functions, taking the principal value of the logarithm
on C∗ \ (−∞, 0].
12 T. Morita
Proposition 3. For any γ ∈ C∗, we have
lim
q→1−
θ
(
qγ u
1−q
)
θ
(
u
1−q
) (1− q)−γ = u−γ
converges uniformly on compact subset of C \ (−∞, 0].
We also remind the formulas for the q-gamma function Γq(·) and the q-exponential func-
tion Eq(·). The q-gamma function 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]. The q-exponential function
Eq(z) =
∑
n≥0
qn(n−1)/2
(q; q)n
zn = (−z; q)∞
satisfies the limit
lim
q→1−
Eq (z(1− q)) = ez.
We set a = qα, b = qβ and x = z/(1− q) in Theorem 1. We introduce the constant
w(α, β; q) := (q; q)∞(1− q)1−α−β.
Consider the limit when q → 1− of each side of the identity of Theorem 1. The limit of the left
hand side of (18) is given by the following lemma.
Lemma 5. For any α, β ∈ C∗, α− β 6∈ Z, we have
lim
q→1−
w(α, β; q)
(
qα+βz
1−q ; q
)
∞
θ
(
− qz
1−q
) 2ϕ1
(
q1−α, q1−β; 0; q,
qα+βz
1− q
)
= (−z)1−α−βe
1
z 2F0(1− α, 1− β;−, z).
Proof. Exploiting the fact
w(α, β; q)
(
qα+βz
1−q ; q
)
∞
θ
(
− qz
1−q
) 2ϕ1
(
q1−α, q1−β; 0; q,
qα+βz
1− q
)
=
θ
(
qα+β
(
−z
1−q
))
θ
(
−z
1−q
) (1− q)−α−β
θ
(
−z
1−q
)
θ
(
q
(
−z
1−q
))(1− q)
× 1
Eq
(
− (1−q)
qα+β−1z
)2ϕ1
(
q1−α, q1−β; 0; q,
qα+βz
1− q
)
,
we obtain the conclusion. �
Consider the right-hand side of (18).
A Connection Formula for the q-Confluent Hypergeometric Function 13
Lemma 6. For any α, β ∈ C∗, (α− β 6∈ Z), we have
lim
q→1−
w(α, β, q)2f1
(
qα, qβ; q,
z
(1− q)
)
=
Γ(β − α)
Γ(1− α)
(−z)−α1F1
(
α;α+ 1− β;
1
z
)
+
Γ(α− β)
Γ(1− β)
(−z)−β1F1
(
β;β + 1− α;
1
z
)
.
Proof. Noting that
w(α, β; q)
(
q1−α; q
)
∞
(qβ−α, q; q)∞
θ
(
− qα+1z
1−q
)
θ
(
− qz
1−q
) 2ϕ1
(
qα, 0; qα+1−β; q,
q1−α−β(1− q)
z
)
=
{
(q1−α; q)∞
(q; q)∞
(1− q)α
}{
(q; q)∞
(qβ−α; q)∞
(1− q)−(1−β+α)
}
×
θ
(
qα+1
(
−z
1−q
))
θ
(
−z
1−q
) (1− q)−α−1
θ
(
−z
1−q
)
θ
(
q
(
−z
1−q
))(1− q)
× 2ϕ1
(
qα, 0; qα+1−β; q,
q1−α−β(1− q)
z
)
,
we prove the lemma. �
Finally, we obtain the proof of Theorem 4.
Acknowledgements
The author would like to give heartfelt thanks to Professor Yousuke Ohyama who provided
carefully considered feedback and many valuable comments. The author also would like to
thank the anonymous referees for their helpful comments.
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. Arts Sci. 49 (1913), 521–568.
[2] Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications,
Vol. 96, 2nd ed., Cambridge University Press, Cambridge, 2004.
[3] Gauss C.F., Disquisitiones generales circa seriem infinitam . . . , in Werke, Bd. 3, Königlichen Gesellschaft
der Wissenschaften zu Göttingen, 1866, 123–162.
[4] Morita T., A connection formula of the Hahn–Exton q-Bessel function, SIGMA 7 (2011), 115, 11 pages,
arXiv:1105.1998.
[5] Ohyama Y., A unified approach to q-special functions of the Laplace type, arXiv:1103.5232.
[6] Ramis J.P., Sauloy J., Zhang C., Local analytic classification of q-difference equations, arXiv:0903.0853.
[7] Watson G.N., The continuation of functions defined by generalized hypergeometric series, Trans. Camb.
Phil. Soc. 21 (1910), 281–299.
[8] 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.
[9] Zhang C., Sur les fonctions q-Bessel de Jackson, J. Approx. Theory 122 (2003), 208–223.
[10] Zhang C., Une sommation discrète pour des équations aux q-différences linéaires et à coefficients analytiques:
théorie générale et exemples, in Differential Equations and the Stokes Phenomenon, World Sci. Publ., River
Edge, NJ, 2002, 309–329.
http://dx.doi.org/10.1017/CBO9780511526251
http://dx.doi.org/10.3842/SIGMA.2011.115
http://arxiv.org/abs/1105.1998
http://arxiv.org/abs/1103.5232
http://arxiv.org/abs/0903.0853
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 and the connection matrix
2.1 Confluent hypergeometric equation
2.2 Local solutions to the q-confluent hypergeometric equation
2.3 The second connection formula
2.4 The connection matrix
2.5 Derivation from Watson's formula
3 The limit q1- of the connection formula
References
|
| id | nasplib_isofts_kiev_ua-123456789-149343 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-01T10:51:06Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Morita, T. 2019-02-21T07:04:56Z 2019-02-21T07:04:56Z 2013 A Connection Formula for the q-Confluent Hypergeometric Function / T. Morita // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 10 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D15; 34M40; 39A13 DOI: http://dx.doi.org/10.3842/SIGMA.2013.050 https://nasplib.isofts.kiev.ua/handle/123456789/149343 We show a connection formula for the q-confluent hypergeometric functions ₂φ₁(a,b;0;q,x). Combining our connection formula with Zhang's connection formula for ₂φ₀(a,b;−;q,x), we obtain the connection formula for the q-confluent hypergeometric equation in the matrix form. Also we obtain the connection formula of Kummer's confluent hypergeometric functions by taking the limit q→1− of our connection formula. The author would like to give heartfelt thanks to Professor Yousuke Ohyama who provided carefully considered feedback and many valuable comments. The author also would like to thank the anonymous referees for their helpful comments en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Connection Formula for the q-Confluent Hypergeometric Function Article published earlier |
| spellingShingle | A Connection Formula for the q-Confluent Hypergeometric Function Morita, T. |
| title | A Connection Formula for the q-Confluent Hypergeometric Function |
| title_full | A Connection Formula for the q-Confluent Hypergeometric Function |
| title_fullStr | A Connection Formula for the q-Confluent Hypergeometric Function |
| title_full_unstemmed | A Connection Formula for the q-Confluent Hypergeometric Function |
| title_short | A Connection Formula for the q-Confluent Hypergeometric Function |
| title_sort | connection formula for the q-confluent hypergeometric function |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149343 |
| work_keys_str_mv | AT moritat aconnectionformulafortheqconfluenthypergeometricfunction AT moritat connectionformulafortheqconfluenthypergeometricfunction |