Symmetry, Integrability and Geometry: Methods and Applications A Connection Formula for the q-Confluent Hypergeometric Function

We show a connection formula for the $q$-confluent hypergeometric functions ${}_2\varphi_1(a,b;0;q,x)$. Combining our connection formula with Zhang's connection formula for ${}_2\varphi_0(a,b;-;q,x)$, 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 $q\to 1^{-}$ of our connection formula.

Since u 1 (x) is a divergent series, the q-Stokes phenomenon appears in Zhang's connection formula. But our connection formula does not have any discontinuity.
Moreover, the basic hypergeometric series r ϕ s is We assume that q ∈ C * satisfies 0 < |q| < 1. The q-difference operator σ q is given by σ q f (x) = f (qx). The theta function of Jacobi is given by following series: We denote by θ q (x) or more shortly θ(x). Jacobi's triple product identity is The theta function has inversion formula θ(x) = xθ(1/x). For all fixed λ ∈ C * , we define a q-spiral [λ; q] := λq Z = {λq k ; k ∈ Z}. We remark that We study connection problems on linear q-difference equations with irregular singular points. Connection problems on linear q-difference equations of the Laplace type [5] with regular singular points are studied by G. D. Birkhoff [1]. Generically, linear q-difference equations has formal power series solutions x α n≥0 a n x n around the origin and x β n≥0 b n x −n around the infinity. But on connection problems on linear q-difference equations, we replace the function x κ with the function θ(x)/θ(kx), where k = q κ . Then, the fundamental system of solutions are given by single valued functions which have single poles at suitable q-spirals. The first example is given by G. N. Watson [6]. But a few examples of irregular singular case are known [7,8,4]. In this paper, we give a connection formula of the q-confluent type function with using q-Borel-Laplace transformations.
More generally, the generalized hypergeometric series is given by The hypergeometric function 2 F 1 (α, β; γ; z) satisfies the second order differential equation Gauss gives the connection formula of the function 2 F 1 (α, β; γ, z). We put z → z/β and take the limit β → ∞ in the equation (3), we obtain the confluent hypergeometric equation (CHGE) Solutions of (4) around the origin are and Solutions around the infinity are given by divergent series and A well-known connection formula between (7), (8) and (5) is given by where −π/2 < arg z < 3π/2. We remark that the connection formula for second solution around the infinity (6) can be derived from the first one.
In section two, we deal with the degenaration of the equation (3) which is slightly different from standard way. It is known that there exists the q-analogue of 2 F 1 (α, β; γ; z), which is introduced by E. Heine in 1847 as follows: The function 2 ϕ 1 (a, b; c; q, x) satisfies the second order q-difference equation (q-HGE) where D q is the q-derivative operator defined for fixed q by By replacing a, b, c by q α , q β , q γ and then letting q → 1 − 0, the equation (9) tends to the hypergeometric equation (3) where |x| < 1. The qhypergeometric equation (9) can be rewritten as follows: If we set x → cx and c → ∞ in (10), we obtain the q-confluent hypergeometric equation (1). The equation (1) is considered as a q-analogue of CHGE. We remark that the first solution u 1 (x) is a divergent series and u 2 (x) is a convergent around the origin. The second solution u 2 (x) can not be derived from u 1 (x) directly. This point is essentially different from the differential equation case. Connection problems on linear q-difference equations between the origin and the infinity are studied by G. D. Birkhoff [1]. In 1910, Watson [6] has shown the connection formula of the series 2 ϕ 1 (a, b; c; q, x) as follows: We remark that we can not set c = 0 directly in this formula.
In 2002, C. Zhang [9] shows one of the connection formula of the q-CHGE as follows: The The q-Borel-Laplace transformations are studied by C. Zhang in [9] (see section two for more details). When we study connection problems of divergent series, this resummation method becomes a powerful tool. We remark that we can find a new parameter λ in the resummation 2 f 0 (a, b; λ, q, x). This parameter brings us new viewpoints for the study of the q-Stokes phenomenon. It is known that there exists two different types of the q-Borel transformation and the q-Laplace transformation. The q-Borel-Laplace transformation of the first kind are defined in [9] and the q-Borel-Laplace transformations of the second kind are studied in [8]. These q-Borel transformations are the formal inverse transformation of each of the q-Laplace transformations.
Zhang has shown a connection formula of the series 2 ϕ 0 (a, b; −; q, x) by the q-Borel-Laplace transformations of the first kind. But the connection formula for the second solution of (1) has not known. In this paper, we show the second connection formula for q-CHGE with using the q-Borel transformation and the q-Laplace transformation of the second kind. Combining with Zhang's connection formula, we obtain the following connection in section two: The set [λ; q] is the q-spiral. The function 2 f 1 (a, b; q, x) is the q-Borel-Laplace transform of 2 ϕ 1 (a, b; 0; q, x). The function S µ (a, b; q, x) is the solution of (1) around the infinity as follows; We remark that the function x −α in the solution v 1 (x) is replaced with the function θ(aµx)/θ(µx), where a = q α . Functions C λ µ (a, b; q, x), C µ (a, b; q, x) are q-elliptic functions.
In section three, we consider the limit q → 1−0 of our connection formula. If we take a limit q → 1 − 0 of our connection formula, we formally obtain the connection formula of the confluent hypergeometric series 2 F 0 .

The connection formula and the connection matrix
We review a q-confluent hypergeometric equation in subsection 2.1. Then we show a connection formula for the q-confluent hypergeometric function, which is different from Zhang's formula.

Confluent hypergeometric equation
For the confluent hypergeometric equation (3), we take another degeneration. We put z → zγ and the take the limit γ → 0. Then we obtain Solutions of (12) around the origin are given by divergent series

Solutions around the infinity arẽ
We consider the following second order q-difference equation This equation can be rewritten as follows: which is called a q-confluent hypergeometric equation.

Local solutions to the q-confluent hypergeometric equation
In the following, we study connection problem of (13). At first we show local solutions to (13) around x = 0 and x = ∞.

Lemma 1. The equation (13) has solutions
around the origin and has solutions around the infinity, provided that a = q α and b = q β .
We find another solution. We set E(x) = 1/θ(−qx) and f (x) = n≥0 a n x n , a 0 = 1 to obtain the solution around the origin. We assume that u(x) = E(x)f (x). We remark that the function E(x) has following property: Therefore, we obtain the equation Since the infinite product (abx; q) ∞ satisfies the following q-difference relation we obtain the second solution. Therefore, solutions of the equation (13) are given by Around x = ∞, we can easily determine local solutions by setting v(x) = θ(aµx) θ(µx) n≥0 a n x −n , a 0 = 1, for any fixed µ ∈ C * and x ∈ C * \ [−µ; q].
Here, u 1 (x) is a divergent series and u 2 (x),v 1 (x) and v 2 (x) are convergent series [2]. Therefore, the q-Stokes phenomenon occurs for u 1 (x) and Zhang has shown a connection formula for u 1 (x) be means of the following q-Borel-Laplace transformations of the first kind [9].
But the connection formula between (15) and (16),(17) is not known. In the next section, we show the second connection formula with using the q-Borel-Laplace transformations of the second kind.

The second connection formula
We introduce the q-Borel transformation and the q-Laplace transformation of the second kind following C. Zhang to obtain the solution of the equation (18).
Definition 2. For f (x) = n≥0 a n t n , the q-Borel transformation is defined by and the q-Laplace transformation is given by Here, r 0 > 0 is enough small number. The q-Borel transformation is considered as a formal inverse of the q-Laplace transformation.
Lemma 2. We assume that the function f can be q-Borel transformed to the analytic function g(ξ) around ξ = 0. Then, we have Proof. We can show 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 , We apply the q-Borel transformation to the equation (18) and using Lemma 3. We check out g(ξ) satisfies the first order difference equation Since g(0) = a 0 = 1, we have the infinite product of g(ξ) as follows; We remark that g(ξ) has single poles at We set 0 < r < r 0 := max 1 |aq| , 1 |bq| and choose the radius r > 0 such that 0 < r < r 0 . By Cauchy's residue theorem, the p-Laplace transform of g(ξ) is where 0 < r < r 0 . We use the following lemma to calculate the residue.
Lemma 4. For any k ∈ N, λ ∈ C * , we have; Summing up all of residues, we obtain f (x) as follows: Therefore, we obtain the following theorem.

The connection matrix
We give the connection matrix for the equation (1). We define new functions S µ (a, b; q, x), C λ µ (a, b; q, x) and C µ (a, b; q, x). For any λ, µ ∈ C * , The function θ(aµx)/θ(µx) satisfies the following q-difference equation which is also satisfied by the function u(x) = x −α , a = q α . We remark that the pair (S µ (a, b; q, x), S µ (b, a; q, x)) gives a fundamental system of solutions of the equation (1) if a, b ∈ Z. We set functions C λ µ (a, b; q, x) and C µ (a, b; q, x) as follow; Definition 3. For any λ, µ ∈ C * , C λ µ (a, b; q, x) is Similarly, C µ (a, b; q, x) is Then C λ µ (a, b; q, x) and C µ (a, b; q, x) are single valued as a function of x. They satisfy the following relation C λ µ (a, b; q, e 2πi x) = C λ µ (a, b; q, x), C λ µ (a, b; q, qx) = C λ µ (a, b; q, x) and C µ (a, b; q, e 2πi x) = C µ (a, b; q, x), C µ (a, b; q, qx) = C µ (a, b; q, x).
Combining Zhang's formula (11) and Theorem 1, we obtain the connection formula in matrix form.
3 The limit q → 1 − 0 of the connection formula In this section, we show the limit q → 1 − 0 of our connection formula. In [9], Zhang has given the limit q → 1 − 0 of as follows: Theorem 3 (Zhang). For any α, β ∈ C * (α − β ∈ Z) and z in any compact domain of C * \ [−∞, 0], we have lim q→1−0 2 f 0 q α , q β ; λ, q, Our limit of the connection formula in theorem 1 is different from the theorem above. The aim of this section is to give the following theorem.