Symmetry, Integrability and Geometry: Methods and Applications 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.

Both J ν (z) and J −ν (z) are linearly independent if ν ∈ Z. It is known that there exists three different q-analogues of the Bessel function.
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: The limits of these q-analogues of the Bessel function are the Bessel function when q → 1 − : The relation between J ν (x; q) and J (2) ν (x; q) was found by Hahn [3] as follows: 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.
The connection formula of J (2) ν (x; q) is obtained by (3) and (2). But it is not known the connection formula of the Hahn-Exton q-Bessel function.
The Hahn-Exton q-Bessel equations (1) has two analytic solutions u(x) = J −ν (xp −ν ) around x = 0 and has one analytic solution z (1/x) = 1 θp(−p ν+2 /x) n≥0 a n x −n , a 0 = 1. We show a connection formula of J ν (x; q) in Section 2 as follows: Here, θ p (·) 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 Here, H 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).

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 = p 2 . The q-difference operator σ q is given by σ q f (x) = f (qx).

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: We denote by θ q (x) or more shortly θ(x). The theta function satisfies Jacobi's triple product identity: The theta function satisfies the q-difference equation as follows The theta function has the inversion formula xθ(

The Hahn-Exton q-Bessel function
The Hahn-Exton q-Bessel function is defined by The function J ν (x; q) satisfies the q-difference equation If we replace ν by −ν and x by xp −ν , we obtain J −ν (xp −ν ; q) which is another solution of (5) around the origin. This solution corresponds to the classical Neumann function Y ν (x) [5]. We consider the behavior of equation (5) around the infinity. We set 1/t, formally t 2 → t and z(t) = y(1/t). Then z(t) satisfies We set E(t) = 1/θ p (−p ν+2 t) and f (t) = n≥0 a n t n , a 0 = 1. We assume that z(t) can be described as Since E(t) satisfies the following q-difference equation we can check out that the function f (t) satisfies the equation

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].
Definition 2. For f (t) = n≥0 a n t n , the p-Borel transformation is defined by g(τ ) = (B p f ) (τ ) := n≥0 a n p − n(n−1) 2 τ n , and the p-Laplace transformation is given by Here, r 0 > 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 function g(τ ) around τ = 0. Then, 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.
Since g(0) = 1, we get an infinite product of g(τ ): Then g(τ ) has single poles at We set and choose the radius r > 0 such that 0 < r < r 0 . By Cauchy's residue theorem, the p-Laplace transform of g(τ ) is Res g(τ )θ p t τ where 0 < r < r 0 . To calculate the residue, we use the following lemma.
Lemma 3. For any k ∈ N, λ ∈ C * , we have Summing up all of the residues, we obtain the convergent series f (t) as follows where xt = 1. Therefore, we acquire the connection formula for z(t) = E(t)f (t).

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 The Hankel function of the second kind is defined by 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 J ν (z): The Hankel functions have asymptotic expansions around z = 0 [4]: as z → ∞. Here, δ is an any small constant, In this sense, (8) and (9) considered as connection formula of the Bessel equation.
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 Here, H 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, h ν 1/{(1 − p) 2 x}; p can be described as follows We consider the limit of each part {·}.
Proof . We can check out as follows Here, Γ q (·) is Jackson's q-gamma function which is defined by This function satisfies lim [2]. Therefore, By Euler's reflection formula of the gamma function, we get .
Therefore, we get the conclusion.
If we replace ν by −ν, we get the limit .
Therefore, we obtain the following relation.
Therefore, we obtain the conclusion. Similarly, we can prove the latter.
We give the proof of Theorem 2.