Uniform Treatment of Darboux's Method and the Heisenberg Polynomials

We show that the set of Heisenberg polynomials furnishes a simple non-trivial example in the uniform treatment of Darboux's method.

The Heisenberg group H N arose in various branches of mathematics, such as harmonic analysis [10,16] and partial differential operators and equations [6,7]; the reason may partially be that the results in Abelian harmonic analysis could be adapted to Heisenberg groups.
The so-called L 2γ -harmonic polynomials are the polynomial solutions p = p(z 1 ,z 1 , · · · z N ,z N , t) of L 2γ p = 0 in H N , where the differential operator L 2γ is defined by and γ being a constant. In particular, L 0 in H 1 is a sub-elliptic Laplacian on R 3 , cf. [6,7]. Greiner [9] initiated the study of L 2γ -spherical harmonic polynomials. It is later shown by Dunkl (cf. [2]) that every such polynomial can be expressed in terms of C (α,β) n . A few facts are known about the Heisenberg polynomials. For example, the representation (1.1) of C (α,β) n can be interpreted as a Gauss hypergeometric function. The polynomials C (α,β) n (z) reduce to the Legendre polynomials on the unit circle when α = β = 1 2 . The readers are referred to [2] for more properties of these polynomials.
In 1984, Dunkl [2] obtained a bi-orthogonality relation for these functions in L 2 (T, | sin θ| α+β dθ), where α > 0, β > 0, and T = { e iθ − π < θ ≤ π}. A 1986-paper [3] of Dunkl addressed the density problem for C (α,β) n (e iθ ) in L 2 . In 1986, Temme [15] considered two sets of bi-orthogonal polynomials, P n (z; α, β) and Q n (z; α, β), closely related to the Heisenberg polynomials, where P n (z; α, β) = F (−n, α + β + 1; 2α + 1; 1 − z) and Q n (z; α, β) = P n (z; α, −β), with F (a, b; c; z) being the Gauss hypergeometric function; cf. [13, p.384]. The connection with the Heisenberg polynomials is The two sets {P n } and {Q n } are bi-orthogonal on the unit circle with respect to the weight function (1 − e iθ ) α+β (1 − e −iθ ) α−β . Based on an integral representation of the Gauss hypergeometric function, uniform asymptotic expansions for P n (z; α, β) were obtained in [15], including error bounds. These expansions are in terms of confluent hypergeometric functions, and show that all points on the unit circle in the z-plane are transition points in the sense that the polynomials have different asymptotic behavior inside and outside the unit disk. To ensure the convergence of the integral representation, extra restrictions on the parameters, namely α + β > −1 and α − β ≥ 0, are required in [15]. It is readily seen that the Heisenberg polynomials are homogeneous functions in z andz, which implies that C . Hence, to study the behavior of the polynomials, it is natural to consider the polynomials on the unit circle. The generating function (1.2) now takes the form In what follows, we will show that the set of Heisenberg polynomials on the unit circle provides a simple non-trivial example in the uniform treatment of Darboux's method given in [18].

Uniform Treatment of Darboux's Method
Darboux was the first to consider the asymptotic behavior of the coefficients a n in the Maclaurin expansion where F (w) has only a finite number of singularities on its circle of convergence, all of which are algebraic in nature. A method was introduced to obtain the asymptotic expansion for a n as n → ∞. The method, named after Darboux, indicates that the contribution to the expansion comes from the singularities on the circle of convergence; cf., e.g., Olver [12] and Wong [17] for the classical Darboux's method.
If the singularities are allowed to move around on the circle of convergence, Darboux's method will continue to work provided that their essential configuration remains the same (although their relative positions vary). However, this method breaks down, when two or more singularities coalesce with each other. A uniform treatment was later given by Fields [5], in which he considered generating functions of the type for |z| < 1, where λ and ∆ are bounded quantities, the branches are chosen such that both with η > 0 not depending on θ. The results in [5] were regarded to be too complicated to apply; see, e.g., [4, p.167]. The problem of uniform treatment was revisited by Wong and Zhao [18], in which they firstly addressed a special case of (2.2) with λ = 0, namely Uniform asymptotic expansion was derived in terms of the Bessel functions J α±1/2 (nθ), valid for θ ∈ [0, π − ε], ε > 0. The general case considered in [18] deals with many coalescing singularities where |z k (θ)| = 1, q ≥ 2, α k are constants, and f (z, θ) is the same function as in (2.2) and (2.3).
In a follow-up paper, Bai and Zhao [1] have extended the essential ideas in [18] to study the classical Jacobi polynomials.
As mentioned in the last section and shown in (1.6), the set of Heisenberg polynomials serves as a good example of the uniform Darboux treatment, which is simple (only two singularities) and non-trivial (the exponents α and β may be different). To derive the asymptotic expansion of C (α,β) n (e iθ ), we apply Theorem 2 of [18]. To this end, we define where Γ 0 is a Hankel-type loop which starts and ends at −∞, and encircles s = ±i in the positive sense; cf. [18, (4.7)]. We further introduce an auxiliary function and a sequence of functions {h k (s, θ)} ∞ k=0 defined inductively by for k = 0, 1, · · · . The coefficients α k (θ) and β k (θ) are determined by requiring all h k (s, θ) and g k (s, θ) to be analytic in D = {s : Re s ≥ − η θ , |s ± i| < 2π θ }. The uniform asymptotic expansion is given in the following theorem:

Special Functions T 1 and T 2
By expanding the slowly varying factor in the integrand of (2.5) in uniformly convergent power series of 1/s and integrating term by term, we obtain where x α+β−1 is positive for real positive x. It is easily seen that x −α−β+1 T 1 (x) is an entire function. From (2.5), it is also readily verified that T 2 (x) = T ′ 1 (x) in the cut plane x ∈ C\(−∞, 0]. The behavior of T 1 (x) at the origin is exhibited by the leading terms in (3.1). The asymptotic behavior of T l (x) as x → +∞ has already been given in [18, (4.45)]. Moreover, with a = 2−α−β and b = β − α, T 1 (x) satisfies the differential equation cf. [18, (4.52)]. This equation is of Laplace type in the sense that the coefficients are linear in x. We proceed to show that T 1 (x) is connected to the confluent hypergeometric function via the following change of variables:

Substituting (3.3) in (3.2) leads to the Kummer equation
Taking into account the first two terms of the infinite series in (3.1), we have where M is the Kummer function; cf. [13, p.322]. (One may also derive this result directly from (2.5) and a Laplace inversion integral of this Kummer function [13, p.327, (13.4.13)].) Accordingly, from T 2 (x) = T ′ 1 (x) one has where M ′ (γ, δ, z) = d dz M (γ, δ, z). Thus, substituting (3.5) and (3.6) in (2.8), we obtain an asymptotic expansion of the Heisenberg polynomials in terms of the Kummer function as follows: Theorem 2. Assume that α and β are real and fixed, z = ρe iθ with ρ > 0 and θ real. Then we have the compound asymptotic expansion ( §4.2 of Chapter 4 in [12]) as n → ∞, uniformly with respect to ρ ∈ (0, ∞) and θ ∈ [0, π − δ], where δ is an arbitrary constant such that 0 < δ ≤ π. Here the coefficients are given by for k = 0, 1, 2, · · · , α k (θ) and β k (θ) being defined as in Section 2. In particular, When the parameters α and β in the above theorem are nonpositive integers, the coefficients c k and d k in (3.7) all vanish; see (3.8). However, the asymptotic relation remains valid since the polynomials also vanish for large values of n; i.e., it is a trivial result.
For completeness, we write down, by using (2.5) and (3.2), the asymptotic expansion in the special case α = β; see also [18]. In this case, we have where J ν (x) is the Bessel function of the first kind; cf. [13, p.217]. The expansion in Theorems 1 and 2 now takes the form: Corollary 1. Assume that α is real and fixed, z = ρe iθ with ρ > 0 and θ real. Then we have the compound asymptotic expansion as n → ∞, uniformly with respect to ρ ∈ (0, ∞) and θ ∈ [0, π − δ], where δ is an arbitrary constant such that 0 < δ ≤ π. Here α k (θ) and β k (θ) are given in (2.7), and the leading coefficients are (3.11)
On the other hand, taking the two leading terms in (3.10)-(3.11) and making use of the asymptotic approximation for the Bessel function [13, p.223, (10.7.8