The Asymptotic Expansion of Kummer Functions for Large Values of the $a$-Parameter, and Remarks on a Paper by Olver

It is shown that a known asymptotic expansion of the Kummer function $U(a,b,z)$ as $a$ tends to infinity is valid for $z$ on the full Riemann surface of the logarithm. A corresponding result is also proved in a more general setting considered by Olver (1956).


Introduction
Recently, the author collaborated on a project [1] investigating the maximal domain in which an integral addition theorem for the Kummer function U (a, b, z) due to Magnus [2,3] is valid. In this work it is important to know the asymptotic expansion of U (a, b, z) as a tends to infinity. Such an expansion is well-known, and, for instance, can be found in Slater's book [8]. Slater's expansion is in terms of modified Bessel functions K ν (z), and it is derived from a paper by Olver [5]. However, there are two problems when we try to use the known result. As Temme [9] pointed out, there is an error in Slater's expansion. Moreover, in all known results the range of validity for the variable z is restricted to certain sectors in the z-plane.
The purpose of this paper is two-fold. Firstly, we correct the error in [8], and we show that the corrected expansion based on [5] agrees with the result in [9] which was obtained in an entirely different way. Secondly, we show that the asymptotic expansion of U (a, b, z) as a tends to infinity is valid for z on the full Riemann surface of the logarithm. This is somewhat surprising because often the range of validity of asymptotic expansions is restricted by Stokes' lines. Olver's results in [5] are valid for a more general class of functions (containing confluent hypergeometric functions as a special case.) He introduces a restriction on arg z, and on [5, p. 76] he writes "In the case of the series with the basis function K µ we establish the asymptotic property in the range | arg z| ≤ 3 2 π. It is, in fact, unlikely that the valid range exceeds this . . . ". However, we show in this paper that the restriction | arg z| ≤ 3 2 π can be removed at least under an additional assumption (2.4).
In Section 2 of this paper we review the results that we need from Olver [5]. We discuss these results in Section 3. In Section 4 we prove that Olver's asymptotic expansion holds on the full Riemann surface of the logarithm. Sections 5, 6 and 7 deal with extensions to more general values of parameters. In Section 8 we specialize to asymptotic expansions of Kummer functions. In Section 9 we make the connection to Temme [9]. This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html arXiv:1601.02263v2 [math.CA] 6 May 2016 2 Olver's work Olver [5, (7.3)] considers the differential equation The function f (z) is even and analytic in a simply-connected domain D containing 0. It is assumed that µ ≥ 0. The goal is to find the asymptotic behavior of solutions of (2.1) as 0 < u → ∞.
Olver [5, (7.4)] starts with a formal solution to (2.1) of the form are defined by A 0 (z) = 1, and then recursively, for s ≥ 0, The integral in (2.3) denotes an arbitrary antiderivative of f (z)B s (z). The functions A s (z), B s (z) are analytic in D, and they are even and odd, respectively. If the domain D is unbounded, Olver [5, p. 77] requires that f (z) = O(|z| −1−α ) as |z| → ∞, where α > 0. In our application to the confluent hypergeometric equation in Section 8 the function f (z) = z 2 does not satisfy this condition. Therefore, throughout this paper, we will take where R 0 is a positive constant. Olver [5, p. 77] introduces various subdomains D , D 1 , D 2 of D.
We may choose D = {z : |z| ≤ R}, where 0 < R < R 0 . The domain D 1 comprises those points z in D which can be joined to the origin by a contour which lies in D and does not cross either the imaginary axis, or the line through z parallel to the imaginary axis. For our special D the contour can be taken as the line segment connecting z and 0, so D 1 = D . The domain D 1 appears in Olver [5,Theorem D(i)]. According to this theorem, (2.1) has a solution W 1 (u, z) of the form Remarks 2.1.
1. The parameter µ is considered fixed. We may write W 1 (u, µ, z) to indicate the dependence of W 1 on µ.
6. Olver has the term z 1+|z| in place of z in front of h 1 in (2.5) but since we assume |z| ≤ R this makes no difference.
For the definition of D 2 we suppose that a is an arbitrary point of the sector | arg a| < 1 2 π and > 0. Then D 2 comprises those points z ∈ D for which | arg z| ≤ 3 2 π, z ≤ a, and a contour can be found joining z and a which satisfies the following conditions: (ii) it lies wholly to the right of the line through z parallel to the imaginary axis, (iii) it does not cross the negative imaginary axis if 1 2 π ≤ arg z ≤ 3 2 π, and does not cross the positive imaginary axis if − 3 2 π ≤ arg z ≤ − π 2 , (iv) it lies outside the circle |t| = |z|.
In our special case D = {z : |z| ≤ R} we choose a = R. If 0 ≤ arg z ≤ 3 2 π and 0 < |z| ≤ R, we choose the contour starting at z moving in positive direction parallel to the imaginary axis until we hit the circle |t| = R. Then we move clockwise along the circle |t| = R towards a. Taking into account condition (iv), we see that D 2 is the set of points z with − 3 2 π + δ ≤ z ≤ 3 2 π − δ, 0 < |z| ≤ R, where δ > 0. The domain D 2 appears in Olver [5,Theorem D(ii)]. According to this theorem, (2.1) has a solution W 2 (u, z) of the form where Note that in (2.8) there is a restriction on arg z.
In the rest of this paper we choose the functions A s (z) such that Then the functions A s (z), B s (z) are uniquely determined.
3 Properties of solutions W 1 and W 2 The differential equation (2.1) has a regular singularity at z = 0 with exponents 1 ± µ. Substituting x = z 2 we obtain an equation which has a regular singularity at x = 0 with exponents 1 2 (1 ± µ). Therefore, for every µ which is not a negative integer, (2.1) has a unique solution W + (z) = W + (u, µ, z) of the form where the c n are determined by c 0 = 1, and If µ is not an integer, then W + (u, µ, z) and W + (u, −µ, z) form a fundamental system of solutions of (2.1). If µ ≥ 0, there is a solution W − (z) linearly independent of W + (z) such that where p is a power series and d is a suitable constant. If µ = 0 we choose p(0) = 1. If µ is not an integer then d = 0.
Unfortunately, it seems impossible to replace W 2 by an easily identifiable solution of (2.1). However, we will now prove several useful properties of W 2 .
Lemma 3.2. Suppose that µ ≥ 0. There is a function β(u) such that

5)
and, for every N = 1, 2, 3, . . . , Proof . We set λ ± = e πi(1±µ) . Equation (2.1) has a fundamental system of solutions W + , W − such that If we apply this result to w = W 2 we see that there is a function β(u) such that (3.5) holds. Let z > 0 and set z 1 = ze πi . We use (2.7) for z 1 in place of z, and [7, (10.34.2)] with m = 1. Then Using (2.7) a second time, we find that We now expand the right-hand side of (3.5) using (2.5), and compare the expansions. Setting z = R and dividing by RI µ (uR), we obtain where we used [7, (10.40.1)] This proves (3.6).

Removal of restriction on arg z
Using β(u) from Lemma 3.2 we define Then we have Moreover, (3.6) shows that W 4 shares the asymptotic expansion (2.7), (2.8) with W 2 . From (4.1) we obtain for every integer m. We will use (4.2) and the asymptotic expansions (2.5), (2.7) for | arg z| ≤ 1 2 π to prove that in (2.8) we can remove the restriction on arg z completely.
Suppose µ = 0. We use [7, (10.31.2)] It follows from (4.13) that there exist positive constants r > 0, D > 0 such that Then we set with G 1 and H 1 the same as before when |uz| > r. The estimates (4.9) are valid with a suitable constant C. The rest of the proof is unchanged. This completes the proof of the theorem.

Extension to complex u
So far we considered only 0 < u → ∞. Now we set u = te iθ , where t > 0 and θ ∈ R. In (2.1) we substitute z = e −iθ x,w(x) = w(z). Then we obtain the differential equation Assuming µ ≥ 0, we can apply Olver's theory to this equation, and obtain functionsW 1 (t, x) andW 2 (t, x). Since we assumed that f (z) is analytic in the disk {z : |z| < R 0 }, the new functioñ f (x) = e −2iθ f (e −iθ x) is analytic in the same disk. Therefore, the domains D 1 , D 2 are the same as before. The functionsÃ s (x),B s (x) that appear in place of A s (z), B s (z) satisfỹ Therefore, the functions e −iθW 1 (t,x) and e −iθW 2 (t,x) have the asymptotic expansions (2.5), (2.6) and (2.7), (2.8) with (t, x) replacing (u, z).
LetW 3 (t, µ, x) be the function W 3 for the differential equation (5.1). Then It follows that W 3 (u, µ, z) can be expanded in the form of the right-hand side of (2.5), and (2.6) holds for 0 < |z| ≤ R and u = te iθ for any fixed real θ.
We would like to connectW 2 to W 2 in a similar manner but this is not possible at this point because W 2 (u, z) is only defined for u > 0, and so we cannot substitute u = te iθ . Proof . Let us denote the right-hand sides of equations (6.1) by A * s (z), B * s (z), respectively. It is easy to show that A * s (z), B * s (z) is a solution of (2.2), (2.3). Since A * 0 (z) = 1 and A * s (0) =Â s (0), this solution must agree withÂ s (z),B s (z).
We now define a 0 (z) = 1 and, for s ≥ 0, We add 4µ z B s (−µ, z) on both sides and get Equation (2.2) is true for a s (z), b s (z) when s = 0. Suppose s ≥ 1. We have Using the definitions of a s (z), b s (z) we get where In (6.7) we replace 4µ z a s (z) through (6.6). Then we obtain where By direct computation, we show H + G = 0 for any function b s−1 (z). Therefore, by integrating (6.8) noting that a s (z) is even and b s (z) is odd, we obtain (2.2) for a s (z), b s (z). We now get (6.4), (6.5) from Lemma 6.1.
Using multiplication of formal series, we can write (6.4), (6.5) as We differentiate (6.5) with respect to z and set z = 0. Then we find or, equivalently, In particular, it follows that A s (z) u 2s , (6.12) 7 Asymptotic expansion of W 3 when µ < 0 In Section 3 we saw that W 3 (u, µ, z) can be written as the right-hand side of (2.5), and (2.6) holds. However, this was proved only when µ ≥ 0. Now we remove this restriction.

Application to the conf luent hypergeometric equation
The confluent hypergeometric differential equation Equation (8.1) agrees with (2.1) when f (z) = z 2 . Let A s , B s be defined as in Section 2 for f (z) = z 2 . In this case, A s (z), B s (z) are polynomials. Throughout this section, we assume that a, b, u, µ satisfy (8.2). The function M (a, b, x) is given by a power series in x and M (a, b, 0) = 1. Therefore, the function W 3 associated with (8.1) is given by Theorem 7.1 implies the following theorem.
Theorem 8.1. Suppose that b ∈ C is not 0 or a negative integer, u = te iθ with t > 0, θ ∈ R, and N ≥ 1, R > 0. Then we can write

4)
where and L 1 , t 1 are positive constants independent of z and u (but possibly depending on b, θ, N , R).
Proof . We use the integral representation [7, (13 where the contour C starts at +∞i and follows the positive imaginary axis, then describes a loop around 0 in positive direction and returns to +∞i. The argument of t starts at 1 2 π and increases to 5 2 π. It will be sufficient to estimate Γ(a)U (a, b, x) in the sector 1 2 π ≤ arg(a − 1) ≤ α 0 , where 1 2 π < α 0 < π. The loop is chosen so that w = t 1+t describes the circle |w| = cos θ 0 , where θ 0 ∈ (0, 1 2 π) is the unique solution of the equation cos θ 0 = e θ 0 tan α 0 .
Then one obtains |w a−1 | ≤ 1 on the contour C which implies the desired estimate.