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### Abstract

The discrete Chebyshev polynomials are orthogonal with respect to a distribution, which is a step function with jumps one unit at the points , N being a fixed positive integer. By using a double integral representation, we have recently obtained asymptotic expansions for in the double scaling limit, namely, and , where and ; see [8]. In this paper, we continue to investigate the behavior of these polynomials when the parameter b approaches the endpoints of the interval (0, 1). While the case is relatively simple (because it is very much like the case when b is fixed), the case is quite complicated. The discussion of the latter case is divided into several subcases, depending on the quantities n, x, and , and different special functions have been used as approximants, including Airy, Bessel, and Kummer functions.

### 1. Introduction

The discrete Chebyshev polynomials can be defined as a special case of the Hahn polynomials [1, p. 174]

• (1)

With in (1), we have

• (2)

see [1, p. 176].

In a recent paper [8], we have studied the asymptotic behavior of as n, in such a way that the ratios

• (3)

satisfy the inequalities

• (4)

In view of the symmetry relation [8, eq. (8)]

• (5)

our study actually covers the entire real-axis or, equivalently, the entire parameter range . For a discussion of the asymptotic behavior of the Hahn polynomials, see [4].

In this paper, we shall investigate the behavior of the polynomials , when the parameter b given in (3) either tends to 0 or tends to 1. The case turns out to be rather simple, because the ultimate expansions and their derivations are similar to those in the case when b is fixed; see (27) for and (34) for .

To derive asymptotic approximation for when , we divide our discussion into several cases, depending on the quantities n, x, and . A summary of our findings is given in Table 1 below.

Some explanation is needed for this table. By “small”, “fixed”, and “large”, we mean, respectively, , and as , where δ is a small positive number and M is a large positive number. By “Airy”, we mean an asymptotic expansion whose associated approximants are the Airy functions and . In the same manner, by “Bessel” and “Kummer”, we mean asymptotic expansions whose associated approximants are, respectively, the Bessel function and the Kummer function . The Kummer function used in this paper is defined by

• (6)

which was introduced in [6, p. 255]. The case involving Bessel functions covers an earlier result of Sharapudinov [10], where, instead of Bessel functions, Jacobi polynomials are used as an approximant. In the case when is large and x is either fixed or small, the series in (2) is itself an asymptotic expansion (in the generalized sense [11, p. 10]) as . Hence, there is no need to seek for another asymptotic representation.

The arrangement of the present paper is as follows. In Section 'Results in Ref. [8]', we recall the major results in [8]; to facilitate our presentation for later sections, we also include here brief derivations of the expansions given in [8]. In Section 'Kummer-type expansion', we consider the case when is small and show that the expansions are of Kummer-type. In Section 'Airy-type expansion', we consider the case when the quantity is fixed; that is, when is bounded away from zero and infinity. There are three subcases, depending on x being large, fixed or small. In all three subcases, we will show that the expansions are of Airy-type; see Table 1. The case when both quantities x and are large is dealt with in Section 'Bessel-type expansion', where we show that the expansion of can be expressed in terms of Bessel functions. The final section is devoted to two remaining cases, namely, (i) when is large and x is either fixed or small, and (ii) .

### 2. Results in Ref. [8]

In [8], we have divided our discussion into two cases: (i) , and (ii) . As mentioned in Section 'Introduction', by virtue of the symmetry relation (5), these two cases cover the entire range .

In case (i), we started with the integral representation

• (7)

where

• (8)

and the curve γ1 starts at , runs along the lower edge of the positive real line toward , encircles the point in the counterclockwise direction and returns to the origin along the upper edge of the positive real line. As a function of two variables, the partial derivatives and vanish at , where

• (9)

and

• (10)

For each fixed , we now find a steepest descent path of the phase function in the variable t, which passes through a saddle point depending on w. (Note that not only the saddle point , but all points t on this steepest descent path depend on w.) The function is a function of w alone. To find the relevant saddle point , we solve the equation , and obtain

• (11)

It can be shown that

• (12)

Define the standard transformation by

• (13)

see [11, p. 88]. Note that we have when , and when . Furthermore, this mapping takes to . Coupling (7) and (13) gives

• (14)

where γ1 is the integration path of the variable w in (7). Note that the first variable of f in (14) is now , instead of t. Hence, the phase function is a function of w alone. Setting , we obtain the saddle points

• (15)

cf. (9) and (10). Motivated by (8), define the new phase function

• (16)

The saddle points of ψ are given by

• (17)

To reduce the double integral in (7) into a canonical form, we define the second mapping by

• (18)

with

• (19)

where η and γ are real numbers depending on the parameters a and b in (8). From (18), we have

• (20)

for , and by L'Hpital's rule,

• (21)

for . With the change of variable defined in (18), the representation in (14) becomes

• (22)

where

• (23)

and is the steepest descent path of in the u-plane. An asymptotic expansion, holding uniformly for , was then derived in [8] by an integration-by-parts technique. To state the result, we define recursively ,

• (24)

and

• (25)

Furthermore, we write

• (26)

The resulting expansion takes the form

• (27)

where is the Kummer function defined in (6), and the coefficients and are given explicitly by

• (28)

Case (ii) was dealt with in a similar manner. We started with the double-integral representation

• (29)

where

• (30)

and the integration path γ2 starts at , traverses along the upper edge of the positive real line toward , encircles the origin in the counterclockwise direction and returns to along the lower edge of the positive real line.

Now we follow the same argument as given in Case (i), and use the same mapping defined in (13), except with replaced by . The result is

• (31)

Because of the shape of the contour γ2, it turns out that the phase function has only one relevant saddle point, namely ; see (15). As a consequence, we define the mapping by

• (32)

with

• (33)

where γ is a constant depending on the parameters a and b in (3). The final expansion is in terms of the gamma function, and we have

• (34)

where are constants that can be given recursively.

### 3. Kummer-type expansion

As , some of the steps in Section 'Results in Ref. [8]' are no longer valid. Let us first examine the mapping given in (13). In this mapping, we have used the fact that for each fixed , the saddle point in (11) is bounded away from and , and the steepest descent path from to , passing through , is mapped onto in the τ-plane. However, from (11), we have, for any fixed w,

• (35)

that is, as , approaches the branch point in the t-plane. Thus, the mapping (13) is no longer suitable in this case. Because we are more interested in the neighborhood of , where the term in the phase function (8) becomes singular, we introduce the mapping

• (36)

where the constant A does not depend on t or τ (but may depend on w); see (38) below. To make the mapping defined in (36) one-to-one and analytic, we prescribe to correspond to , which is the saddle point of ; that is,

• (37)

This gives

• (38)

Note that we have when , and when . Furthermore, from (36), we have

• (39)

where we have used L'Hpital's rule for and

cf. (11). Coupling (7) and (36), we have, instead of (14),

• (40)

Next, let us examine the mapping (18). We note that this mapping still works in the present case; the only difference is that for fixed , the constant η in the mapping (18) is bounded away from 0, whereas when , η approaches 0. More precisely, we have . This can roughly be seen from the saddle points (15) and (17), together with their correspondence relation (19) under the mapping given in (18). To prove this rigorously, we give the following lemma.

Lemma 1. Let η be the constant defined in the mapping (18). If (or, equivalently, ), then we have

• (41)

Proof. From (11), we have, for any fixed w,

• (42)

where the term holds uniformly when either w or is small. If , from (15) we have and both bounded. Thus,

• (43)

To obtain η, we use (18) and (19). First, substituting (43) in (8) gives

• (44)

Here, we have made use of the fact that if lie on the upper edge of the cut along the interval , then we have . Similarly, if lie on the lower edge of the cut, then . Next, we have from (16) and (17)

• (45)

Note that and . Formula (41) now follows from a combination of (18), (19), (44), and (45).▪

What Lemma 1 says is that if and or , then we have , that is, is large.

Coupling (40) and (18), we have

• (46)

where

• (47)

is given by (20) and (21) and is given by (39). Following the same integration-by-parts procedure outlined in Section 'Results in Ref. [8]', we let in (47), and define recursively by (24), (25), and (26). The final result is

• (48)

where

• (49)
• (50)

and

• (51)

Note that because , the series in (49) and (50) are asymptotic. Moreover, because is large in this case, (48) is indeed a compound asymptotic expansion as .

Now, we consider the subcase in which and . In this case, the expansion in (48) is still asymptotic as long as or, equivalently, . If , then the series in (2) is itself an asymptotic expansion as . This completes our discussion of all three cases listed under the condition “ small” in Table 1.

### 4. Airy-type expansion

For the case , where γ is a small positive number and M is a large positive number, the saddle points in (15) are bounded away from the singularities , , and , and coalesce with each other when . Therefore, to derive an asymptotic expansion uniformly for , we need the cubic transformation (see (57) below) introduced by Chester, Friedman, and Ursell [2]. The resulting expansion is in terms of the Airy function .

Following the same argument as in Section 'Kummer-type expansion', we again use the mapping from in (36), and start with the integral representation (40)

• (52)

where the integration path is described in the line below (8). To proceed further, we divide γ1 into two parts, and denote the part in the upper half of the plane by , and the other part in the lower half of the plane by . Recall from (8) that the phase function in (52) is given by

• (53)

which has a cut in w-plane. Put

• (54)

which has cuts and [1, ∞). From (53) and (54), we have

• (55)

for , and

• (56)

for . We now make the standard transformation

• (57)

with the correspondence between the critical points of the two sides prescribed by

• (58)

If , then are both real; if , then are complex conjugates and purely imaginary. The values of A and ζ can be obtained by using (57) and (58). We also have

• (59)

where we have again used L'Hpital's rule for . Let us first consider the case , and deform the image of the contour under the mapping defined in (57) to the steepest descent path of in the u-plane which passes through . We denote the path by C2. Similarly, we deform the image of the contour , and denote the steepest descent path passing through by C3; see Figure 1. Next, we consider the case . Note that here the saddle points are real, and that the contours C2 and C3 pass through both of them; see Figure 2. Clearly, in both cases, C3 is the reflection of C2 with respect to the real axis in the u-plane.

Let C1 denote the dotted curve shown in Figures 1 and 2, and recall the identities and . A combination of (52), (55), (56), and (57) then gives

• (60)

where

• (61)

, are given, respectively, by (39) and (59).

#### 4.1. When ζ is bounded

Following the standard integration-by-parts procedure [11, p. 368], we define recursively

• (62)
• (63)

Furthermore, expand

• (64)

From (60), (62), (63), and (64), it follows

• (65)

where

• (66)

and the coefficients are given by

• (67)

In the present case, or, equivalently, are bounded. Thus, ζ is also bounded in view of (58). Therefore, it is easy to prove that (65) is asymptotic; for details, see [11, p. 371-372].

#### 4.2. When

When , by (15) the saddle points approach , respectively, along the line . In view of (58), this is equivalent to saying that . To prove that (65) is also asymptotic in this case, we rewrite the expansion in (65) as

• (68)

where is the same as in (66) and

• (69)

Thus, it is sufficient to first prove the boundedness of the coefficients and , and then establish the asymptotic nature of the error term in (68), namely, to prove that there exist positive constants , , , and such that

• (70)

where

• (71)

and for . Note that in the present case, .

From (59), we recall that depends on ζ. The value of ζ is obtained by solving the two equations gotten from (57) with w and u replaced, respectively, by and ; see [11, p. 367]. Thus, ζ and hence both depend on the parameters a and b in (3). As a and b approach zero, ζ may tend to infinity. When the saddle points are bounded, the mapping defined by the cubic transformation (57) is analytic, and its derivative is bounded. However, in the present case, the saddle points go to infinity as . Hence, the coefficients , and the function given recursively in (62) and (63) may all blow up as , where is defined by (61). Because the coefficients and in (65) are related to and via (67) and (64), to prove that the expansion in (68) is asymptotic when , we must first give estimates for the coefficient functions and . To this end, we shall adopt a method introduced by Olde Daalhuis and Temme [5]. To begin with, we define

• (72)

Using (62) and the Cauchy residue theorem, it can be verified that

• (73)

where Γ is the contour consisting of two circles, centering at , both with radius R, where R could be as large as possible until the circles reach the singularities of in the u-plane.

We note that are removable singularities of . However, blows up at the points , where is any integer; see (59). To find their image points in the u-plane under the mapping (57), we take as an example and denote its image point by . The image point of can be treated in a similar manner. By (57), we have

• (74)

From (57) and (58), we also have

• (75)

Subtracting (75) from (74) gives

• (76)

Because and , from (76) it follows that . Therefore, there exists a constant , independent of a, b, and ζ, such that the interior of the two circles with centers at and radius is free of the singularities of in the u-plane. Because is now analytic inside the contour Γ, there exists a constant such that

• (77)

for u in the domain enclosed by the contour Γ, where denotes the maximum of the two functions . Note that as functions of τ, are analytic in the neighborhood of steepest descent path in the τ-plane.

We further introduce rational functions and , , defined recursively by

• (78)

where A0 and B0 are given in (72). By induction, we can show that and are expressible as

• (79)

where and are constants independent of u and ζ. As in (73), by Cauchy's theorem, we have from Equations (62) and (63)

• (80)

where we have used integration-by-parts to derive the second equality. The second term in the second equality vanishes because is as and all poles of that function lie inside Γ; see (79). Similarly, we also have

• (81)

Using (79), it is easy to obtain the estimates

• (82)

for u on and inside the contour Γ and . Here and thereafter, is used as a generic symbol for constants independent of u, ζ, a, and b. Substituting (82) into (80) and (81) gives

• (83)

and

• (84)

Therefore, and are both bounded for τ in the neighborhood of steepest descent path in the τ-plane, and of course for τ in the neighborhood of . Thus, we have the boundedness of the coefficients and .

To estimate , we use the rational functions , , defined recursively by

• (85)

These functions were also introduced by Olde Daalhuis and Temme [5]. They showed by induction that can be written as

• (86)

where do not depend on u, w, and ζ. Similar to (80), we have

• (87)

where Γ is the same contour used in (80) and w lies inside two disks centered at and with radius . It is easy to verify from (86) that

• (88)

and from (87) that

• (89)

Substituting (89) into (71) gives (70); for details, see [5, pp. 311–312]. Note that to make the expansion (68) asymptotic, we require to be large.

### 5. Bessel-type expansion

In the case of Hahn Polynomials given in (1), Sharapudinov [10] has given an asymptotic formula involving Jacobi polynomials when the parameters satisfy α, and , where c is a positive constant. The value of the variable x is also required to be large; more precisely, and is a small number. Although discrete Chebyshev polynomials given in (2) is a special case of the Hahn polynomials, the values of the parameters are ; that is, Sharapudinov's result does not include our case. However, because the leading term in the uniform asymptotic expansion of the Jacobi polynomials is a Bessel function (see [7, p. 451]), the work of Sharapudinov did inspire us to look for an asymptotic expansion for involving Bessel functions, when the parameters a and b in (3) satisfy and the variable x is large. Our method differs completely from that of Sharapudinov.

Returning to (7), and making the change of variable , we have

• (90)

where

• (91)

the curve γ3 starts at , runs along the lower edge of the positive real line toward , encircles the point in the clockwise direction and returns to along the upper edge of the positive real line. Note that because , there is no need to have a cut from the origin to infinity.

The saddle point of in the t-plane is given by

• (92)

cf. (11), where the v-plane is cut along two line segments joining 0 to the two conjugate points , and the branch of the square root is chosen so that as . From (92), we have

• (93)

where c is a positive constant, and for ,

• (94)

where 0+ and 0 mean limits approaching 0 from the right-hand side of the cut and the left-hand side of the cut, respectively. Moreover, easy calculation shows that for any v, is not a real number on the cut (1, ∞) in the t-plane. Following the same argument given prior to (36), we introduce the mapping

• (95)

with the correspondence between the saddle points and given by

• (96)

Coupling (95) and (96) yields

• (97)

The zeros of are given by

• (98)

where is the relevant saddle point given in (92). By straightforward calculation, we have

• (99)

Note that we have when , and when . For any fixed v, we can deform the original interval of integration into a steepest descent path , passing through . Also note that is not on , unless . Moreover, if , then and is the real interval [0, 1]. In our case, there is only one point on the path γ3 in the v-plane (see (90)), where it crosses the real line. Let us denote this point by v0. When , the mapping which we have introduced in (95) becomes singular at the point , because in (99) blows up. However, for this particular case, we only need to slightly modify the path by replacing part of the original path near this point by a small half circle as shown in Figure 3. The corresponding integration path in the τ-plane following the mapping (95) also needs to be modified. However, this small modification on the integration path will not affect the following argument and calculation. With this in mind, we will simply ignore this particular case, and proceed with the assumption that the integration path in the τ-plane is always for all w, and that will not blow up in the neighborhood of the path.

Thus, from (90) and (95), we have

• (100)

Here, we rewrite the phase function as

• (101)

Recalling the statements following (91) and (92), we know that there are only two cuts in the v-plane: one along the infinite interval and the other along the bent line joining the conjugate points and passing through the origin. To find the saddle points of , we set

and obtain

• (102)

Because (90) is obtained from (7) by making the change of variable , (102) can also be derived from (15). Note that in this case, . Furthermore, because , the quantity inside the square root is positive. Hence, are distinct, and approach .

Define

where is some constant to be determined. The saddle points of are

• (103)

Make the transformation

• (104)

with

• (105)

Note the fact that , where are given in (9) and (10). This can be seen from (12) and the change of variable that we have made. Thus, substituting (105) into (104) gives

• (106)

and

• (107)

when and . Moreover, from (104) we have

• (108)

Here, we have made use of the equality .

Coupling (100) and (104), the integral representation of becomes

• (109)

where

• (110)

and the contour starts from , encircles the point in the counterclockwise direction and returns to . For , we define recursively

• (111)
• (112)

where given in (110). Furthermore, we expand and at , and write

• (113)

It is easy to see that

• (114)

From (109), (111), and (112), we have

• (115)

where

• (116)

and

• (117)

In (115), we have made use of the integral representations of Bessel function and Gamma function

and

see [7], (10.9.19) and (5.9.1). Using (112) and integration by parts, we can rewrite in (117) as

• (118)

Repeating the procedure above, we obtain

• (119)

where

• (120)
• (121)
• (122)

Because involves and depends on the parameters a and b in (3), the coefficients and in (120) and (121) also depend on a and b. For an estimate on these coefficients, see (146) below.

To facilitate the application of expansion (119), we recall that the constants γ and m are explicitly given in (106) and (107), and note that the leading coefficient can be (asymptotically) calculated by using (113) and (114). Indeed, we have

and

as and . Furthermore, Equation (128) gives

• (123)

where .

#### 5.1. The mapping in (104)

In the case under discussion, and . Thus, because , we have , and the saddle points in (102) are complex.

Theorem 1. When , , and , the mapping defined in (104) is one-to-one and analytic for in the v-plane and in the u-plane, where the image of the boundary of in the Z-plane is given in (125), and is the image of under the mapping (104).

Proof. As in the cases of Charlier polynomials [9] and Meixner polynomials [3], we introduce an intermediate variable Z defined by

• (124)

where is the relevant saddle point given in (92).

We consider the upper half of the v-plane. Because the functions and are both symmetric with respect to the real line, the case for the lower half of the v-plane can be handled in the same manner. To avoid multivaluedness around the saddle point, we divide the upper half of the v-plane into two parts by using the steepest descent path through , that is, the point denoted by D; see Figure 4. Call the two parts Region I and Region III. Note that there are two branch cuts: one along the infinite interval [1, ∞), and the other along the line segment joining to , which was introduced in (92). We denoted the point by the letter T in Figure 4.

As v traverses along the boundary of Region I, the image point Z traverses along the corresponding boundary of a region in the Z-plane; see Figure 5. In Figure 6, we draw the boundary of the region, corresponding to Region I, in the u-plane. The image point traverses along the boundary of the same region shown in Figure 5. Note that we have used an arc to avoid the cut . The boundary curves and in the v-plane are rather arbitrary; for convenience, we choose them to be the ones whose images in the Z-plane are as indicated in Figure 5.

From (104), it is readily seen that the mapping is the composite function of and . We have just verified that this mapping is one-to-one on the boundary of Region I. By the same argument, one can prove that this mapping is also one-to-one on the boundary of Region III. For the image of Region III in the intermediate Z-plane, see Figure 7. In the u-plane, we let denote the union of the Regions I–IV, with outer boundary , and inner boundary ; see Figure 6. Under the mapping , we express the images of parts of the outer and inner boundaries of in the Z-plane as follows:

• (125)

where is any fixed constant, and θ is a generic symbol for a constant in (0, 1). Furthermore, we denote by the corresponding region of in the v-plane.

By Theorem 1.2.2 of [13, p. 12], the mapping is one-to-one in the interior of both Regions I and III in the upper half of the v-plane. As explained earlier, the one-to-one property of this mapping in the lower half of the v-plane can be established by using the symmetry of the functions with respect to the real axis. Note that the only possible singular points of the mapping in are at . Because the images of these points in the u-plane (i.e., ) are bounded, the mapping is in fact one-to-one and analytic in .▪

#### 5.2. Analyticity of

We now investigate the function given in (110). By (99) and (108), we have

• (126)

when and , and

• (127)

when and , where and is defined in the previous subsection. Note that is excluded from . It is easy to see that is bounded by a constant, independent of b and a, as for and . Therefore, is analytic in τ for τ in the neighborhood of [0, ∞). Moreover, is also analytic in u for , because the only singularities in are both removable; indeed, we have

• (128)

when and , and by the equation following (108) we also have

• (129)

when and . To reach (129), we have written as and move inside the square root in at ; see (108).

For the analysis to be used in the next subsection, we now give an estimate for when u lies on the boundaries of , that is, the outer boundary and the inner boundary shown in Figure 6. For convenience, let us denote the outer boundary by CO and the inner boundary by CI.

First, we show that for u on the inner boundary CI,

• (130)

where and C, C1, C2 are used, here and thereafter, as generic symbols for constants independent of u, v, m, a, and b. Recall from (107) that as . To prove (130), we take the part of the inner boundary CI as an illustration. Using (125), we obtain

• (131)

Because , from the equality in (131) we have . Therefore,

• (132)

Rewriting (131) gives

• (133)

which lead to . Substituting (132) into the right-hand side of the last inequality, and combining the resulting inequality with (132), we obtain (130) immediately. Similarly, for u on the outer boundary CO, we have the estimates

• (134)

Hence, it follows from (130), (134), and (126) that

• (135)

#### 5.3. Error bounds for the remainder

To prove the asymptotic nature of the expansion in (119), we need to give precise estimates for its coefficients , in (120) and (121), and the error term given in (122), because the derivative in (108) may blow up as approach 0, just like the case in Section 'Airy-type expansion'. Because the coefficients and are related to and by (113), (120), and (121), let us first estimate and . To this end, we define recursively

and

for . By induction, it can be shown that

• (136)

for , where and are constants independent of m and u (but dependent of τ); see [12]. Furthermore, by (114), (111), and (112), it can be proved that

• (137)

and

• (138)

see also [12]. Define ; because in this case both n and , and , we have , , if , and if .

Using (136), (130), and (134), it is easily verified that

• (139)

for u on the inner boundary CI, and

• (140)

for u on the outer boundary CO, where C is used again as a generic symbol for constants independent of u, a, and b. Also, we have the estimates

• (141)

To see this, we take one part of CI, namely, as an example; see (124) and (125), where Z in (124) satisfies . Using the second equality in (125), and in polar coordinates, , the last equation is equivalent to

• (142)

Thus,

• (143)

Because in this case , we have . By (130), and

• (144)

from which the first inequality in (141) follows. The second inequality can be established in a similar manner.

Because and is analytic in τ, we have for some constant and for τ in the neighborhood of the origin. By a combination of (137), (138), (139), (140), (141), and (135), we obtain

• (145)

for τ near . Furthermore, by (113), (120), (121), and (145), we have

• (146)

To estimate the remainder in (122), we split the loop contour into two parts; see Figure 8. The bounded part of the contour, denoted by Γ1, is contained in a subdomain of , which has a distance from the outer boundary of (i.e., CO), and has a distance from the inner boundary of (i.e., CI), c1 and c2 being two constants independent of u, a and b. Note that may become large, whereas may approach zero. The unbounded part of the contour, denoted by Γ2, is the rest of the loop outside the subdomain . Put

• (147)
• (148)

As in [5] and [12], it can be shown that is exponentially small in comparison with . To estimate , we also follow [5] and [12] by first deriving an integral representation for . Define recursively

• (149)

for . Using induction, one can easily verify that

• (150)

where are some constants depending only on i, j, and p. Similar to (87), for we have from (111), (112), and (149) the integral representation

• (151)

From (150), it is easy to see that

• (152)

Coupling the above inequalities with (141) and (151), we obtain

• (153)

for and τ in the neighborhood of .

If is bounded, one can easily show from (147) and (153) that

• (154)

If , we divide the integration path Γ1 into three pieces: the steepest descent path L1 through in the upper half of the u-plane, the steepest descent path L2 through in the lower half of the u-plane, and the circular arc , denoted by L3, which joins the steepest descent paths L1 and L2 at and , respectively; see Figure 8. Hence,

• (155)

where , , denotes the integral over the subcontour . Applying the steepest descent method [11], p.84], and using (153), it can be easily verified that

• (156)

For I3, because

I2 is exponentially small in comparison with I1, 2. Therefore,

• (157)

It is well-known that the Bessel function is bounded when t is bounded, and that as ,

• (158)

Coupling the estimates (154) and (157), and together with (158) and (146), we can find a constant M independent of a and b such that

• (159)

which establishes the asymptotic nature of the expansion in (119), given that . In conclusion, in the case and , (119) is an asymptotic expansion.

### 6. Remaining cases

We are now left with only two easy cases to consider.

#### 6.1. , large and x bounded

In this case, the series in (2) is itself an asymptotic expansion, because

is an asymptotic sequence when ; see Table 1 and also [11, p. 10].

#### 6.2.

The case is relatively easy, compared with the case . Let us start with the integral representation (29). For fixed , the phase function in (29) has a saddle point

• (160)

cf. (11). Note that the only difference between in (8) and in (30) is the choice of branch cuts in the w-plane due to the term in and the term in . Similar to (35), for any fixed , we have as ; in particular, . By the same reasoning given for (36), we introduce the mapping defined by

• (161)

with

• (162)

Coupling (162) and (161) yields . From (161), we have

• (163)

where we have used L'Hspital's rule for and

• (164)

cf. (39). Furthermore, it follows from (161) and (29) that

• (165)

cf. (40). Solving the equation , we obtain the saddle points

• (166)

cf. (15). The relevant saddle point on the integral path γ2 is the negative saddle point .

The following procedure is the same as that given in [8]. Recall the Hankel integral for the Gamma function

• (167)

where the contour is a loop starting at , encircling the origin in the counterclockwise direction and returning to . With and u replaced by , we obtain

• (168)

where . Make the transformation defined by

• (169)

with

• (170)

where γ is a constant to be determined. We have from (169)

• (171)

where we again have used L'Hspital's rule for .

By (165) and (169), we have

• (172)

where

• (173)

Define recursively

• (174)
• (175)

where , and expand into a Maclaurin series

• (176)

By an integration-by-parts procedure, we obtain

• (177)

where

• (178)

and

• (179)

Estimation of the error term given in (179) is exactly the same as what we have done in [8].

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