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Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results in Ref.
  5. 3. Kummer-type expansion
  6. 4. Airy-type expansion
  7. 5. Bessel-type expansion
  8. 6. Remaining cases
  9. References

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

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results in Ref.
  5. 3. Kummer-type expansion
  6. 4. Airy-type expansion
  7. 5. Bessel-type expansion
  8. 6. Remaining cases
  9. References

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

  • math image(1)

With inline image in (1), we have

  • math image(2)

see [1, p. 176].

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

  • display math(3)

satisfy the inequalities

  • display math(4)

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

  • display math(5)

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

In this paper, we shall investigate the behavior of the polynomials inline image, when the parameter b given in (3) either tends to 0 or tends to 1. The case inline image 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 inline image and (34) for inline image.

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

Table 1. Asymptotic Expansions in Different Cases when inline image and inline image
Image

Some explanation is needed for this table. By “small”, “fixed”, and “large”, we mean, respectively, inline image, inline image and inline image as inline image, 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 inline image and inline image. In the same manner, by “Bessel” and “Kummer”, we mean asymptotic expansions whose associated approximants are, respectively, the Bessel function inline image and the Kummer function inline image. The Kummer function used in this paper is defined by

  • display math(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 inline image 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 inline image. 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 inline image is small and show that the expansions are of Kummer-type. In Section 'Airy-type expansion', we consider the case when the quantity inline image is fixed; that is, when inline image 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 inline image are large is dealt with in Section 'Bessel-type expansion', where we show that the expansion of inline image can be expressed in terms of Bessel functions. The final section is devoted to two remaining cases, namely, (i) when inline image is large and x is either fixed or small, and (ii) inline image.

2. Results in Ref. [8]

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results in Ref.
  5. 3. Kummer-type expansion
  6. 4. Airy-type expansion
  7. 5. Bessel-type expansion
  8. 6. Remaining cases
  9. References

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

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

  • display math(7)

where

  • display math(8)

and the curve γ1 starts at inline image, runs along the lower edge of the positive real line toward inline image, encircles the point inline image 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 inline image and inline image vanish at inline image, where

  • display math(9)

and

  • display math(10)

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

  • display math(11)

It can be shown that

  • display math(12)

Define the standard transformation inline image by

  • display math(13)

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

  • math image(14)

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

  • display math(15)

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

  • display math(16)

The saddle points of ψ are given by

  • display math(17)

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

  • display math(18)

with

  • display math(19)

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

  • math image(20)

for inline image, and by L'Hinline imagepital's rule,

  • display math(21)

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

  • math image(22)

where

  • display math(23)

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

  • display math(24)

and

  • display math(25)

Furthermore, we write

  • display math(26)

The resulting expansion takes the form

  • math image(27)

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

  • display math(28)

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

  • display math(29)

where

  • display math(30)

and the integration path γ2 starts at inline image, traverses along the upper edge of the positive real line toward inline image, encircles the origin in the counterclockwise direction and returns to inline image 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 inline image defined in (13), except with inline image replaced by inline image. The result is

  • math image(31)

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

  • display math(32)

with

  • display math(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

  • display math(34)

where inline image are constants that can be given recursively.

3. Kummer-type expansion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results in Ref.
  5. 3. Kummer-type expansion
  6. 4. Airy-type expansion
  7. 5. Bessel-type expansion
  8. 6. Remaining cases
  9. References

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

  • display math(35)

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

  • display math(36)

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

  • display math(37)

This gives

  • display math(38)

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

  • math image(39)

where we have used L'Hinline imagepital's rule for inline image and

  • display math

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

  • math image(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 inline image, the constant η in the mapping (18) is bounded away from 0, whereas when inline image, η approaches 0. More precisely, we have inline image. This can roughly be seen from the saddle points (15) and (17), together with their correspondence relation (19) under the mapping inline image 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 inline image (or, equivalently, inline image), then we have

  • display math(41)

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

  • display math(42)

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

  • math image(43)

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

  • display math(44)

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

  • display math(45)

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

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

Coupling (40) and (18), we have

  • math image(46)

where

  • display math(47)

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

  • math image(48)

where

  • display math(49)
  • display math(50)

and

  • display math(51)

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

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

4. Airy-type expansion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results in Ref.
  5. 3. Kummer-type expansion
  6. 4. Airy-type expansion
  7. 5. Bessel-type expansion
  8. 6. Remaining cases
  9. References

For the case inline image, where γ is a small positive number and M is a large positive number, the saddle points inline image in (15) are bounded away from the singularities inline image, inline image, and inline image, and coalesce with each other when inline image. Therefore, to derive an asymptotic expansion uniformly for inline image, 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 inline image.

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

  • math image(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 inline image, and the other part in the lower half of the plane by inline image. Recall from (8) that the phase function in (52) is given by

  • display math(53)

which has a cut inline image in w-plane. Put

  • display math(54)

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

  • display math(55)

for inline image, and

  • display math(56)

for inline image. We now make the standard transformation

  • display math(57)

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

  • display math(58)

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

  • math image(59)

where we have again used L'Hinline imagepital's rule for inline image. Let us first consider the case inline image, and deform the image of the contour inline image under the mapping inline image defined in (57) to the steepest descent path of inline image in the u-plane which passes through inline image. We denote the path by C2. Similarly, we deform the image of the contour inline image, and denote the steepest descent path passing through inline image by C3; see Figure 1. Next, we consider the case inline image. Note that here the saddle points inline image 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 inline image and inline image. A combination of (52), (55), (56), and (57) then gives

  • math image(60)

where

  • display math(61)

inline image, inline image 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

  • display math(62)
  • display math(63)

Furthermore, expand

  • display math(64)

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

  • math image(65)

where

  • math image(66)

and the coefficients are given by

  • math image(67)

In the present case, inline image or, equivalently, inline image 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 inline image

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

  • math image(68)

where inline image is the same as in (66) and

  • display math(69)

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

  • math image(70)

where

  • math image(71)

and inline image for inline image. Note that in the present case, inline image.

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

  • display math(72)

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

  • math image(73)

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

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

  • display math(74)

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

  • display math(75)

Subtracting (75) from (74) gives

  • display math(76)

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

  • display math(77)

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

We further introduce rational functions inline image and inline image, inline image, defined recursively by

  • math image(78)

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

  • math image(79)

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

  • math image(80)

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

  • display math(81)

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

  • display math(82)

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

  • display math(83)

and

  • display math(84)

Therefore, inline image and inline image 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 inline image. Thus, we have the boundedness of the coefficients inline image and inline image.

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

  • math image(85)

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

  • display math(86)

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

  • display math(87)

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

  • display math(88)

and from (87) that

  • display math(89)

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

5. Bessel-type expansion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results in Ref.
  5. 3. Kummer-type expansion
  6. 4. Airy-type expansion
  7. 5. Bessel-type expansion
  8. 6. Remaining cases
  9. References

In the case of Hahn Polynomials inline image given in (1), Sharapudinov [10] has given an asymptotic formula involving Jacobi polynomials when the parameters satisfy α, inline image and inline image, where c is a positive constant. The value of the variable x is also required to be large; more precisely, inline image and inline image is a small number. Although discrete Chebyshev polynomials inline image given in (2) is a special case of the Hahn polynomials, the values of the parameters are inline image; 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 inline image involving Bessel functions, when the parameters a and b in (3) satisfy inline image and the variable x is large. Our method differs completely from that of Sharapudinov.

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

  • display math(90)

where

  • display math(91)

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

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

  • display math(92)

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

  • display math(93)

where c is a positive constant, and for inline image,

  • math image(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, inline image 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

  • display math(95)

with the correspondence between the saddle points inline image and inline image given by

  • display math(96)

Coupling (95) and (96) yields

  • display math(97)

The zeros of inline image are given by

  • display math(98)

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

  • math image(99)

Note that we have inline image when inline image, and inline image when inline image. For any fixed v, we can deform the original interval of integration inline image into a steepest descent path inline image, passing through inline image. Also note that inline image is not on inline image, unless inline image. Moreover, if inline image, then inline image and inline image 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 inline image, the mapping which we have introduced in (95) becomes singular at the point inline image, because inline image 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 inline image for all w, and that inline image will not blow up in the neighborhood of the path.

image

Figure 1. C2 and C3 inline image.

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image

Figure 2. C2 and C3 inline image.

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

  • math image(100)

Here, we rewrite the phase function inline image as

  • math image(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 inline image and the other along the bent line joining the conjugate points inline image and passing through the origin. To find the saddle points of inline image, we set

  • display math

and obtain

  • display math(102)

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

Define

  • display math

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

  • display math(103)

Make the transformation

  • math image(104)

with

  • display math(105)

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

  • math image(106)

and

  • math image(107)

when inline image and inline image. Moreover, from (104) we have

  • math image(108)

Here, we have made use of the equality inline image.

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

  • math image(109)

where

  • display math(110)

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

  • display math(111)
  • display math(112)

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

  • display math(113)

It is easy to see that

  • math image(114)

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

  • math image(115)

where

  • math image(116)

and

  • math image(117)

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

  • display math

and

  • display math

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

  • math image(118)

Repeating the procedure above, we obtain

  • math image(119)

where

  • display math(120)
  • display math(121)
  • display math(122)

Because inline image involves inline image and inline image depends on the parameters a and b in (3), the coefficients inline image and inline image 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

  • display math

and

  • display math

as inline image and inline image. Furthermore, Equation (128) gives

  • math image(123)

where inline image.

5.1. The mapping inline image in (104)

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

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

  • math image(124)

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

We consider the upper half of the v-plane. Because the functions inline image and inline image 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 inline image, 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 inline image to inline image, which was introduced in (92). We denoted the point inline image by the letter T in Figure 4.

As v traverses along the boundary inline image 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 inline image traverses along the boundary of the same region shown in Figure 5. Note that we have used an arc inline image to avoid the cut inline image. The boundary curves inline image and inline image 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 inline image is the composite function of inline image and inline image. 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 inline image denote the union of the Regions I–IV, with outer boundary inline image, and inner boundary inline image; see Figure 6. Under the mapping inline image, we express the images of parts of the outer and inner boundaries of inline image in the Z-plane as follows:

  • display math(125)
  • display math

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

By Theorem 1.2.2 of [13, p. 12], the mapping inline image 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 inline image in inline image are at inline image. Because the images of these points in the u-plane (i.e., inline image) are bounded, the mapping is in fact one-to-one and analytic in inline image.▪

image

Figure 3. The indentation of the integration path in the t-plane for inline image.

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image

Figure 4. The upper half of the v-plane.

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image

Figure 5. The image of Region I in the Z-plane.

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image

Figure 6. The upper half of the u-plane.

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5.2. Analyticity of inline image

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

  • math image(126)

when inline image and inline image, and

  • math image(127)

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

  • math image(128)

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

  • display math(129)

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

For the analysis to be used in the next subsection, we now give an estimate for inline image when u lies on the boundaries of inline image, that is, the outer boundary inline image and the inner boundary inline image 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,

  • display math(130)

where inline image 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 inline image as inline image. To prove (130), we take the part inline image of the inner boundary CI as an illustration. Using (125), we obtain

  • math image(131)

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

  • display math(132)

Rewriting (131) gives

  • display math(133)

which lead to inline image. 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

  • display math(134)

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

  • display math(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 inline image, inline image in (120) and (121), and the error term inline image given in (122), because the derivative inline image in (108) may blow up as inline image approach 0, just like the case in Section 'Airy-type expansion'. Because the coefficients inline image and inline image are related to inline image and inline image by (113), (120), and (121), let us first estimate inline image and inline image. To this end, we define recursively

  • display math

and

  • display math

for inline image. By induction, it can be shown that

  • math image(136)

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

  • display math(137)

and

  • display math(138)

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

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

  • display math(139)

for u on the inner boundary CI, and

  • display math(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

  • display math(141)

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

  • display math(142)

Thus,

  • display math(143)

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

  • display math(144)

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

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

  • display math(145)

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

  • display math(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 inline image of inline image, which has a distance inline image from the outer boundary of inline image (i.e., CO), and has a distance inline image from the inner boundary of inline image (i.e., CI), c1 and c2 being two constants independent of u, a and b. Note that inline image may become large, whereas inline image may approach zero. The unbounded part of the contour, denoted by Γ2, is the rest of the loop outside the subdomain inline image. Put

  • display math(147)
  • display math(148)
image

Figure 7. The image of Region II in the Z-plane.

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image

Figure 8. The contour in u-plane.

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

  • math image(149)

for inline image. Using induction, one can easily verify that

  • display math(150)

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

  • display math(151)

From (150), it is easy to see that

  • math image(152)

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

  • display math(153)

for inline image and τ in the neighborhood of inline image.

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

  • display math(154)

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

  • display math(155)

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

  • display math(156)

For I3, because

  • math image

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

  • display math(157)

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

  • display math(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

  • display math(159)

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

6. Remaining cases

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results in Ref.
  5. 3. Kummer-type expansion
  6. 4. Airy-type expansion
  7. 5. Bessel-type expansion
  8. 6. Remaining cases
  9. References

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

6.1. inline image, inline image large and x bounded

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

  • display math

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

6.2. inline image

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

  • display math(160)

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

  • display math(161)

with

  • display math(162)

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

  • math image(163)

where we have used L'Hinline imagespital's rule for inline image and

  • display math(164)

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

  • math image(165)

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

  • math image(166)

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

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

  • display math(167)

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

  • display math(168)

where inline image. Make the transformation inline image defined by

  • display math(169)

with

  • display math(170)

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

  • math image(171)

where we again have used L'Hinline imagespital's rule for inline image.

By (165) and (169), we have

  • math image(172)

where

  • display math(173)

Define recursively

  • display math(174)
  • display math(175)

where inline image, and expand inline image into a Maclaurin series

  • display math(176)

By an integration-by-parts procedure, we obtain

  • display math(177)

where

  • display math(178)

and

  • display math(179)

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

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results in Ref.
  5. 3. Kummer-type expansion
  6. 4. Airy-type expansion
  7. 5. Bessel-type expansion
  8. 6. Remaining cases
  9. References
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    R. Beals and R. Wong, Special Functions, A Graduate Text, Cambridge University Press, Cambridge, 2010.
  • 2
    C. Chester, B. Friedman, and F. Ursell, An extension of the method of steepest descents, Proc. Cambridge Philos. Soc. 53:599611 (1957).
  • 3
    X.-S. Jin and R. Wong, Uniform asymptotic expansions for Meixner polynomials, Constr. Approx. 14:113150 (1998).
  • 4
    Y. Lin and R. Wong, Global asymptotics of the Hahn polynomials, Analysis and Applications, 11(3), 1350018 (47 pages) (2013).
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    A. B. Olde Daalhuis and N. M. Temme, Uniform Airy-type expansions of integrals, SIAM J. Math. Anal. 25:304321 (1994).
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    F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974. (Reprinted by A. K. Peters, Wellesley, MA, 1997.)
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    F. W. J. Olver, D. W. Lozier, C. W. Clark, and R. F. Boisvert, NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010.
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    Rui Bo and R. Wong, Uniform asymptotic expansion of Charlier polynomials. Meth. Applic. Anal. 1:294313 (1994).
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    I. I. Sharapudinov, Asymptotic properties of orthogonal Hahn polynomials in a discrete variable, Sbornik: Mathematics, 681:111132 (1991).
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    R. Wong, Asymptotic Approximations of Integrals, Academic Press, Boston, 1989. (Reprinted by SIAM, Philadelphia, PA, 2001)
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    R. Wong and Y. Q. Zhao, Estimates for the error term in a uniform asymptotic expansion of the Jacobi polynomials. Anal. Appl. (Singap.) 1:213241 (2003).
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    R. Wong, Lecture Notes on Applied Analysis, World Scientific, Singapore, 2010.