In the case of Hahn Polynomials given in (1), Sharapudinov  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.
The saddle point of in the t-plane is given by
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
where c is a positive constant, and for ,
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
with the correspondence between the saddle points and given by
Coupling (95) and (96) yields
The zeros of are given by
where is the relevant saddle point given in (92). By straightforward calculation, we have
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
Here, we rewrite the phase function as
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
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 .
Coupling (100) and (104), the integral representation of becomes
and the contour starts from , encircles the point in the counterclockwise direction and returns to . For , we define recursively
where given in (110). Furthermore, we expand and at , and write
It is easy to see that
From (109), (111), and (112), we have
In (115), we have made use of the integral representations of Bessel function and Gamma function
see , (10.9.19) and (5.9.1). Using (112) and integration by parts, we can rewrite in (117) as
Repeating the procedure above, we obtain
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.
5.2. Analyticity of
We now investigate the function given in (110). By (99) and (108), we have
when and , and
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
when and , and by the equation following (108) we also have
when and . To reach (129), we have written as and move inside the square root in at ; see (108).
First, we show that for u on the inner boundary CI,
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
Because , from the equality in (131) we have . Therefore,
Rewriting (131) gives
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
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
for . By induction, it can be shown that
for , where and are constants independent of m and u (but dependent of τ); see . Furthermore, by (114), (111), and (112), it can be proved that
see also . Define ; because in this case both n and , and , we have , , if , and if .
Using (136), (130), and (134), it is easily verified that
for u on the inner boundary CI, and
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
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
Because in this case , we have . By (130), and
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
for τ near . Furthermore, by (113), (120), (121), and (145), we have
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
As in  and , it can be shown that is exponentially small in comparison with . To estimate , we also follow  and  by first deriving an integral representation for . Define recursively
for . Using induction, one can easily verify that
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
From (150), it is easy to see that
Coupling the above inequalities with (141) and (151), we obtain
for and τ in the neighborhood of .
If is bounded, one can easily show from (147) and (153) that
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,
where , , denotes the integral over the subcontour . Applying the steepest descent method , p.84], and using (153), it can be easily verified that
For I3, because
I2 is exponentially small in comparison with I1, 2. Therefore,
It is well-known that the Bessel function is bounded when t is bounded, and that as ,
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
which establishes the asymptotic nature of the expansion in (119), given that . In conclusion, in the case and , (119) is an asymptotic expansion.