LARGE-PARAMETER ASYMPTOTIC EXPANSIONS FOR THE LEGENDRE AND ALLIED FUNCTIONS
GERGO˝ NEMES AND ADRI B. OLDE DAALHUIS
Abstract. Surprisingly, apart from some special cases, simple asymptotic expansions for µ µ the associated Legendre functions Pν (z) and Qν (z) for large degree ν or large order µ are not available in the literature. The main purpose of the present paper is to fill this gap by deriving simple (inverse) factorial expansions for these functions and provide sharp and realistic bounds on their error terms. Analogous results for the Ferrers functions and the closely related Gegenbauer function are also included. In the cases that ν is an integer or 2µ is an odd integer, many of these new expansions terminate and provide finite representations in terms of simple functions. Most of these representations appear to be new. It is well known that the hypergeometric series can be regarded as a large-c asymptotic expansion for the hypergeometric function F (a, b; c; z). We also derive computable bounds for the remainder term of this expansion. To our best knowledge, no such estimates have been given in the literature prior to this paper.
1. Introduction and main results µ In this paper, we derive asymptotic expansions for the associated Legendre functions Pν (z) µ µ µ and Qν (z), and the Ferrers functions Pν (x) and Qν (x) in the case that either the degree ν or the order µ becomes large. These functions are solutions of the associated Legendre differential equation d2w(z) dw(z) µ2 (1.1) (1 z2) 2z + ν(ν + 1) w(z) = 0. − dz2 − dz − 1 z2 − The equation (1.1) occurs in the theory of spherical harmonics, potential theory, quantum me- chanics and in other branches of applied mathematics (see, for example, [4, Ch. 14]). Over the past several decades, a great deal of effort has gone into deriving asymptotic expansions for these functions. Most of these (uniform) expansions are in terms of non-elementary functions (Airy functions, Bessel functions, parabolic cylinder functions, etc.) and often have complicated coefficients (see the end of this section for a short summary). In contrast, the expansions we present in this paper are just (inverse) factorial expansions, with simple, elementary coefficients. In addition, we provide, in Sections2–5, sharp and realistic error bounds for the new expan- sions. Numerical examples demonstrating the sharpness of these bounds and the accuracy of the asymptotic expansions are given in Section 11. Note, however, that unlike their uniform coun- terparts, our expansions break down near the transition points z = 1 of the equation (1.1), except for the large-µ asymptotic expansions for the associated Legendre± functions which require z to be bounded instead. We find it surprising that, apart from some special cases, these simple arXiv:1905.07204v2 [math.CA] 16 Oct 2019 expansions have not been given in the literature before. The reason might be that previous authors were mainly focusing on deriving asymptotic results that are uniform in character, and hence, inevitably posses a rather complicated structure. Some of our results are special cases
2010 Mathematics Subject Classification. 41A60, 33C05, 33C45. Key words and phrases. asymptotic expansions, error bounds, Ferrers functions, Gegenbauer function, hyper- geometric function, Legendre functions. 1 2 G. NEMES AND A. B. OLDE DAALHUIS of the well-known large-c asymptotic expansion of the hypergeometric function F (a, b; c; z), for which we also provide, for the first time, computable error bounds. A common feature of our large-ν expansions below is that they terminate when 2µ equals an odd integer, thereby giving simple, closed-form expressions for the associated Legendre functions and the Ferrers functions. Most of these representations appear to be new. Similarly, the large-µ expansions terminate in the case that ν is an integer and, again, yield simple, finite representations. We begin with the definition of the functions under consideration. The associated Legendre functions of complex degree ν and order µ are defined in terms of the (regularised) hypergeometric function as follows (cf. [4, 14.3(ii)]): § z + 1µ/2 ν + 1, ν 1 z P µ(z) = F − ; − ν z 1 1 µ 2 − − for z C ( , 1], and ∈ \ −∞ µ/2 ν+µ+2 ν+µ+1 ! π1/2Γ(ν + µ + 1) z2 1 , 1 πiµ µ − F 2 2 (1.2) e− Qν (z) = ν+1 ν+µ+1 3 ; 2 2 z ν + 2 z for z C ( , 1] and ν + µ C Z<0. For analytic continuation to other Riemann sheets, see [4, 14.24∈ ].\ The−∞ Ferrers functions∈ of\ complex degree ν and order µ are defined as follows (cf. [4, 14.3(i)§ ]): § 1 + xµ/2 ν + 1, ν 1 x (1.3) Pµ(x) = F − ; − ν 1 x 1 µ 2 − − for 1 < x < 1, and −
π 1 + xµ/2 ν + 1, ν 1 x Qµ(x) = cos(πµ) F − ; − ν 2 sin(πµ) 1 x 1 µ 2 − − ! Γ(ν + µ + 1) 1 xµ/2 ν + 1, ν 1 x − F − ; − −Γ(ν µ + 1) 1 + x 1 + µ 2 − for 1 < x < 1 and ν + µ C Z<0. When µ is an integer, the right-hand side is replaced by its limiting− value. ∈ \ We shall also study the closely related Gegenbauer function. The Gegenbauer function of complex degree ν and order λ may be defined as follows (cf. [4, Eq. 14.3.21] and [4, Eq. 14.3.22]):
1 (λ) √πΓ(ν + 2λ) 2 λ (1.4) C (z) = P − 1 (z) ν λ 1 ν+λ Γ(λ)Γ(ν + 1) 2√z2 1 − 2 − 2 − for z C ( , 1], and ∈ \ −∞ 1 (λ) √πΓ(ν + 2λ) 2 λ (1.5) C (x) = P − 1 (x) ν λ 1 ν+λ Γ(λ)Γ(ν + 1) 2√1 x2 − 2 − 2 − for 1 < x < 1. − µ Our initial focus is on the large-ν asymptotics. In the case that ν is large, Qν (z) is the recessive solution of differential equation (1.1). Our main tools are the following convenient representations for this function. LARGE-PARAMETER ASYMPTOTICS FOR THE LEGENDRE AND ALLIED FUNCTIONS 3
(1) The associated Legendre function of the second kind can be represented in terms of the modified Bessel function of the second kind as
Z + πiµ µ 1 ∞ ν t cosh ξ (1.6) e− Q (cosh ξ) = t e− K (t sinh ξ)dt, ν Γ(ν µ + 1) µ − 0 provided that ξ > 0 and ν > µ 1 (compare [6, Eq. (6.8.29)] and (1.2)). An immediate consequence of the factor r 1 1 ξ ! π 1 + µ, µ e− πiµ µ (ν+ 2 )ξ 2 2 (1.7) e− Qν (cosh ξ) = e− Γ(ν + µ + 1)F 3− ; − , 2 sinh ξ ν + 2 2 sinh ξ (cf. [7, Eq. (3.2.44)]) in which, again, ν + µ C Z<0 and we take ξ 1 where ∈ \ ∈ D (1.8) = ξ : ξ > 0, ξ < πp , p > 0. Dp { < |= | } Note that the function cosh ξ is a biholomorphic bijection between 1 and C ( , 1]. Any hypergeometric function for which a quadratic transformationD exists\ can−∞ be expressed in terms of the associated Legendre functions or the Ferrers functions (cf. [4, 15.9(iv)]). Con- sequently, there are several representations in terms of hypergeometric functions.§ For large parameter asymptotic approximations of hypergeometric functions see, e.g., [21] and [8]. These results can be used to obtain asymptotic expansions for the associated Legendre functions or the Ferrers functions. In general, the hypergeometric series a, b a, b X∞ (a) (b) F ; z = Γ(c)F ; z = n n zn c c (c) n! n=0 n converges only for z < 1, but it can be regarded as a large-c asymptotic series in much larger complex z-domains| (cf.| [4, 15.12(ii)]). Thus, the representation (1.7) is very convenient since it is possible to obtain a large-ν§asymptotic expansion by replacing the hypergeometric function with its hypergeometric series. This asymptotic expansion will be valid in the sector arg ν π δ | | ≤ − (< π) and for ξ 1, sinh ξ ε (> 0). Error bounds for this expansion will then follow from the more general∈ results D | we shall| ≥ prove for the hypergeometric series (see Section2). Before stating our main results, we introduce some notation. We define, for any n 0, ≥ 1 1 (4µ2 12)(4µ2 32) (4µ2 (2n 1)2) µ + µ a (µ) = − − ··· − − = 2 − n 2 n , n 8nn! ( 2)n n! − and let ( p sin(πµ) if ξ > 0, √πΓ 2 + 1 (1.9) C(ξ, µ) = πiµ χ(p) = p 1 , p > 0. ie∓ if 0 < ξ < π, Γ + ± ±= 2 2 The main large parameter asymptotic expansions we prove in this paper are as follows. 4 G. NEMES AND A. B. OLDE DAALHUIS Legendre functions for large ν and fixed µ. Assume that ξ , sinh ξ ε (> 0) and ∈ D1 | | ≥ µ C is bounded. Then the associated Legendre functions have the inverse factorial expansions ∈ ν+ 1 ξ 1 n ( 2 ) ∞ ξ µ e X Γ ν n + 2 e− P (cosh ξ) an(µ) − − ν ∼ √2π sinh ξ Γ(ν µ + 1) sinh ξ (1.10) n=0 − (ν+ 1 )ξ 1 ξ n C(ξ, µ)e− 2 X∞ Γ ν n + e + a (µ) − 2 , √2π sinh ξ n Γ(ν µ + 1) sinh ξ n=0 − 1 ξ n r 1 ∞ πiµ µ π (ν+ )ξ X Γ ν n + 2 e (1.11) e− Q (cosh ξ) e− 2 a (µ) − , ν ∼ 2 sinh ξ n Γ(ν µ + 1) sinh ξ n=0 − as ν + , with ν being bounded. These results are direct consequences of Theorems 3.1 r ξ n πiµ µ π (ν+ 1 )ξ X∞ Γ(ν + µ + 1) e− (1.12) e− Q (cosh ξ) e− 2 an(µ) , ν ∼ 2 sinh ξ Γ ν + 3 + n sinh ξ n=0 2 as ν + in the sector arg ν π δ (< π). This result is a direct consequence of Theorems 3.3| and| → 3.4∞, where we provide| sharp| ≤ and− realistic error bounds for the truncated version of the expansion. For more information regarding factorial series, see, e.g., [3]. Note that Theorem 3.4 1 implies (1.12) for bounded values of µ satisfying 2 < µ. This restriction can be removed by appealing to the connection formula (1.32). − < 2 ξ We observe that when e < > 2 cos (2 ξ), the infinite series on the right-hand side of (1.12) converges to its left-hand side. This is no= surprise because this infinite series is just the power 2 ξ 1 series for the hypergeometric function in (1.7). Note that when e− < > 2 cos (2 ξ) and ν+ 2 / Z, the infinite series on the right-hand side of (1.11) is also convergent, however,= its sum is not∈ the associated Legendre function on the left-hand side. We mention that if 2µ equals an odd integer, then the expansions (1.10), (1.11) and (1.12) terminate and represent the corresponding function exactly. This generalises the cases 2µ = 1 given in [4, 14.5(iii)]. The exact representations resulting from (1.12) were also found earlier± by Cohl et al. [§2, Eq. (5.5)]. µ To obtain a large-ν asymptotic expansion for the associated Legendre function Pν (z) that holds in the sectors (0 <) δ arg ν π δ (< π), we can combine (1.12) with the connection formula ≤ | | ≤ − µ µ πiµ µ Q−ν 1(cosh ξ) Qν− (cosh ξ) e− cos(πν)P (cosh ξ) = − − − ν Γ(ν µ + 1)Γ( ν µ) − − − (combine [4, Eq. 14.9.12] with [4, Eq. 14.3.10]). LARGE-PARAMETER ASYMPTOTICS FOR THE LEGENDRE AND ALLIED FUNCTIONS 5 Legendre functions for large µ and fixed ν. To obtain large-µ asymptotic expansions for the associated Legendre functions, we can combine the results above with the identities r 2 sinh ξ ν 1 µ − − 2 Pν (coth ξ) = sin(πµ)Γ(ν + µ + 1)Pµ 1 (cosh ξ)(1.13) π − 2 πiν 1 ie− ν+ 2 Qµ 1 (cosh ξ) , − Γ(ν µ + 1) − 2 r − π sinh ξ ν 1 µ πiµ − − 2 (1.14) Qν (coth ξ) = e Γ(ν + µ + 1)Pµ 1 (cosh ξ). 2 − 2 These identities follow from Whipple’s formulae [4, 14.9(iv)] and the equations [4, Eq. 14.3.10], § (1.30) and (1.31). For example, if ξ 1 , sinh ξ ε (> 0) and ν C is bounded, (1.10) and ∈ D 2 | | ≥ ∈ (1.11) yield the inverse factorial expansions ξ n sin(πµ) X∞ e− P µ(coth ξ) eµξ a ν + 1 Γ(µ n) − ν ∼ π n 2 − sinh ξ n=0 (1.15) n ∞ ξ 1 sin(πν) µξ X 1 e + C ξ, µ e− a ν + Γ(µ n) , − − 2 π n 2 − sinh ξ n=0 n ∞ ξ πiµ µ 1 µξ X 1 e− e− Q (coth ξ) e a ν + Γ(µ n) − ν ∼ 2 n 2 − sinh ξ n=0 (1.16) n ∞ ξ 1 1 µξ X 1 e + C ξ, ν e− a ν + Γ(µ n) , − − 2 2 n 2 − sinh ξ Bn=0= ⇡ i C = ⇡ i+ + 2 + 2 1 as µ + , with µ being bounded. Examination< → ∞ of the= remainder terms in the limiting case ξ 0, ξ = 0 (using Theorems < → = 6 3.1 and 3.2), shows that these inverse factorial expansionsA =0 are actually valid on the boundaryC =+ 0 1 iξ arcsin ε, π as well. Bounds for the error terms of these expansions follow from Theorems ∓ ∈ 2 3.1 and 3.2, the details are left to the reader. Note that the function coth ξ is a⇠ biholomorphic bijection between the domains 1 and z : z > 0, z (0, 1] , see Figure1. In particular, the 2 ⇡ ⇡ D { < B 6∈= 2 i } C = i+ domain z : z < 0, z = 0 is not covered by the expansions (1.15) and (1.16). 2 1 { < = 6 } Figure 1: Test. A ⇡ ⇡ B+ = i C+ = i+ 2 2 1 z C =1 A =0 C0 =+ B =0 A 1 0 ⇠ ⇡ ⇡ B = 2 i C = i+ 2 1 Figure 1: Test. A+ Figure 2: Test. A 1 Figure 1. The mapping z = cothFigureξ from 3: Test.to z : z > 0, z (0, 1] . The D 2 { < 6∈ } boundary iξ (0, π ) is mapped to the imaginary axis iz > 0. ∓ ∈ 2 z ± C =1 B =0 A0 1 A+ Figure 2: Test. 2 1 6 G. NEMES AND A. B. OLDE DAALHUIS In order to cover the whole Riemann sheet C ( , 1], we employ the analytic continuation formulae \ −∞ µ πi πiν µ 2 πiµ µ P ze± = e± P (z) sin(π(ν + µ))e− Q (z), ν ν − π ν µ πi πiν µ Q ze± = e∓ Q (z), ν − ν (see, e.g., [4, Eq. 14.3.10], [4, Eq. 14.24.1] and [4, Eq. 14.24.2]) together with the above inverse factorial expansions, to obtain n ∞ ξ µ πi 1 sin(πν) µξ X 1 e− P e± coth ξ C ξ, µ + e a ν + Γ(µ n) − ν ∼ − 2 π n 2 − sinh ξ n=0 (1.17) n ∞ ξ sin(πµ) µξ X 1 e + e− a ν + Γ(µ n) , π n 2 − sinh ξ n=0 n ∞ ξ πiµ µ πi 1 1 µξ X 1 e− e− Q e± coth ξ C ξ, ν + e a ν + Γ(µ n) − ν ∼ − 2 2 n 2 − sinh ξ n=0 (1.18) n ∞ ξ 1 µξ X 1 e + e− a ν + Γ(µ n) , 2 n 2 − sinh ξ n=0 as µ + , with µ being bounded and ξ > 0. Again, construction of error bounds is possible< → and∞ we leave= the details to the interested±= reader. It is seen from Figure1, for example, +πi π that in the case ξ 1 and ξ > 0, we have arg e coth ξ ( , π). ∈ D 2 = ∈ 2 We note that if ν is an integer, then the expansions (1.15)–(1.18) terminate and represent the corresponding function exactly. Ferrers functions for large ν and fixed µ. Suppose that (0 <) ε ζ π ε (< π) and ≤ ≤ − µ C is bounded, and let ∈ (1.19) α = ν n + 1 ζ + µ + n 1 π , β = ν + n + 1 ζ + µ + n 1 π . µ,ν,n − 2 − 2 2 µ,ν,n 2 − 2 2 Then the Ferrers functions have the inverse factorial expansions r 1 2 X∞ Γ ν n + cos(αµ,ν,n) (1.20) Pµ(cos ζ) a (µ) − 2 , ν ∼ π sin ζ n Γ(ν µ + 1) sinn ζ n=0 − r 1 π X∞ Γ ν n + sin(αµ,ν,n) (1.21) Qµ(cos ζ) a (µ) − 2 , ν ∼ − 2 sin ζ n Γ(ν µ + 1) sinn ζ n=0 − as ν + , with ν being bounded. These results follow from Theorem 4.1, where we give sharp< and→ realistic∞ error= bounds for the truncated versions of the expansions. Under the same assumptions, the Ferrers functions posses the factorial expansions r µ 2 X∞ Γ(ν + µ + 1) cos(βµ,ν,n) (1.22) P (cos ζ) an(µ) , ν ∼ π sin ζ Γ ν + 3 + n sinn ζ n=0 2 r µ π X∞ Γ(ν + µ + 1) sin(βµ,ν,n) (1.23) Q (cos ζ) an(µ) , ν ∼ − 2 sin ζ Γ ν + 3 + n sinn ζ n=0 2 as ν + in the sector arg ν π δ (< π). These results follow from Theorems 4.2 and 4.3, where| | → we provide∞ sharp and| realistic| ≤ − error bounds for the truncated version of the expansions. 1 Note that Theorem 4.3 implies (1.22) and (1.23) for bounded values of µ satisfying 2 < µ. This restriction can be removed by appealing to the connection formulae (1.33) and (−1.34).< We LARGE-PARAMETER ASYMPTOTICS FOR THE LEGENDRE AND ALLIED FUNCTIONS 7 remark that the expansion (1.22) (without error estimates and conditions on ζ, µ and ν) is posed as an exercise in the book of Whittaker and Watson [23, Exer. 38, p. 335] (see also [10, p. 169]). π 5π In the case that 6 < ζ < 6 , the infinite series on the right-hand sides of (1.22) and (1.23) π 5π converge to the Ferrers functions on the left-hand sides. Note that when 6 < ζ < 6 and 1 ν + 2 / Z, the infinite series in (1.20) and (1.21) are also convergent, however, their sums are not the∈ Ferrers functions on the left-hand sides. We mention that if 2µ equals an odd integer, then the expansions (1.20)–(1.23) terminate and represent the corresponding function exactly. This generalises the cases 2µ = 1 given in [4, 14.5(iii)]. ± § Ferrers functions for large µ and fixed ν. Assume that (0 <) ε ζ π ε (< π) and ≤ ≤ − ν C is bounded. Then the Ferrers functions have the inverse factorial expansions ∈ sin(πµ) X∞ n Pµ(cos ζ) cotµ 1 ζ a ν + 1 Γ(µ n) 2 sin2 1 ζ ν ∼ π 2 n 2 − 2 (1.24) n=0 sin(πν) X∞ n tanµ 1 ζ a ν + 1 Γ(µ n) 2 cos2 1 ζ , − π 2 n 2 − 2 n=0 cos(πµ) X∞ n Qµ(cos ζ) cotµ 1 ζ a ν + 1 Γ(µ n) 2 sin2 1 ζ ν ∼ 2 2 n 2 − 2 (1.25) n=0 cos(πν) X∞ n tanµ 1 ζ a ν + 1 Γ(µ n) 2 cos2 1 ζ , − 2 2 n 2 − 2 n=0 as µ + , with µ being bounded. The proof of these results is given in Section 10. Bounds for< the→ error∞ terms of= these expansions follow from Theorems 3.1 and 3.2, we leave the details to the reader. We note that if ν is an integer, then the expansions (1.24) and (1.25) terminate and represent the corresponding function exactly. If combined with Theorems 2.1 and 2.2, formula (1.3) yields the truncated version, together µ with error bounds, of the known factorial expansion of the Ferrers function Pν− (x) (see [4, Eq. µ µ 14.15.1]). The analogous expansions for Pν (x) and Qν± (x) may then be obtained by means of connection formulae, see [4, 14.15(i)]. § Gegenbauer function for large ν and fixed λ. Suppose that ξ , sinh ξ ε (> 0), ∈ D1 | | ≥ (0 <) ε ζ π ε (< π) and λ C is bounded, and let ≤ ≤ − ∈ ( π cos(πλ) if ξ > 0, γλ,ν,n = (ν + λ n)ζ (λ n) 2 , (1.26) K(ξ, λ) = πiλ − − − π e± if 0 < ξ < π, δλ,ν,n = (ν + λ + n)ζ (λ n) . ±= − − 2 Then the Gegenbauer function has the inverse factorial expansions n ξ(ν+λ) ∞ ξ (λ) e X 1 Γ(ν + λ n) e− Cν (cosh ξ) an λ − − ∼ (2 sinh ξ)λ − 2 Γ(λ)Γ(ν + 1) sinh ξ (1.27) n=0 ξ(ν+λ) ξ n e− X∞ Γ(ν + λ n) e + K(ξ, λ) a λ 1 − , λ n − 2 Γ(λ)Γ(ν + 1) sinh ξ (2 sinh ξ) n=0 ∞ (λ) 2 X 1 Γ(ν + λ n) cos(γλ,ν,n) (1.28) C (cos ζ) a λ − , ν ∼ λ n − 2 Γ(λ)Γ(ν + 1) sinn ζ (2 sin ζ) n=0 8 G. NEMES AND A. B. OLDE DAALHUIS as ν + , with ν being bounded. These results are direct consequences of Corollaries 5.1 2 X∞ Γ(ν + 2λ) cos(δλ,ν,n) (1.29) C(λ)(cos ζ) a λ 1 , ν ∼ λ n − 2 Γ(λ)Γ (ν + λ + n + 1) sinn ζ (2 sin ζ) n=0 as ν + in the sector arg ν π δ (< π). This result follows from (1.5) and (1.22). In Corollaries| | → ∞ 5.3 and 5.4, we| give| sharp ≤ − and realistic error bounds for the truncated version of the expansion. We remark that in the special case that ν is a positive integer and 0 < λ < 1, the expansion (1.29) (together with the error estimate (5.4) below) was also given by Szeg˝o[18, 1 Theorem 8.21.11, p. 197]. If, in addition, λ = 2 , the expansion reduces to Stieltjes’ asymptotic expansion for the Legendre polynomials [18, Theorem 8.21.5, p. 195]. π 5π In the case that 6 < ζ < 6 , the infinite series on the right-hand side of (1.29) converge to π 5π the Gegenbauer function on the left-hand side. Note that when 6 < ζ < 6 and ν + λ / Z, the infinite series in (1.28) is also convergent, however, its sum is not the Gegenbauer function∈ on the left-hand side. We mention that if λ is an integer, then the expansions (1.27)–(1.29) terminate and represent the corresponding function exactly. The cases in which ν or µ is large. The asymptotic expansions we gave in this section are valid either in half-planes− or− in sectors not including the negative real axis. For the reader’s convenience, we reproduce here a series of known connection relations satisfied by the Legendre functions and the Ferrers functions. By combining the above asymptotic expansions with these connection relations, it is possible to cover larger regions in the parameter spaces. According to [4, 14.9(iii)] and [4, Eq. 14.3.10], § µ µ (1.30) P ν (z) = Pν 1(z), − − µ πiµ µ µ (1.31) Q ν (z) = e cos(πν)Γ(ν + µ)Γ(µ ν + 1)Pν− 1(z) + Qν 1(z), − − − − − µ Γ(ν µ + 1) µ 2 πiµ µ P − (z) = − P (z) e− sin(πµ)Q (z) , ν Γ(ν + µ + 1) ν − π ν µ 2πiµ Γ(ν µ + 1) µ (1.32) Q− (z) = e− − Q (z). ν Γ(ν + µ + 1) ν LARGE-PARAMETER ASYMPTOTICS FOR THE LEGENDRE AND ALLIED FUNCTIONS 9 Similarly, according to [4, 14.9(i)], § µ µ P ν (x) = Pν 1(x), − − µ µ µ sin(π(ν µ))Q ν (x) = π cos(πν) cos(πµ)Pν 1(x) + sin(π(ν + µ))Qν 1(x), − − − − − µ Γ(ν µ + 1) µ 2 µ (1.33) P− (x) = − cos(πµ)P (x) sin(πµ)Q (x) , ν Γ(ν + µ + 1) ν − π ν µ Γ(ν µ + 1) π µ µ (1.34) Q− (x) = − sin(πµ)P (x) + cos(πµ)Q (x) . ν Γ(ν + µ + 1) 2 ν ν Earlier asymptotic results. We close this section by providing a brief summary of existing asymptotic results in the literature. Thorne [19] obtained uniform asymptotic expansions for µ µ Pν (z) and Qν (z) with the assumptions that z is complex, µ is a negative integer, ν is a large 1 positive integer and 0 < µ/ ν + 2 < 1 is fixed. Olver [14] derived uniform expansions for µ µ − Pν (x) and Qν (x) under the conditions that 0 x < 1, µ is real and negative, ν is a large positive 1 ≤ 1 real number, and 0 < µ/ ν + 2 1 is fixed. He also treated the case when µ and ν + 2 are both purely imaginary.− Olver [15≤, 12 and 13] also gave uniform asymptotic expansions µ µ µ§ §µ with error bounds for Pν (z), Qν (z), Pν (x) and Qν (x) with the assumptions that z is complex, 1 < x < 1, µ is real and fixed and ν is a large positive real number (see also [20]). Similar − µ µ results were obtained by Shivakumar and Wong [17] for Pν (z) and by Frenzen [9] for Qν (z) 1 under the conditions that z 1, µ < 2 is bounded and ν is a large positive real number. Their expansions are also supplied≥ with computable error bounds. Dunster [5] completed the works of Thorne and Olver by deriving uniform asymptotic expansions for the Legendre and Ferrers 1 functions with the assumptions that ν + 2 is real and positive, µ is a large negative real number 1 and µ/ ν + 2 > 1. Recently, Cohl et al. [1] gave uniform asymptotic approximations for the Legendre− and Ferrers functions under the conditions that µ is real and bounded, ν is large and 1 is either real and positive or ν + 2 is purely imaginary. 2. Hypergeometric function for large c and fixed a, b: error bounds In the following two theorems, we provide computable error bounds for the hypergeometric series. The results in Theorem 2.1 are especially useful when c + and c is bounded, whereas that in Theorem 2.2 is useful when c + in the larger< domain→ ∞ arg c= π δ (< π). | | → ∞ | | ≤ − Theorem 2.1. Let N be an arbitrary non-negative integer. Let a, b and c be complex numbers satisfying N > max( a, b), c > b and c / Z 0. Then −< −< < < ∈ ≤ N 1 a, b a, b X− (a) (b) (2.1) F ; z = Γ(c)F ; z = n n zn + R(F )(z, a, b, c), c c (c) n! N n=0 n where the remainder term satisfies the estimate (F ) Γ( b + N)Γ(c + N)Γ( c b) (a)N (b)N N ( a) arg( z) (2.2) R (z, a, b, c) < < − < z max(1, e = − ), N ≤ Γ(b + N)Γ( c + N)Γ(c b) (c) N! < − N π provided z 0 and with the convention arg( z) 2 . With the extra assumption a < 1, we also have< ≤ | − | ≤ < (F ) sin(πa) Γ( b)Γ(c)Γ( c b) R (z, a, b, c) < < − < N ≤ sin(π a) Γ(b)Γ( c)Γ(c b) < < − (2.3) 1 if z 0, ( a)N ( b)N < ≤ 2 < < zN z if 0 < z z , × ( c) N! × z < ≤ | | N =1 2 < 1 z if z > z , | − | < | | 10 G. NEMES AND A. B. OLDE DAALHUIS provided z C [1, + ). If a is an integer or b is a non-positive integer, then the limiting ∈ \ ∞ < < (F ) value has to be taken in this bound. In addition, the remainder term RN (z, a, b, c) does not exceed the corresponding first neglected term in absolute value and has the same sign provided that z is negative, a, b and c are real, and N > max( a, b), c > b and c / Z 0. − − ∈ ≤ 1 Theorem 2.2. Let z be a complex number such that z 2 , and N be an arbitrary non-negative integer. Let a, b and c be complex numbers satisfying< ≤a > 0, N > b, arg(c b) < π and < −< | − | c / Z 0. Then the expansion (2.1) holds with the remainder estimate ∈ ≤ (a) (b) (F ) N N N RN (z, a, b, c) AN z . ≤ (c)N N! Here π a + σ b N a N Γ( b + N)Γ(c + N) e |= | | ||= | 1 + K< A = A (z, a, b, c, σ) = + a , N N < b+N a Γ(b + N)Γ(c b) ((c b) cos(θ + σ))< − − in which θ = arg(c b), θ + σ < π , σ π and − | | 2 | | ≤ 2 q 2 1 2 ( z) + 4 K = K(z, σ) = min , max 1, = . cos σ 1 z 2 − < π In the case that θ < 2 we take θ + σ = 0, that is, cos(θ + σ) = 1. Note that AN = (1) as c + . | | O | | → ∞ 3. Legendre functions for large ν and fixed µ: error bounds In the following two theorems, we provide computable error bounds for the inverse factorial µ µ expansions (1.10) and (1.11) of the associated Legendre functions Pν (z) and Qν (z), respectively. For the definition of the domain , see (1.8). D1 Theorem 3.1. Let ξ 1 and N, M be arbitrary non-negative integers. Let µ and ν be complex numbers satisfying µ∈ D< min N + 1 ,M + 1 and ν > max N 1 ,M 1 . Then |< | 2 2 < − 2 − 2 (3.1) (ν+ 1 )ξ N 1 ξ n ! e 2 X− e− P µ(cosh ξ) = a (µ)Γ ν n + 1 − + R(P 1)(ξ, µ, ν) ν Γ(ν µ + 1)√2π sinh ξ n − 2 sinh ξ N − n=0 (ν+ 1 )ξ M 1 ξ m ! C(ξ, µ)e− 2 X− e + a (µ)Γ ν m + 1 + R(P 2)(ξ, µ, ν) , Γ(ν µ + 1)√2π sinh ξ m − 2 sinh ξ M − m=0 where C(ξ, µ) is defined in (1.9), and the remainder terms satisfy the estimates ξ N (P 1) cos(πµ) 1 e− R (ξ, µ, ν) aN ( µ) Γ ν N + N ≤ cos(π µ) | < | < − 2 sinh ξ ( < 1 if (e2ξ) 1, 2ξ 1 < 2ξ ≤ × min 1 e− csc(2 ξ) , 1 + χ N + if (e ) > 1, − | = | 2 < and ξ M (P 2) cos(πµ) 1 e R (ξ, µ, ν) aM ( µ) Γ ν M + . M ≤ cos(π µ) | < | < − 2 sinh ξ < If 2 µ is an odd integer, then the limiting values have to be taken in these bounds. The square roots< are defined to be positive for positive real ξ and are defined by continuity elsewhere. In (P 2) addition, the remainder term RM (ξ, µ, ν) does not exceed the corresponding first neglected term LARGE-PARAMETER ASYMPTOTICS FOR THE LEGENDRE AND ALLIED FUNCTIONS 11 in absolute value and has the same sign provided that ξ is positive, µ and ν are real, and µ < M + 1 and ν > M 1 . | | 2 − 2 Theorem 3.2. Let ξ 1 and N be an arbitrary non-negative integer. Let µ and ν be complex ∈ D 1 1 numbers satisfying µ < N + 2 and ν > N 2 . Then (3.2) |< | < − q 1 π (ν+ 2 )ξ N 1 n ! 2 sinh ξ e− − ξ πiµ µ X 1 e (Q) e− Q (cosh ξ) = a (µ)Γ ν n + + R (ξ, µ, ν) , ν Γ(ν µ + 1) n − 2 sinh ξ N − n=0 where the remainder term satisfies the estimate ξ N (Q) cos(πµ) 1 e (3.3) R (ξ, µ, ν) aN ( µ) Γ ν N + . N ≤ cos(π µ) | < | < − 2 sinh ξ < If 2 µ is an odd integer, then the limiting value has to be taken in this bound. The square root is defined< to be positive for positive real ξ and is defined by continuity elsewhere. In addition, the (Q) remainder term RN (ξ, µ, ν) does not exceed the corresponding first neglected term in absolute value and has the same sign provided that ξ is positive, µ and ν are real, and µ < N + 1 and | | 2 ν > N 1 . − 2 A combination of Theorems 2.1 and 2.2 with the representation (1.7) leads to error bounds for µ the factorial expansion (1.12) of the associated Legendre function Qν (z) as given in the following two theorems. Theorem 3.3. Let N be an arbitrary non-negative integer. Let µ and ν be complex numbers 1 satisfying µ < N + 2 and ν > µ 1. Then (3.4) |< | < −< − r N 1 ξ n ! πiµ µ π (ν+ 1 )ξ X− an(µ) e− (Q) e− Q (cosh ξ) = e− 2 Γ(ν+µ+1) + Rb (ξ, µ, ν) , ν 2 sinh ξ Γ ν + 3 + n sinh ξ N n=0 2 where the remainder term satisfies the estimate 1 (Q) Γ 2 µ + N Γ( ν + µ + 1) ( µ) arg(eξ sinh ξ) Rb (ξ, µ, ν) − < < < max 1, e = N ≤ Γ 1 µ + N Γ(ν + µ + 1) 2 − ξ N aN (µ) e− | | , × Γ ν + 3 + N sinh ξ < 2 provided (e2ξ) 1 and with the convention arg(eξ sinh ξ) π . With the extra assumption < ≥ ≤ 2 µ < 1 , we also have < 2 1 ξ N (Q) cos(πµ) Γ µ Γ( ν + µ + 1) aN ( µ) e− Rb (ξ, µ, ν) 2 − < < < | < | N ≤ cos(π µ) Γ 1 µ Γ(ν + µ + 1) Γ ν + 3 + N sinh ξ < 2 − < 2 1 if (e2ξ) 1, 2ξ < ≥2ξ 1 e− csc(2 ξ) if 0 < (e ) < 1, × − 2ξ | = | < 1 e− if √2 cos ξ < 1, | − | | = | provided ξ 1. If 2 µ is an odd integer, then the limiting value has to be taken in this bound. The square∈ root D is defined< to be positive for positive real ξ and is defined by continuity elsewhere. (Q) In addition, the remainder term RbN (ξ, µ, ν) does not exceed the corresponding first neglected term in absolute value and has the same sign provided that ξ is positive, µ and ν are real, and µ < N + 1 and ν > µ 1. | | 2 − − 12 G. NEMES AND A. B. OLDE DAALHUIS Theorem 3.4. Let ξ 1 and N be an arbitrary non-negative integer. Let µ and ν be complex 1∈ D 1 numbers satisfying 2 < µ < N + 2 and arg(ν + µ + 1) < π. Then the expansion (3.4) holds with the remainder− estimate< | | ξ N (Q) aN (µ) e− (3.5) RbN (ξ, µ, ν) BN 3 . ≤ Γ ν + 2 + N sinh ξ Here 1 (π+ σ ) µ N 2 + µ 1 3 e | | |= | 1 + 1 L < N Γ N + 2 µ Γ ν + 2 + N 2 +µ BN = BN (ξ, µ, ν, σ) = 1 + 1− < 1 , + µ Γ N + µ Γ(ν + µ + 1) N+ 2 µ 2 2 − ((ν + µ + 1) cos(θ + σ)) −< in which θ = arg(ν + µ + 1), θ + σ < π , σ π and | | 2 | | ≤ 2 q 2 2 1 + ( coth ξ) L = L(ξ, σ) = min , max 1, = . cos σ coth ξ < π The fractional powers are taking their principal values. In the case that θ < 2 we take θ+σ = 0, that is, cos(θ + σ) = 1. Note that B = (1) as ν + . | | N O | | → ∞ 4. Ferrers functions for large ν and fixed µ: error bounds In the following theorem, we give computable error bounds for the inverse factorial expansions (1.20) and (1.21) of the Ferrers functions. Theorem 4.1. Let 0 < ζ < π and N be an arbitrary non-negative integer. Let µ and ν be complex numbers satisfying µ < N + 1 and ν > N 1 . Then |< | 2 < − 2 q 2 N 1 ! π sin ζ X− cos(αµ,ν,n) P (4.1) Pµ(cos ζ) = a (µ)Γ ν n + 1 + R( )(ζ, µ, ν) , ν Γ(ν µ + 1) n − 2 sinn ζ N − n=0 q π N 1 ! 2 sin ζ X− sin(αµ,ν,n) Q (4.2) Qµ(cos ζ) = − a (µ)Γ ν n + 1 + R( )(ζ, µ, ν) , ν Γ(ν µ + 1) n − 2 sinn ζ N − n=0 where αµ,ν,n is defined in (1.19), and the remainder terms satisfy the estimate (P) (Q) cos(πµ) 1 cosh( (αµ,ν,N )) (4.3) R (ζ, µ, ν) , R (ζ, µ, ν) aN ( µ) Γ ν N + = . N N ≤ cos(π µ) | < | < − 2 sinN ζ < With the modified conditions µ < N + 1 and ν > N + 1 , we also have |< | 2 < 2 (P) cos(πµ) 1 cos(αµ,ν,N ) R (ζ, µ, ν) aN ( µ) Γ ν N + | | N ≤ cos(π µ) | < | < − 2 sinN ζ (4.4) < cos(πµ) 1 sin(αµ,ν,N ) + aN+1( µ) Γ ν N | | cos(π µ) | < | < − − 2 sinN+1 ζ < and (Q) cos(πµ) 1 sin(αµ,ν,N ) R (ζ, µ, ν) aN ( µ) Γ ν N + | | N ≤ cos(π µ) | < | < − 2 sinN ζ (4.5) < cos(πµ) 1 cos(αµ,ν,N ) + aN+1( µ) Γ ν N | |. cos(π µ) | < | < − − 2 sinN+1 ζ < If 2 µ is an odd integer, then the limiting values have to be taken in these bounds. < LARGE-PARAMETER ASYMPTOTICS FOR THE LEGENDRE AND ALLIED FUNCTIONS 13 In the following two theorems, we give computable error bounds for the factorial expansions (1.22) and (1.23) of the Ferrers functions. The results in Theorem 4.2 are especially useful when ν + and ν is bounded, whereas those in Theorem 4.3 are useful when ν + in the Γ 1 µ Γ( ν + µ + 1) (P) cos(πµ) 2 aN ( µ) cos(βµ,ν,N ) RbN (ζ, µ, ν) −1 < < < | < | ≤ cos(π µ) Γ µ Γ(ν + µ + 1) Γ ν + 3 + N sinN ζ < 2 − < 2 ! ( a ( µ) sin(β ) sec ζ if 0 < ζ π or 3π ζ < π, + | N+1 < µ,ν,N | | | ≤ 4 4 ≤ Γ ν + 5 + N sinN+1 ζ × 2 sin ζ if π < ζ < 3π , < 2 4 4 and (4.10) Γ 1 µ Γ( ν + µ + 1) (Q) cos(πµ) 2 aN ( µ) sin(βµ,ν,N ) RbN (ζ, µ, ν) −1 < < < | < | ≤ cos(π µ) Γ µ Γ(ν + µ + 1) Γ ν + 3 + N sinN ζ < 2 − < 2 ! ( a ( µ) cos(β ) sec ζ if 0 < ζ π or 3π ζ < π, + | N+1 < µ,ν,N | | | ≤ 4 4 ≤ Γ ν + 5 + N sinN+1 ζ × 2 sin ζ if π < ζ < 3π . < 2 4 4 If 2 µ is an odd integer, then the limiting values have to be taken in these bounds. < Theorem 4.3. Let 0 < ζ < π and N be an arbitrary non-negative integer. Let µ and ν be 1 1 complex numbers satisfying 2 < µ < N + 2 and arg(ν + µ + 1) < π. Then the expansions (4.6) and (4.7) hold with the− remainder< estimates | | a (µ) cosh( (β )) (4.11) R(P)(ζ, µ, ν) , R(Q)(ζ, µ, ν) B N µ,ν,N , bN bN N 3 = N ≤ Γ(ν + 2 + N) sin ζ a (µ) cos(β ) a (µ) sin(β ) (4.12) R(P)(ζ, µ, ν) B N µ,ν,N + B N+1 µ,ν,N bN N 3 N N+1 5 N+1 ≤ Γ ν + 2 + N sin ζ Γ ν + 2 + N sin ζ and a (µ) sin(β ) a (µ) cos(β ) (4.13) R(Q)(ζ, µ, ν) B N µ,ν,N + B N+1 µ,ν,N . bN N 3 N N+1 5 N+1 ≤ Γ ν + 2 + N sin ζ Γ ν + 2 + N sin ζ 14 G. NEMES AND A. B. OLDE DAALHUIS Here 1 µ 1 3 (π+ σ ) µ N 1 − 2 −< e | | |= | 1 + 1 2 cos σ N Γ N + 2 µ Γ ν + 2 + N 2 +µ BN = BN (µ, ν, σ) = 1 + 1− < 1 , + µ Γ N + µ Γ(ν + µ + 1) N+ 2 µ 2 2 − ((ν + µ + 1) cos(θ + σ)) −< π π in which θ = arg(ν +µ+1), θ+σ < 2 , σ < 2 . The fractional powers are taking their principal | π | | | values. In the case that θ < 2 we take θ + σ = 0, that is, cos(θ + σ) = 1. Note that BN = (1) as ν + . | | O | | → ∞ 5. Gegenbauer function for large ν and fixed λ: error bounds As a corollary of Theorem 3.1 and the relation (1.4), we provide error bounds for the inverse factorial expansion (1.27) of the Gegenbauer function as given in the following theorem. Corollary 5.1. Let ξ 1 and N, M be arbitrary non-negative integers. Let λ and ν be complex ∈ D1 1 1 numbers satisfying λ < min N + ,M + and ν > max (N λ, M λ). Then < − 2 2 2 < − < − < (5.1) ξ(ν+λ) N 1 ξ n ! e X− Γ(ν + λ n) e− C(λ)(cosh ξ) = a λ 1 − − + R(C1)(ξ, λ, ν) ν λ n − 2 Γ(ν + 1) sinh ξ N Γ(λ) (2 sinh ξ) n=0 ξ(ν+λ) M 1 ξ m ! e− X− Γ(ν + λ m) e + K(ξ, λ) a λ 1 − + R(C2)(ξ, λ, ν) , λ m − 2 Γ(ν + 1) sinh ξ M Γ(λ) (2 sinh ξ) m=0 where K(ξ, λ) is defined in (1.26), and the remainder terms satisfy the estimates ξ N (C1) sin(πλ) 1 Γ( ν + λ N) e− R (ξ, λ, ν) aN λ < < − N ≤ sin(π λ) < − 2 Γ(ν + 1) sinh ξ ( < | | 1 if (e2ξ) 1, 2ξ 1 < 2ξ ≤ × min 1 e− csc(2 ξ) , 1 + χ N + if (e ) > 1, − | = | 2 < and ξ M (C2) sin(πλ) 1 Γ( ν + λ M) e R (ξ, λ, ν) aM λ < < − . M ≤ sin(π λ) < − 2 Γ(ν + 1) sinh ξ < | | If λ is an integer, then the limiting values have to be taken in these bounds. The fractional powers< are defined to be positive for positive real ξ and are defined by continuity elsewhere. (C2) In addition, the remainder term RM (ξ, λ, ν) does not exceed the corresponding first neglected term in absolute value and has the same sign provided that ξ is positive, λ and ν are real, and λ 1 < M + 1 and ν > M λ. − 2 2 − (λ) Note that if ν is a positive integer, Cν (cosh ξ) is a polynomial in cosh ξ. Thus, we can extend this result to the larger set f1 = ξ : ξ > 0, π < ξ π . The function cosh ξ is a continuous D { < − = ≤ } bijection between f1 and C [ 1, 1]. An immediate corollaryD of\ Theorem− 4.1 and the relation (1.5) are the following error bounds for the inverse factorial expansion (1.28) of the Gegenbauer function. Corollary 5.2. Let 0 < ζ < π and N be an arbitrary non-negative integer. Let λ and ν be complex numbers satisfying λ 1 < N + 1 and ν > N λ. Then < − 2 2 < − < N 1 ! 2 X− Γ(ν + λ n) cos(γλ,ν,n) (5.2) C(λ)(cos ζ) = a λ 1 − + R(C)(ζ, λ, ν) , ν λ n − 2 Γ(ν + 1) sinn ζ N Γ(λ) (2 sin ζ) n=0 LARGE-PARAMETER ASYMPTOTICS FOR THE LEGENDRE AND ALLIED FUNCTIONS 15 where γλ,ν,n is defined in (1.26), and the remainder term satisfies the estimate (C) sin(πλ) 1 Γ( ν + λ N) cosh( (γλ,ν,N )) R (ζ, λ, ν) aN λ < < − = . N ≤ sin(π λ) < − 2 Γ(ν + 1) sinN ζ < | | With the modified conditions λ 1 < N + 1 and ν > N λ + 1, we also have < − 2 2 < − < (C) sin(πλ) 1 Γ( ν + λ N) cos(γλ,ν,N ) R (ζ, λ, ν) aN λ < < − | | N ≤ sin(π λ) < − 2 Γ(ν + 1) sinN ζ < | | sin(πλ) 1 Γ( ν + λ N 1) sin(γλ,ν,N ) + aN+1 λ < < − − | |. sin(π λ) < − 2 Γ(ν + 1) sinN+1 ζ < | | If λ is an integer, then the limiting values have to be taken in these bounds. < An immediate corollary of Theorems 4.2, 4.3 and the relation (1.5) are the following error bounds for the factorial expansion (1.29) of the Gegenbauer function. Corollary 5.3. Let 0 < ζ < π and N be an arbitrary non-negative integer. Let λ and ν be complex numbers satisfying 0 < λ < N + 1 and ν > 1. Then (5.3) < < − N 1 ! − (λ) 2 X 1 Γ(ν + 2λ) cos(δλ,ν,n) (C) C (cos ζ) = an λ + Rb (ζ, λ, ν) , ν Γ(λ)(2 sin ζ)λ − 2 Γ(ν + λ + n + 1) sinn ζ N n=0 where δλ,ν,n is defined in (1.26), and the remainder term satisfies the estimates (C) sin(πλ) Γ( λ)Γ( ν + 1) 1 Γ(ν + 2λ) Rb (ζ, λ, ν) < < aN λ | | N ≤ sin(π λ) Γ(λ)Γ(ν + 1) < − 2 Γ( ν + λ + N + 1) < < < (5.4) ( π 3π cosh( (δλ,ν,N )) sec ζ if 0 < ζ 4 or 4 ζ < π, = N | | π ≤ 3π ≤ × sin ζ × 2 sin ζ if 4 < ζ < 4 , and (C) sin(πλ) Γ( λ)Γ( ν + 1) 1 Γ(ν + 2λ) cos(δλ,ν,N ) Rb (ζ, λ, ν) < < aN λ | | | | N ≤ sin(π λ) Γ(λ)Γ(ν + 1) < − 2 Γ( ν + λ + N + 1) sinN ζ < < < 1 Γ(ν + 2λ) sin(δλ,ν,N ) + aN+1 λ | | | | < − 2 Γ( ν + λ + N + 2) sinN+1 ζ < < ( π 3π sec ζ if 0 < ζ 4 or 4 ζ < π, | | π ≤ 3π ≤ × 2 sin ζ if 4 < ζ < 4 . If λ is an integer, then the limiting values have to be taken in these bounds. < Corollary 5.4. Let 0 < ζ < π and N be an arbitrary non-negative integer. Let λ and ν be complex numbers satisfying N < λ < 1 and arg(ν + 1) < π. Then the expansion (5.3) holds with the remainder estimates− < | | (C) 1 Γ(ν + 2λ) cosh( (δλ,ν,N )) Rb (ζ, λ, ν) KN aN λ = N ≤ − 2 Γ(ν + λ + N + 1) sinN ζ and (C) 1 Γ(ν + 2λ) cos(δλ,ν,N ) Rb (ζ, λ, ν) KN aN λ N ≤ − 2 Γ(ν + λ + N + 1) sinN ζ 1 Γ(ν + 2λ) sin(δλ,ν,N ) + KN+1 aN+1 λ . − 2 Γ(ν + λ + N + 2) sinN+1 ζ 16 G. NEMES AND A. B. OLDE DAALHUIS Here λ 1 (π+ σ ) λ N 1 < − N Γ( λ + N)Γ(ν + λ + N + 1) e | | |= | 1 + 1 λ 2 cos σ − KN = KN (λ, ν, σ) = + < , 1 λ Γ(λ + N)Γ(ν + 1) ((ν + 1) cos(θ + σ)) λ+N − < π π in which θ = arg(ν + 1), θ + σ < 2 , σ < 2 . The fractional powers are taking their principal | π | | | values. In the case that θ < 2 we take θ + σ = 0, that is, cos(θ + σ) = 1. Note that KN = (1) as ν + . | | O | | → ∞ 6. Proof of Theorems 2.1 and 2.2 In this section, we prove the bounds for the large-c expansion of the hypergeometric function given in Theorems 2.1–2.2. Throughout the section, we will assume that c is not zero or a negative integer. We begin with the proof of the estimate (2.2). Starting with the standard integral representation [4, Eq. 15.6.1] and using Taylor’s formula with integral remainder for a (1 zt)− , we obtain the truncated expansion (2.1) with − (F ) (a)N (b)N N NΓ(c + N) RN (a, b, c, z) = z (c)N N! Γ(b + N)Γ(c b) (6.1) − Z 1 Z 1 N 1 b+N 1 c b 1 (1 u) − t − (1 t) − − − dudt, × − (1 ztu)a+N 0 0 − provided that c > b, N > b, arg( z) < π. With the added conditions N > a, arg( z) π ,< we find< −< | − | −< | − | ≤ 2 Z 1 N 1 Z 1 N 1 (1 u) − (1 u) − ( a) arg(1 ztu) − a+N du − a+N e = − du 0 (1 ztu) ≤ 0 1 ztu < − | − | Z 1 ( a) arg( z) N 1 ( a) arg(1+rei arg(−z)) max(1, e = − ) (1 u) − du sup e = . ≤ 0 − r>0 ≤ N Thus Z 1 Z 1 N 1 NΓ(c + N) b+N 1 c b 1 (1 u) − t − (1 t) − − − dudt Γ(b + N)Γ(c b) − (1 ztu)a+N − 0 0 − Z 1 Γ(c + N) b+N 1 c b 1 ( a) arg( z) t< − (1 t)< −< − dt max(1, e = − ) ≤ Γ(b + N)Γ(c b) 0 − − Γ( b + N)Γ(c + N)Γ( c b) ( a) arg( z) = < < − < max(1, e = − ). Γ(b + N)Γ( c + N)Γ(c b) < − In arriving at the penultimate expression, use has been made of the standard integral represen- tation of the beta function [4, Eq. 5.12.1]. This completes the proof of (2.2). Now suppose that z < 0, a, b and c are real, and N > max( a, b) and c > b. From (6.1) and the mean value theorem of integration, we can assert that − − (a) (b) NΓ(c + N) R(F )(a, b, c, z) = N N zN N (c) N! Γ(b + N)Γ(c b) N − Z 1 Z 1 b+N 1 c b 1 N 1 t − (1 t) − − θN (t, z, a) (1 u) − dudt × 0 − 0 − Z 1 (a)N (b)N N Γ(c + N) b+N 1 c b 1 = z t − (1 t) − − θ (t, z, a)dt (c) N! Γ(b + N)Γ(c b) − N N − 0 (a)N (b)N N = z ΘN (z, a, b, c), (c)N N! LARGE-PARAMETER ASYMPTOTICS FOR THE LEGENDRE AND ALLIED FUNCTIONS 17 where 0 < θN (t, z, a) < 1 and 0 < ΘN (z, a, b, c) < 1. Thus, under these assumptions, the (F ) remainder term RN (a, b, c, z) does not exceed the corresponding first neglected term in absolute value and has the same sign. We continue with the proof of the inequality (2.3). An integral representation for the beta function [4, Eq. 5.12.2] gives Z π/2 (a)n 2 sin(πa) 2a 1 2n (6.2) = tan − u sin udu, n! π 0 provided n > a and a < 1. Hence, by summation and analytic continuation, we have −< < N 1 Z π/2 2a 1 2N a X− (a)n n n N N 2 sin(πa) tan − u sin u (1 zt)− = t z + z t du − n! π 1 zt sin2 u n=0 0 − provided that z C [1, + ), N > a and a < 1. Substitution into the standard integral representation [4∈, Eq.\ 15.6.1∞] followed−< by term-by-term< integration, yields (2.1) with Z 1 Z π/2 2a 1 2N (F ) 2 sin(πa) Γ(c) N b+N 1 c b 1 tan − u sin u R (z, a, b, c) = z t − (1 t) − − dudt, N π Γ(b)Γ(c b) − 1 zt sin2 u − 0 0 − under the assumptions that z C [1, + ), N > max( a, b), a < 1 and c > b. Note that for 0 < s < 1, ∈ \ ∞ −< −< < < < 1 if z 0, 2 2 2 ( z)2 < ≤ 2 = 1 zs = z s 2( z)s + 1 z 2 if 0 < z z , | − | | | − < ≥ | | < ≤ | | 1 z 2 if z > z 2 . | − | < | | Consequently, Z 1 (F ) 2 sin(πa) Γ(c) N b+N 1 c b 1 RN (z, a, b, c) | | z t< − (1 t)< −< − dt ≤ π Γ(b)Γ(c b) 0 − − Z π/2 1 if z 0, 2 a 1 2N z < ≤ 2 tan < − u sin udu if 0 < z z , × × z < ≤ | | 0 =1 2 1 z if z > z . | − | < | | The t-integral can be computed using the standard formula for the beta function [4, Eq. 5.12.1], and the explicit form of the u-integral follows from (6.2). This completes the proof of the bound (2.3). We conclude this section with the proof of Theorem 2.2. We will focus on the closed half-plane z 1 . Combining formula (6.1) with the identity < ≤ 2 (F ) (a)N (b)N N (F ) RN (a, b, c, z) = z + RN+1(a, b, c, z) (c)N N! and the standard integral representation [4, Eq. 15.6.1], we obtain the truncated expansion (2.1), in which we present the remainder term as N+1 Z 1 (F ) (a)N (b)N N Γ(c)(a)N+1 z c b 1 b+N a + N + 1, 1 R (z, a, b, c) = z + (1 t) − − t F ; zt dt, N (c) N! Γ(b)Γ(c b)(N + 1)! − N + 2 N − 0 18 G. NEMES AND A. B. OLDE DAALHUIS provided that N > b 1 and c > b. Using the identities −< − < < a + N + 1, 1 a 1 a, N + 1 F ; zt = (1 zt)− F − ; zt N + 2 − N + 2 N + 1 a a, N a, N + 1 = (1 zt)− F − ; zt F − ; zt azt − N + 1 − N + 1 N + 1 a a, N = (1 zt)− F − ; zt 1 , azt − N + 1 − (F ) we decompose the remainder RN (z, a, b, c) as follows: (F ) (a)N (b)N N (a + 1)N (b)N N (F ) RN (z, a, b, c) = z z + ReN (z, a, b, c) (c)N N! − (c)N N! N (a)N (b)N N (F ) = − z + ReN (z, a, b, c), a (c)N N! with Z 1 c b 1 b+N 1 (F ) Γ(c)(a + 1) (1 t) − − t − a, N Re (z, a, b, c) = N zN − F − ; zt dt. N Γ(b)Γ(c b)N! (1 zt)a N + 1 − 0 − The integral on the right-hand side is convergent provided that N > b and c > b. Now, τ −< < < by employing the substitution t = 1 e− , we deduce − Z + (b c)τ τ b+N 1 (F ) Γ(c)(a + 1)N N ∞ e − (1 e− ) − a, N τ Re (z, a, b, c) = z − F − ; z 1 e− dτ. N Γ(b)Γ(c b)N! (1 z (1 e τ ))a N + 1 − − 0 − − − (F ) N With this integral representation one can show that RN (z, a, b, c) = c − as c + π O | | | | → ∞ in the sector arg c < 2 . To extend this result to the larger sector arg c < π, we deform the | | π | | iσ contour of integration by rotating it through an angle σ, σ 2 . Thus, by writing τ = te , we arrive at | | ≤ Z + (F ) Γ(c)(a + 1)N N ∞ (b c)teiσ b+N 1 Re (z, a, b, c) = z e − t − f(t)dt, N Γ(b)Γ(c b)N! − 0 with iσ !b+N 1 te − a iσ 1 e− teiσ − a, N teiσ f(t) = e − 1 z 1 e− F − ; z 1 e− . t − − N + 1 − Let θ = arg(c b) ( π, π), θ + σ < π . Since σ π , we have − ∈ − | | 2 | | ≤ 2 teiσ iσ t Z iσ 1 e− e τe − = e− dτ 1, t t 0 ≤ and hence b+N 1 teiσ ! − 1 e− teiσ − exp ( b) arg 1 e− t ≤ − = − teiσ exp b arg 1 e− exp ( b σ ) . ≤ |= | − ≤ |= | | | LARGE-PARAMETER ASYMPTOTICS FOR THE LEGENDRE AND ALLIED FUNCTIONS 19 The last inequality may be proved as follows. Assume that σ < π and t > 0. Then | | 2 teiσ sin(t sin σ ) sin(t sin σ ) arg 1 e− = arctan | | arctan | | − et cos σ cos(t sin σ ) ≤ et cos σ 1 | | − | | | | − t sin σ t cos σ arctan | | = arctan | | tan σ arctan(tan σ ) = σ . ≤ et cos σ 1 et cos σ 1 | | ≤ | | | | | | − | | − π The case σ = 2 follows by continuity. For the hypergeometric function, we use the integral representation| | (6.3) a iσ 1 te iσ a iσ Z 1 τz 1 e− te − a, N te N 1 − − 1 z 1 e− F − ; z 1 e− = N τ − iσ dτ. − − N + 1 − 1 z 1 e te 0 − − − Since σ π , we have | | ≤ 2 a a teiσ teiσ < teiσ 1 τz 1 e− 1 τz 1 e− 1 τz 1 e− − − − − − − iσ = iσ exp ( a) arg iσ 1 z 1 e te 1 z 1 e te − = 1 z 1 e te − − − − − − − − − a teiσ < 1 τz 1 e− − − a π iσ e|= | . ≤ 1 z 1 e te − − − We continue bounding the last expression under the assumption that a > 0. For fixed complex w and 0 τ 1, the function τ 1 τw has only one critical point,< a local minimum. Hence, ≤ ≤ 7→ | − | max0 τ 1 1 τw = max(1, 1 w ). Consequently, ≤ ≤ | − | | − | iσ te 1 τz 1 e− iσ 1 − − te − iσ max 1, 1 z 1 e− . 1 z 1 e te ≤ − − − − − π teiσ 1 Since t 0 and σ 2 , we have for w = 1 e− that w 1 1. Taking z 2 , it follows ≥ | | ≤ − | − | ≤ 1 < π≤ that 1 zw = 0 can only happen in the two extreme cases, z = 2 and σ = 2 . We consider these two− cases separately. < | | 1 (1) In the case that z < 2 , it follows from the maximum modulus principle that the minimum of 1 zw occurs when< σ = π . We consider the case σ = π and take | − | | | 2 2 2! it 2 1 2 sin t G(x, y, t) = 1 (x + iy)(1 e− ) = 2(1 cos t) x + y + . − − − 2 − 2(1 cos t) − sin t 2 The minimum occurs at y = 2(1− cos t) , that is, 1 cos t = 4y2+1 . Hence, − − 2 1 x G(x, y, t) 2 − . 2 1 ≥ y + 4 Thus q 2 1 1 ( z) + teiσ − 4 1 z 1 e− = , − − ≤ 1 z 2 − < which holds for σ π and tends to infinity when z approaches the boundary z = 1 . | | ≤ 2 < 2 20 G. NEMES AND A. B. OLDE DAALHUIS (2) In the case that σ < π , it follows from maximum modulus principle that the minimum | | 2 of 1 zw occurs when z = 1 . We take z = 1 + iy and | − | < 2 2 2 1 teiσ H(y, t, σ) = 1 ( + iy)(1 e− ) . − 2 − We expand the right-hand side, obtain a quadratic expression in y and, hence, the minimum 2t cos σ2 1 e− H(y, t, σ) − . ≥ 4 (1 2 cos(t sin σ)e t cos σ + e 2t cos σ) − − − 2t cos σ2 t cos σ 2 t cos σ 2 t cos σ 2 Note that 1 e− = (1 e− ) (1 + e− ) (1 e− ) . Thus − − ≥ − t cos σ t cos σ e 2 + e− 4H(y, t, σ) − cos2 σ, ≥ et cos σ 2 cos(t sin σ) + e t cos σ ≥ − − where the second inequality can be proved as follows. The inequality is equivalent to cos2 σ cos(t sin σ) 1 + sin2 σ cosh(t cos σ) 0. − ≥ Using 1 = cos2 σ + sin2 σ we can re-write this inequality as sin2 σ sinh2 1 t cos σ cos2 σ sin2 1 t sin σ , 2 ≥ 2 sinh2 A sin2 B 2 which holds since for all positive A and B we have A2 1 B2 . Thus 4H(y, t, σ) cos σ, that is, ≥ ≥ ≥ 1 teiσ − 2 1 z 1 e− , − − ≤ cos σ 1 π which holds for z 2 and tends to infinity when σ approaches the boundary σ = 2 . < ≤ | | a a π Therefore, in summary, the absolute value of the left-hand side of (6.3) is at most K< e|= | , in which we can take q 2 1 ( z) + 4 2 K = max 1, = or K = . 1 z cos σ 2 − < Consequently, Z + (F ) Γ(c)(a + 1)N N ∞ c b t cos(θ+σ) b+N 1 Re (z, a, b, c) z e−| − | t< − f(t) dt N ≤ Γ(b)Γ(c b)N! | | − 0 Z + Γ(c)(a + 1)N N ∞ c b t cos(θ+σ) b+N 1 π a + σ b a z e−| − | t< − e |= | | ||= |K< dt ≤ Γ(b)Γ(c b)N! 0 − π a + σ b Γ( b + N)Γ(c)(a + 1)N N a = e |= | | ||= | < z K< , Γ(b)Γ(c b)N!((c b) cos(θ + σ)) b+N − − < and therefore, (F ) N (a)N (b)N N (F ) RN (z, a, b, c) z + ReN (z, a, b, c) ≤ a (c)N N! N (a)N (b)N N π a + σ b Γ( b + N)Γ(c)(a + 1)N N a z + e |= | | ||= | < z K< ≤ a (c) N! Γ(b)Γ(c b)N!((c b) cos(θ + σ)) b+N N − − < π a + σ b N a ! N Γ( b + N)Γ(c + N) e |= | | ||= | 1 + K< (a)N (b)N = + < a zN . a Γ(b + N)Γ(c b) ((c b) cos(θ + σ)) b+N (c) N! − − < N LARGE-PARAMETER ASYMPTOTICS FOR THE LEGENDRE AND ALLIED FUNCTIONS 21 7. Proof of Theorems 3.1 and 3.2 In this section, we prove the error bounds stated in Theorems 3.1–3.2. To this end, we require an estimate for the remainder term of the well-known asymptotic expansion of the modified Bessel function of the second kind given in the following lemma. Lemma 7.1. Let N be a non-negative integer and let µ be an arbitrary complex number such that µ < N + 1 . Then |< | 2 r N 1 ! π w X− an(µ) (K) (7.1) K µ(w) = e− + r (w, µ) , ± 2w wn N n=0 where the remainder term satisfies the estimate ( cos(πµ) a ( µ) 1 if arg w π , (K) N 2 rN (w, µ) | Here N and M are arbitrary non-negative integers. Assuming that ν > max N 1 ,M 1 , < − 2 − 2 the substitution of (7.3) (with w = t sinh ξ) into (7.2) gives (3.1) with C(ξ, µ) = Ce(arg(sinh ξ), µ) and Z + (P 1) 1 ∞ 1 −ξ (K) (ν+ 2 )ξ ν 2 te RN (ξ, µ, ν) = e− t − e− reN (t sinh ξ, µ)dt, 0 Z + (P 2) 1 ∞ 1 ξ (K) (ν+ 2 )ξ ν 2 te RM (ξ, µ, ν) = e t − e− rM (t sinh ξ, µ)dt. 0 We perform a change of integration variable from t to s by t = seξ in the case of the first integral ξ and by t = se− in the case of the second integral. By an application of Cauchy’s theorem, the contours of integration can then be deformed back into the positive real axis and we arrive at the representations Z + (P 1) ∞ 1 (K) ν 2 s ξ (7.6) RN (ξ, µ, ν) = s − e− reN se sinh ξ, µ ds 0 and Z + (P 2) ∞ 1 (K) ν 2 s ξ (7.7) RM (ξ, µ, ν) = s − e− rM se− sinh ξ, µ ds, 0 respectively. Now observe that the function eξ sinh ξ can be regarded as an analytic function from 1 1 the domain 1 to the sector z : arg z < 2π and z + 2 > 2 . It maps the part of 1 lying D | | 1 1 D in the upper half-plane into the sector z : 0 < arg z < 2π and z + 2 > 2 , and it maps the 1 1 part of 1 lying in the lower half-plane into the sector z : 2π < arg z < 0 and z + 2 > 2 . D ξ − Similarly, the function e− sinh ξ can be viewed as an analytic function from the domain 1 1 1 D to the disk z : z < . Thus, we can employ analytic continuation in ξ to extend our − 2 2 results into the domain 1. The formula (1.9) for C(ξ, µ) follows immediately from (7.4) and the properties of the hyperbolicD sine function. ξ By Lemma 7.1 and the properties of the function e− sinh ξ, we can assert that Z + (P 2) ∞ 1 (K) ν 2 s ξ RM (ξ, µ, ν) s< − e− rM (se− sinh ξ, µ) ds ≤ 0 ξ M cos(πµ) 1 e aM ( µ) Γ ν M + , ≤ cos(π µ) | < | < − 2 sinh ξ < 1 1 provided that ξ 1, µ < M + 2 and ν > M 2 . If 2 µ is an odd integer, then the limiting value has to be taken∈ D in|< this| bound. Similarly,< by− Lemma <7.1 and the properties of the function eξ sinh ξ, we can infer that ξ N (P 1) cos(πµ) 1 e− R (ξ, µ, ν) aN ( µ) Γ ν N + N ≤ cos(π µ) | < | < − 2 sinh ξ ( < 1 if π ϑ mod 2π π, 2 ≤ | | ≤ × min csc ϑ , 1 + χ N + 1 if ϑ mod 2π < π , | | 2 | | 2 ξ 1 1 with ϑ = arg(e sinh ξ) ( 2π, 2π), provided that ξ 1, µ < N + 2 and ν > N 2 . Again, if 2 µ is an odd∈ integer,− then the limiting value∈ has D to|< be| taken in this bound.< Since− < ξ 2 ξ (e sinh ξ) e < sin(2 ξ) tan ϑ = = = = , | | (eξ sinh ξ) e2 ξ cos(2 ξ) 1 < < = − LARGE-PARAMETER ASYMPTOTICS FOR THE LEGENDRE AND ALLIED FUNCTIONS 23 2 ξ π whenever e < cos(2 ξ) = 1 (i.e., ϑ mod π), and = 6 6≡ 2 √1 + tan2 ϑ csc ϑ = , | | tan ϑ | | 2ξ π it follows that csc ϑ = 1 e− csc(2 ξ) . Finally, ϑ mod 2π π occurs whenever | | − | = | 2 ≤ | | ≤ (eξ sinh ξ) 0, i.e., (e2ξ) 1, and ϑ mod 2π < π occurs whenever (eξ sinh ξ) > 0, i.e., < ≤ < ≤ | | 2 < (e2ξ) > 1. < 1 1 Now assume that ξ > 0, µ and ν are real, and µ < M + 2 and ν > M 2 . From Lemma 7.1 and the mean value theorem of integration, we can| | infer that − Z + (P 2) ∞ 1 aM (ν) ν 2 s RM (ξ, µ, ν) = s − e− ξ M θM (ξ, µ)ds 0 (se− sinh ξ) M eξ = a (ν)Γ ν M + 1 Θ (ξ, µ, ν), M − 2 sinh ξ M where 0 < θM (ξ, µ) < 1 and 0 < ΘM (ξ, µ, ν) < 1. This completes the proof of Theorem 3.1. We conclude this section with the proof of Theorem 3.2. Let N be an arbitrary non-negative 1 integer. Assuming that ν > N 2 , the substitution of (7.3) (with w = t sinh ξ) into (1.6) gives (3.2) with < − Z + (Q) 1 ∞ 1 ξ (K) (ν+ 2 )ξ ν 2 te RN (ξ, µ, ν) = e t − e− rN (t sinh ξ, µ)dt. 0 (Q) (P 2) (Q) Hence, RN (ξ, µ, ν) = RN (ξ, µ, ν) and the estimates for RN (ξ, µ, ν) follow from those for (P 2) RN (ξ, µ, ν). 8. Proof of Theorem 4.1 In this section, we prove the error bounds stated in Theorem 4.1. The Ferrers function of the first kind is related to the associated Legendre function of the first kind via the limit π µ 2 iµ µ (8.1) Pν (x) = lim e Pν (x + iε), ε 0+ → where 1 < x < 1 [4, Eq. 14.23.1]. Suppose that 0 < ζ < π. Since − (8.2) cosh(ε + iζ) = cosh ε cos ζ + i sinh ε sin ζ, we obtain, using (8.1), that π µ 2 iµ µ (8.3) Pν (cos ζ) = lim e Pν (cosh(ε + iζ)). ε 0+ → 1 Assuming that ν > µ 1 and ν > N 2 , and taking into account (7.5), (7.6) and (7.7), the substitution< of (3.1<) (with− M =< N) into− the right-hand side of (8.3) yields (4.1) with (P) 1 ν+ 1 ζ+ µ 1 π i (P 1) π i R (ζ, µ, ν) = e(( 2 ) ( − 2 ) 2 ) lim R ε + e 2 ζ, µ, ν N 2 ε 0+ N → 1 ν+ 1 ζ+ µ 1 π i (P 2) π i + e−(( 2 ) ( − 2 ) 2 ) lim R ε + e 2 ζ, µ, ν 2 ε 0+ N → (8.4) Z + 1 1 π ∞ 1 (K) π 1 ((ν+ 2 )ζ+(µ 2 ) 2 )i ν 2 s (ζ 2 )i = 2 e − s − e− rN (se − sin ζ, µ)ds 0 Z + 1 1 π ∞ 1 (K) π 1 ((ν+ 2 )ζ+(µ 2 ) 2 )i ν 2 s (ζ 2 )i + 2 e− − s − e− rN (se− − sin ζ, µ)ds. 0 24 G. NEMES AND A. B. OLDE DAALHUIS It follows from Lemma 7.1 that Z + (P) ∞ 1 (K) π 1 (αµ,ν,N ) ν 2 s (ζ 2 )i RN (ζ, µ, ν) 2 e−= s< − e− rN (se − sin ζ, µ) ds ≤ 0 Z + (α ) ∞ ν 1 s (K) ζ π i (8.5) 1 µ,ν,N 2 ( 2 ) + 2 e= s< − e− rN (se− − sin ζ, µ) ds 0 cos(πµ) 1 cosh( (αµ,ν,N )) aN ( µ) Γ ν N + = , ≤ cos(π µ) | < | < − 2 sinN ζ < 1 1 provided that 0 < ζ < π, µ < N + 2 and ν > N 2 . If 2 µ is an odd integer, then the |< | < − < (P) limiting value has to be taken in this bound. This proves (4.3) for RN (ζ, µ, ν). To prove (4.4), we first re-write (8.4) in the form Z + (P) ∞ 1 (K) π 1 ν 2 s iζN N (ζ 2 )i RN (ζ, µ, ν) = 2 cos(αµ,ν,N ) s − e− e ( i) rN (se − sin ζ, µ) 0 − π iζN N (K) (ζ 2 )i +e− i rN (se− − sin ζ, µ) ds Z + ∞ 1 (K) π i ν 2 s iζN N (ζ 2 )i + 2 sin(αµ,ν,N ) s − e− e ( i) rN (se − sin ζ, µ) 0 − iζN N (K) (ζ π )i e− i r (se− − 2 sin ζ, µ) ds. − N 1 (K) N (K) Suppose that ν > N + 2 . Next, we employ the relation rN (w, µ) = aN (µ)w− + rN+1(w, µ) inside the second< integral, to obtain Z + (P) ∞ 1 (K) π 1 ν 2 s iζN N (ζ 2 )i RN (ζ, µ, ν) = 2 cos(αµ,ν,N ) s − e− e ( i) rN (se − sin ζ, µ) 0 − π iζN N (K) (ζ 2 )i +e− i rN (se− − sin ζ, µ) ds Z + ∞ 1 (K) π i ν 2 s iζN N (ζ 2 )i + 2 sin(αµ,ν,N ) s − e− e ( i) rN+1(se − sin ζ, µ) 0 − iζN N (K) (ζ π )i e− i r (se− − 2 sin ζ, µ) ds. − N+1 It follows from Lemma 7.1 that (P) cos(πµ) 1 cos(αµ,ν,N ) R (ζ, µ, ν) aN ( µ) Γ ν N + | | N ≤ cos(π µ) | < | < − 2 sinN ζ (8.6) < cos(πµ) 1 sin(αµ,ν,N ) + aN+1( µ) Γ ν N | |, cos(π µ) | < | < − − 2 sinN+1 ζ < 1 1 provided that 0 < ζ < π, µ < N + 2 and ν > N + 2 . Again, if 2 µ is an odd integer, then the limiting value has to be|< taken| in this bound.< < The Ferrers function of the second kind is related to the associated Legendre function of the second kind via the limit π π µ 1 πiµ 2 iµ µ 2 iµ µ (8.7) Qν (x) = lim e− e− Qν (x + iε) + e Qν (x iε) , ε 0+ 2 → − where 1 < x < 1 (combine [4, Eq. 14.23.5] with [4, Eq. 14.3.10]). Suppose that 0 < ζ < π. From (−8.2) and (8.7), we can assert that π π µ 1 πiµ 2 iµ µ 2 iµ µ (8.8) Qν (cos ζ) = lim e− e− Qν (cosh(ε + iζ)) + e Qν (cosh(ε iζ)) . ε 0+ 2 → − LARGE-PARAMETER ASYMPTOTICS FOR THE LEGENDRE AND ALLIED FUNCTIONS 25 1 Assuming that ν > µ 1 and ν > N 2 , and taking into account (7.7), the substitution of (3.2) into the right-hand< < − side of (<8.8) yields− (4.2) with (Q) 1 ν+ 1 ζ+ µ+ 1 π i (Q) π i R (ζ, µ, ν) = e−(( 2 ) ( 2 ) 2 ) lim R (ε + e 2 ζ, µ, ν) N 2 ε 0+ N − → 1 ((ν+ 1 )ζ+(µ+ 1 ) π )i (Q) π i e 2 2 2 lim R (ε + e− 2 ζ, µ, ν) 2 ε 0+ N − → (8.9) Z + 1 1 π ∞ 1 (K) π 1 ((ν+ 2 )ζ+(µ+ 2 ) 2 )i ν 2 s (ζ 2 )i = 2 e− s − e− rN (se− − sin ζ, µ)ds − 0 Z + 1 1 π ∞ 1 (K) π 1 ((ν+ 2 )ζ+(µ+ 2 ) 2 )i ν 2 s (ζ 2 )i 2 e s − e− rN (se − sin ζ, µ)ds. − 0 (Q) Now, a procedure analogous to (8.5) yields the estimate (4.3) for RN (ζ, µ, ν). To prove (4.5), we first re-write (8.9) in the form Z + (Q) ∞ 1 (K) π 1 ν 2 s iNζ N (ζ 2 )i RN (ζ, µ, ν) = 2 sin(αµ,ν,N ) s − e− e− i rN (se− − sin ζ, µ) 0 iNζ N (K) ζ π i +e ( i) r (se( − 2 ) sin ζ, µ) ds − N Z + ∞ 1 (K) π i ν 2 s iNζ N (ζ 2 )i + 2 cos(αµ,ν,N ) s − e− e− i rN (se− − sin ζ, µ) 0 iNζ N (K) ζ π i e ( i) r (se( − 2 ) sin ζ, µ) ds. − − N 1 (K) N (K) Suppose that ν > N + 2 . Next, we employ the relation rN (w, µ) = aN (µ)w− + rN+1(w, µ) inside the second< integral, to obtain Z + (Q) ∞ 1 (K) π 1 ν 2 s iNζ N (ζ 2 )i RN (ζ, µ, ν) = 2 sin(αµ,ν,N ) s − e− e− i rN (se− − sin ζ, µ) 0 iNζ N (K) ζ π i +e ( i) r (se( − 2 ) sin ζ, µ) ds − N Z + ∞ 1 (K) π i ν 2 s iNζ N (ζ 2 )i + 2 cos(αµ,ν,N ) s − e− e− i rN+1(se− − sin ζ, µ) 0 iNζ N (K) ζ π i e ( i) r (se( − 2 ) sin ζ, µ) ds. − − N+1 (Q) Now, a procedure analogous to (8.6) gives the bound (4.5) for RN (ζ, µ, ν). 9. Proof of Theorems 4.2 and 4.3 In this section, we prove the error bounds stated in Theorems 4.2 and 4.3. The associated Legendre function of the first kind can be expressed in terms of the hypergeometric function as follows: 1 1 ξ ! √2π sinh ξ 1 1 + µ, µ e− µ ( 2 µ)πi (ν+ 2 )ξ 2 2 Pν (cosh ξ) = e± − − F 3− ; − Γ(µ + ν + 1) ν + 2 2 sinh ξ (9.1) 1 1 ξ ! 1 + µ, µ e (ν+ 2 )ξ 2 2 + e F 3− ; , ν + 2 2 sinh ξ for ξ 1 (0, + ). The upper or lower sign is taken in (9.1) according as ξ is in the upper or lower∈ half-plane D \ (combine∞ [4, Eq. 14.9.12] with (1.7) and connection formula [4, Eq. 15.10.21]). 26 G. NEMES AND A. B. OLDE DAALHUIS Assume that 0 < ζ < π. From (9.1) and (8.3), we deduce 1 1 iζ ! Γ(ν + µ + 1) 1 1 π + µ, µ ie− µ ((ν+ 2 )ζ+(µ 2 ) 2 )i 2 2 Pν (cos ζ) = e− − F 3− ; √2π sin ζ ν + 2 2 sin ζ 1 1 iζ !! 1 1 π + µ, µ ie ((ν+ 2 )ζ+(µ 2 ) 2 )i 2 2 +e − F 3− ; − ν + 2 2 sin ζ (cf. [10, p. 168]). We now substitute the hypergeometric functions by means of the truncated expansion (2.1). In this way, we obtain (4.6) with 1 1 π ((ν+ 2 )ζ+(µ 2 ) 2 )i iζ (P) e− − (F ) ie− 1 1 3 RbN (ζ, µ, ν) = 3 RN , 2 + µ, 2 µ, ν + 2 2Γ ν + 2 2 sin ζ − (9.2) 1 1 π ((ν+ 2 )ζ+(µ 2 ) 2 )i iζ e − (F ) ie 1 1 3 + 3 RN − , 2 + µ, 2 µ, ν + 2 . 2Γ ν + 2 2 sin ζ − Consequently, (βµ,ν,N ) iζ (P) e= (F ) ie− 1 1 3 RbN (ζ, µ, ν) 3 RN , 2 + µ, 2 µ, ν + 2 ≤ 2 Γ ν + 2 sin ζ − (9.3) 2 (βµ,ν,N ) iζ e−= (F ) ie 1 1 3 + 3 RN − , 2 + µ, 2 µ, ν + 2 . 2 Γ ν + 2 2 sin ζ − Now, under the assumptions of Theorem 4.2, we can apply (2.3) of Theorem 2.1 to the right-hand side of this inequality to obtain 1 (P) cos(πµ) Γ µ Γ( ν + µ + 1) Rb (ζ, µ, ν) 2 − < < < N ≤ cos(π µ) Γ 1 µ Γ(ν + µ + 1) < 2 − ( a ( µ) cosh( (β )) sec ζ if 2 sin2 ζ 1, | N < | = µ,ν,N | | ≤ × Γ ν + 3 + N sinN ζ × 2 sin ζ if 2 sin2 ζ > 1. < 2 2 π 3π 2 Noting that 2 sin ζ 1 occurs whenever 0 < ζ 4 or 4 ζ < π, and 2 sin ζ > 1 occurs π 3≤π ≤ ≤(P) whenever 4 < ζ < 4 , the proof of the estimate (4.8) for RbN (ζ, µ, ν) is complete. With the assumptions of Theorem 4.3, Theorem 2.2 is applicable to the right-hand side of the inequality (P) (9.3), and yields the estimate (4.11) for RbN (ζ, µ, ν). To prove (4.9) and (4.12), we first re-write (9.2) in the form iζ (P) cos(βµ,ν,N ) iζN N (F ) ie− 1 1 3 RbN (ζ, µ, ν) = 3 e i RN , 2 + µ, 2 µ, ν + 2 2Γ ν + 2 2 sin ζ − iζ iζN N (F ) ie 1 1 3 +e− ( i) R − , + µ, µ, ν + − N 2 sin ζ 2 2 − 2 iζ i sin(βµ,ν,N ) iζN N (F ) ie− 1 1 3 3 e i RN , 2 + µ, 2 µ, ν + 2 − 2Γ ν + 2 2 sin ζ − iζ iζN N (F ) ie 1 1 3 e− ( i) R − , + µ, µ, ν + . − − N 2 sin ζ 2 2 − 2 Next, we employ the relation a (µ) (9.4) R(F ) w, 1 + µ, 1 µ, ν + 3 = ( 2)N N wN + R(F ) w, 1 + µ, 1 µ, ν + 3 N 2 2 − 2 − 3 N+1 2 2 − 2 ν + 2 N LARGE-PARAMETER ASYMPTOTICS FOR THE LEGENDRE AND ALLIED FUNCTIONS 27 inside the second large parenthesis, to deduce iζ (P) cos(βµ,ν,N ) iζN N (F ) ie− 1 1 3 RbN (ζ, µ, ν) = 3 e i RN , 2 + µ, 2 µ, ν + 2 2Γ ν + 2 2 sin ζ − iζ iζN N (F ) ie 1 1 3 +e− ( i) R − , + µ, µ, ν + − N 2 sin ζ 2 2 − 2 iζ i sin(βµ,ν,N ) iζN N (F ) ie− 1 1 3 3 e i RN+1 , 2 + µ, 2 µ, ν + 2 − 2Γ ν + 2 2 sin ζ − iζ iζN N (F ) ie 1 1 3 e− ( i) R − , + µ, µ, ν + . − − N+1 2 sin ζ 2 2 − 2 Thus, iζ (P) cos(βµ,ν,N ) (F ) ie− 1 1 3 RbN (ζ, µ, ν) | 3 | RN , 2 + µ, 2 µ, ν + 2 ≤ 2 Γ ν + 2 2 sin ζ − iζ (F ) ie + R − , 1 + µ, 1 µ, ν + 3 N 2 sin ζ 2 2 − 2 (9.5) iζ sin(βµ,ν,N ) (F ) ie− 1 1 3 + | 3 | RN+1 , 2 + µ, 2 µ, ν + 2 2 Γ ν + 2 2 sin ζ − iζ (F ) ie + R − , 1 + µ, 1 µ, ν + 3 . N+1 2 sin ζ 2 2 − 2 Under the assumptions of Theorem 4.2, we can apply (2.3) of Theorem 2.1 to the right-hand side of this inequality to obtain the bound (4.9). With the assumptions of Theorem 4.3, Theorem 2.2 is applicable to the right-hand side of the inequality (9.5), leading to the estimate (4.12). (Q) We conclude this section with the proof of the estimates for the reminder term RbN (ζ, µ, ν). From (1.7) and (8.8), we deduce r 1 1 iζ ! i π 1 1 π + µ, µ ie− µ ((ν+ 2 )ζ+(µ 2 ) 2 )i 2 2 Qν (cos ζ) = Γ(ν + µ + 1) e− − F 3− ; −2 2 sin ζ ν + 2 2 sin ζ 1 1 iζ !! 1 1 π + µ, µ ie ((ν+ 2 )ζ+(µ 2 ) 2 )i 2 2 e − F 3− ; − − ν + 2 2 sin ζ (cf. [10, p. 168]). We now substitute the hypergeometric functions by means of the truncated expansion (2.1). In this way, we obtain (4.7) with 1 1 π ((ν+ 2 )ζ+(µ 2 ) 2 )i iζ (Q) ie− − (F ) ie− 1 1 3 RbN (ζ, µ, ν) = 3 RN , 2 + µ, 2 µ, ν + 2 2Γ ν + 2 2 sin ζ − (9.6) 1 1 π ((ν+ 2 )ζ+(µ 2 ) 2 )i iζ ie − (F ) ie 1 1 3 3 RN − , 2 + µ, 2 µ, ν + 2 . − 2Γ ν + 2 2 sin ζ − 28 G. NEMES AND A. B. OLDE DAALHUIS (P) We can now proceed analogously to the case of RbN (ζ, µ, ν) and derive the bounds (4.8) and (Q) (4.11) for RbN (ζ, µ, ν). To prove (4.10) and (4.13), we first re-write (9.6) in the form iζ (Q) sin(βµ,ν,N ) iζN N (F ) ie− 1 1 3 RbN (ζ, µ, ν) = 3 e i RN , 2 + µ, 2 µ, ν + 2 2Γ ν + 2 2 sin ζ − iζ iζN N (F ) ie 1 1 3 +e− ( i) R − , + µ, µ, ν + − N 2 sin ζ 2 2 − 2 iζ i cos(βµ,ν,N ) iζN N (F ) ie− 1 1 3 + 3 e i RN , 2 + µ, 2 µ, ν + 2 2Γ ν + 2 2 sin ζ − iζ iζN N (F ) ie 1 1 3 e− ( i) R − , + µ, µ, ν + . − − N 2 sin ζ 2 2 − 2 Next, we employ the relation (9.4) inside the second large parenthesis, to deduce iζ (Q) sin(βµ,ν,N ) iζN N (F ) ie− 1 1 3 RbN (ζ, µ, ν) = 3 e i RN , 2 + µ, 2 µ, ν + 2 2Γ ν + 2 2 sin ζ − iζ iζN N (F ) ie 1 1 3 +e− ( i) R − , + µ, µ, ν + − N 2 sin ζ 2 2 − 2 iζ i cos(βµ,ν,N ) iζN N (F ) ie− 1 1 3 + 3 e i RN+1 , 2 + µ, 2 µ, ν + 2 2Γ ν + 2 2 sin ζ − iζ iζN N (F ) ie 1 1 3 e− ( i) R − , + µ, µ, ν + . − − N+1 2 sin ζ 2 2 − 2 (P) We can now proceed similarly to the case of RbN (ζ, µ, ν) and derive the bounds (4.10) and (4.13) (Q) for RbN (ζ, µ, ν). 10. Ferrers functions for large µ and fixed ν: proof To obtain large-µ asymptotic expansions for the Ferrers functions, we can combine the expan- sions (1.10) and (1.11) with the identities 2µ−4ν−3 πi r e 4 2 ν+ 1 Pµ 2 ν (cos ζ) = Q µ 1 ( i cot ζ) Γ(ν µ + 1) π sin ζ − − 2 − − r 2µ−1 2 ν+ 1 4 πi 2 = sin(πµ)e Γ(µ ν) Pµ 1 ( i cot ζ) − π sin ζ − 2 − r sin(π(ν + µ)) 2µ−4ν−3 2 ν+ 1 4 πi 2 e Γ(µ ν) Qµ 1 ( i cot ζ), − π − π sin ζ − 2 − r 2µ+1 1 2µ+1 1 µ π πi ν 2 πi ν 2 Q 4 − − 4 − − ν (cos ζ) = Γ(ν + µ + 1) e− Pµ 1 ( i cot ζ) + e Pµ 1 (i cot ζ) 8 sin ζ − 2 − − 2 r π 2µ+1 ν+ 1 2µ+1 ν+ 1 4 πi 2 4 πi 2 = Γ(µ ν) e− Pµ 1 ( i cot ζ) + e Pµ 1 (i cot ζ) − 8 sin ζ − 2 − − 2 (ν+ 1 )πi e− 2 3−2µ ν+ 1 2µ−3 ν+ 1 4 πi 2 4 πi 2 + Γ(µ ν) cos(πν) e Qµ 1 ( i cot ζ) + e Qµ 1 (i cot ζ) . − √2π sin ζ − 2 − − 2 These identities follow by combining (8.1) and (8.7) with Whipple’s formulae (1.13) and (1.14) and then using the connection formulae [4, Eq. 14.9.12] and [4, Eq. 14.9.15]. LARGE-PARAMETER ASYMPTOTICS FOR THE LEGENDRE AND ALLIED FUNCTIONS 29 1 π Let ε be an arbitrary fixed positive number. Since i cot ζ = cosh log cot 2 ζ 2 i , it ± π ± follows from the expansions (1.10) and (1.11) that for ε ζ < 2 , the asymptotic expansions (1.24) and (1.25) hold as µ + , with µ being bounded.≤ Examination of the remainder < →π ∞ = terms in the limiting case ζ 2 (using Theorems 3.1 and 3.2), shows that these inverse factorial → π expansions are actually valid for ε ζ 2 . To extend these expansions to the larger interval ε ζ π ε, we can proceed as follows.≤ ≤ The Ferrers functions satisfy the connection relations ≤ ≤ − Pµ( x) = cos(π(ν + µ))Pµ(x) 2 sin(π(ν + µ))Qµ(x), ν − ν − π ν Qµ( x) = cos(π(ν + µ))Qµ(x) π sin(π(ν + µ))Pµ(x), ν − − ν − 2 ν for 0 x < 1 (see, for example, [4, Eq. 14.9.8] and [4, Eq. 14.9.10]). We employ these relations ≤ π with x = cos ζ = cos(π ζ), 2 < ζ π ε, and use the previously established asymptotic expansions− for the right-hand− sides of the≤ resulting− equalities. After simplification, we find that µ µ π Pν (cos ζ) and Qν (cos ζ), 2 < ζ π ε, possess the same asymptotic expansions as they do for ε ζ π . ≤ − ≤ ≤ 2 11. Numerical examples In this section, we provide some numerical examples to demonstrate the sharpness of our 6 error bounds and the accuracy of our asymptotic approximations. Taking ν = 10, µ = 5 and 4 πiµ µ ξ = 5 , we obtain e− Qν (cosh ξ) 0.00168049. If we take N = 2 terms on the right-hand side (RHS) of (1.11), we obtain the≈ approximation 0.00169164, that is, in Theorem 3.2 we (Q) have RN (ξ, µ, ν) 8642.4139, and the RHS of (3.3) is approximately 10618.854. Note that 8642/10619 0.81,≈ which is close to 1, demonstrating the sharpness of the error bound (3.3). ≈ 6 Figure 2. (LHS of (3.3))/(RHS of (3.3)) for the cases ν = 10 + 5it, µ = 5 , 4 ξ = 5 and N = 1 (black) and N = 5 (grey). 6 4 In Figure2, we take ν = 10 + 5it, µ = 5 and ξ = 5 , and plot (LHS of (3.3))/(RHS of (3.3)) in the cases that N = 1 (black) and N = 5 (grey), respectively. The graph illustrates that our error bound (3.3) is sharp, and that the result in Theorem 3.2 is indeed useful when ν + . iψ 6 4 < → ∞ In Figure3, we take ν = 10e , µ = 5 and ξ = 5 , and plot (LHS of (3.5))/(RHS of (3.5)) in the cases that N = 1 (black) and N = 5 (grey), respectively. The graph illustrates that our error bound (3.5) is reasonably sharp, and that the result in Theorem 3.4 holds when ν + in large sectors of the form arg ν π δ (< π). | | → ∞ |6 | ≤ − In Figure4, we take µ = 5 , ξ = 0.01 + it and N = 1, and plot (LHS of (3.5))/(RHS of (3.5)) in the cases that ν = 10 (black) and ν = 10e2i (grey). These are extreme cases. We observe that when ν = 10, the error bound is still sharp. Even in the case that ν = 10e2i, we see that the bound is realistic near the endpoints ξ = 0, πi. In Figure5, we illustrate how the accuracy± of the large- ν approximations (3.1) and (3.2) µ µ for the associated Legendre functions Pν (z) and Qν (z) diminishes as z approaches 1, but they 30 G. NEMES AND A. B. OLDE DAALHUIS iψ 6 4 Figure 3. (LHS of (3.5))/(RHS of (3.5)) for the cases ν = 10e , µ = 5 , ξ = 5 and N = 1 (black) and N = 5 (grey). 6 Figure 4. (LHS of (3.5))/(RHS of (3.5)) for the cases µ = 5 , ξ = 0.01 + it, N = 1 and ν = 10 (black) and ν = 10e2i (grey). are uniformly valid for z 1 + ε (> 1). Likewise, in Figure6, we illustrate how the accuracy ≥ µ of the large-µ approximation (1.16) for the associated Legendre function Qν (z) decreases as z approaches + , but is uniformly valid for z (1,Z]. Similarly, in Table1, we illustrate how ∞ ∈ µ the accuracy of the large-ν approximations (4.1) and (4.2) for the Ferrers functions Pν (x) and µ Qν (x) diminishes as x approaches 1. 2 100 100 2 1 10 10 4 10 2 10 6 10 3 10 8 10 4 10 10 10 Figure 5. Relative errors of the approximations (3.1) (black) and (3.2) (grey) 6 for the cases ν = 10, µ = 5 , N = M = 2, z = cosh ξ (1, 2) (left) and z = cosh ξ (1, 50) (right). ∈ ∈ 1 LARGE-PARAMETER ASYMPTOTICS FOR THE LEGENDRE AND ALLIED FUNCTIONS 31 3 100 1 10 2 10 3 10 4 10 6 Figure 6. Relative error of the approximation (1.16) for the case ν = 5 , µ = 10, z = coth ξ (1, 10) and taking 2 terms in both sums on the right-hand side of (1.16). ∈ µ µ x = cos ζ Pν (cos ζ) RHS of (4.1) relative error Qν (cos ζ) RHS of (4.2) relative error 0.9 5.58976 5.57356 0.00291 13.1050 13.1067 0.00013 − − 0.95 1.86065 1.82948 0.01704 18.4721 18.4407 0.00170 − − 0.99 8.96386 8.72156 0.02778 24.8288 24.7007 0.00519 − − 0.999 31.5013 32.1480 0.02012 35.7525 30.3175 0.17927 − − − − Table 1. Relative errors of the approximations (4.1) and (4.2) for the case 6 ν = 20, µ = 5 and N = 2. Acknowledgement The authors’ research was supported by a research grant (GRANT11863412/70NANB15H221) from the National Institute of Standards and Technology. The first author was also supported by a Premium Postdoctoral Fellowship of the Hungarian Academy of Sciences. The authors thank the referees for very helpful comments and suggestions for improving the presentation. References [1] H. S. Cohl, T. H. 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An introduc- tion to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, Reprint of the fourth (1927) edition. E-mail address: [email protected] Alfred´ Renyi´ Institute of Mathematics, Realtanoda´ utca 13–15, Budapest H-1053, Hungary E-mail address: [email protected] School of Mathematics and Maxwell Institute for Mathematical Sciences, The University of Edinburgh, Edinburgh EH9 3FD, United Kingdom