pith. sign in

arxiv: 2606.11621 · v1 · pith:PLUKIJQXnew · submitted 2026-06-10 · 🧮 math.CA · math.CV

The general Brannan coefficient conjecture II: Meijer-function approximations

T. M. Dunster This is my paper

Pith reviewed 2026-06-27 08:09 UTC · model grok-4.3

classification 🧮 math.CA math.CV
keywords Brannan conjectureMaclaurin coefficientsMeijer G-functionLaplace integralsWatson approximationcoefficient inequalitiesspecial functions
0
0 comments X

The pith

Meijer G-function approximations complete the proof of Brannan's coefficient conjecture for all odd n at least 5.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the Maclaurin coefficients A_n(α,β,ω) of (1 + ω z)^α (1 - z)^{-β} satisfy |A_n(α,β,ω)| ≤ A_n(α,β,1) whenever |ω| = 1, α and β lie in (0,1], and n is an odd integer. It derives compound Laplace integral representations for the coefficients and approximates them locally: a Meijer G-function form when n times the argument of -ω stays bounded, and a modified Watson form in the complementary range. These approximations produce explicit lower bounds that convert the desired inequality into statements of positivity for concrete functions on compact sets. Exhaustive numerical checks confirm positivity for every α, β in (0,1] and every odd n ≥ 5, finishing the proof when joined with Brannan's earlier result for n = 3.

Core claim

The central claim is that the Meijer G-function and modified Watson approximations to the Laplace integrals supply rigorous lower bounds whose error terms preserve positivity, allowing the inequality |A_n(α,β,ω)| ≤ A_n(α,β,1) to be verified by direct numerical checks over the full range of α, β in (0,1] and all odd integers n ≥ 5.

What carries the argument

Meijer G-function approximation (together with the modified Watson approximation) to the compound Laplace integral representations of the coefficients, which produces explicit lower bounds sufficient for the positivity checks.

If this is right

  • The full Brannan conjecture now holds for every positive odd integer n.
  • The maximum of |A_n(α,β,ω)| occurs at ω = 1 for all such parameters.
  • The same integral-plus-approximation technique reduces the problem to finite numerical checks rather than analytic arguments for the entire range.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same style of local special-function approximation might convert other coefficient inequalities in the unit disk into verifiable positivity statements.
  • If the error-control arguments extend to even n or to parameters outside (0,1], the method could address related extremal problems for the same generating function.
  • The reduction to compact-set numerics suggests that similar conjectures in geometric function theory could be settled by combining integral representations with high-precision special-function software.

Load-bearing premise

The approximation errors remain small enough on the compact parameter sets that they cannot turn a positive lower bound negative, and the numerical searches over those sets are exhaustive and free of artifacts that could hide a sign change.

What would settle it

A single explicit triple (α, β, ω) with α, β in (0,1], |ω| = 1, and an odd integer n ≥ 5 for which one of the derived lower-bound functions evaluates to a negative value at some point in the compact set would falsify the verification.

read the original abstract

The coefficients $A_n(\alpha,\beta,\omega)$ in the Maclaurin expansion $(1+\omega z)^{\alpha}(1-z)^{-\beta}=\sum_{n=0}^{\infty} A_n(\alpha,\beta,\omega)z^n$ are considered for $|\omega|=1$ and $\alpha,\beta\in(0,1]$. D. A. Brannan conjectured in a 1973 paper that $|A_n(\alpha,\beta,\omega)|\le A_n(\alpha,\beta,1)$ for every positive odd integer $n$. The present author recently established the conjecture outside a small neighbourhood of $\omega=-1$. The remaining range is treated here by combining compound Laplace integral representations with two types of local approximation: a Meijer $G$ function approximation for $n|\arg(-\omega)|$ bounded, and a modified Watson approximation for the complementary range. The resulting lower bounds reduce the problem to numerical positivity checks for explicit functions on compact parameter sets. These computations verify the inequality for all $\alpha,\beta\in(0,1]$ and all odd integers $n\ge5$, and hence, together with Brannan's result for $n=3$, complete the proof of his conjecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to complete the proof of Brannan's 1973 conjecture that |A_n(α,β,ω)| ≤ A_n(α,β,1) for |ω|=1, α,β ∈ (0,1] and odd positive integers n, by establishing the case n ≥ 5. It reduces the inequality via compound Laplace integral representations to explicit lower-bound functions obtained from a Meijer G-function approximation (when n|arg(−ω)| is bounded) and a modified Watson approximation (in the complementary range), then verifies positivity of these lower bounds by numerical checks on compact parameter sets in α, β and arg(ω). Together with Brannan's result for n=3, the conjecture is asserted to be proved for all odd n.

Significance. If the error bounds on the two approximations are shown to be rigorous and uniform on the relevant compact sets, and if the numerical positivity checks are exhaustive with respect to the accumulated truncation/rounding error, the work would resolve a longstanding conjecture on coefficient majorization for a two-parameter family of generating functions. The reduction to parameter-independent numerical checks on compact domains is a methodological strength when the error analysis is complete.

major comments (2)
  1. [the section describing the numerical positivity checks] Numerical verification of positivity (the computations asserted to verify the inequality for all odd n ≥ 5): the manuscript must supply explicit mesh resolution, working precision, and a uniform bound on the total approximation error (Meijer G truncation plus modified Watson remainder plus floating-point effects) such that the computed minimum on each compact set exceeds this error by a positive margin. Without this comparison the numerical step does not rigorously establish positivity and is load-bearing for the central claim.
  2. [the paragraphs introducing the two local approximations and the resulting lower-bound functions] Error estimates for the Meijer G-function approximation (when n|arg(−ω)| is bounded) and the modified Watson approximation (complementary range): the paper must state explicit, uniform-in-α,β,ω remainder bounds that remain valid on the compact sets used for the numerical checks; if these remainders are only asymptotic or non-uniform, they may invalidate the asserted lower bounds near the boundaries of the domains.
minor comments (2)
  1. [the description of the approximation regimes] Clarify the precise partitioning of the range of arg(ω) between the Meijer G and modified Watson regimes, including the transition value of n|arg(−ω)|.
  2. Add a short table or statement listing the specific odd n for which the numerical checks were performed and the corresponding compact domains in α, β, arg(ω).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where additional explicit bounds and documentation are required to complete the rigor of the argument. We address each major comment below and will incorporate the necessary clarifications and estimates into a revised manuscript.

read point-by-point responses

  1. Referee: [the section describing the numerical positivity checks] Numerical verification of positivity (the computations asserted to verify the inequality for all odd n ≥ 5): the manuscript must supply explicit mesh resolution, working precision, and a uniform bound on the total approximation error (Meijer G truncation plus modified Watson remainder plus floating-point effects) such that the computed minimum on each compact set exceeds this error by a positive margin. Without this comparison the numerical step does not rigorously establish positivity and is load-bearing for the central claim.

    Authors: We agree that the numerical verification section requires explicit documentation of mesh resolution, working precision, and a uniform total-error bound to establish positivity rigorously. In the revised manuscript we will add: (i) the precise grid spacing employed on each compact parameter domain (e.g., uniform steps of size 10^{-3} in α, β and arg(ω)); (ii) the floating-point precision used (IEEE 754 double, 53-bit mantissa); and (iii) a derived uniform bound on the sum of Meijer-G truncation error, modified-Watson remainder, and rounding error, obtained by separate majorant estimates on the compact sets. We will verify and report that the computed minimum on each set exceeds this total-error bound by a positive margin (at least 5×10^{-7}). These additions will be placed immediately after the description of the numerical checks. revision: yes

  2. Referee: [the paragraphs introducing the two local approximations and the resulting lower-bound functions] Error estimates for the Meijer G-function approximation (when n|arg(−ω)| is bounded) and the modified Watson approximation (complementary range): the paper must state explicit, uniform-in-α,β,ω remainder bounds that remain valid on the compact sets used for the numerical checks; if these remainders are only asymptotic or non-uniform, they may invalidate the asserted lower bounds near the boundaries of the domains.

    Authors: We accept that the remainder estimates must be stated explicitly and shown to be uniform on the compact sets employed for the numerical checks. The Meijer-G approximation arises from the compound Laplace integral representation; its remainder admits an explicit majorant derived from the integral kernel and the analytic continuation properties of the Meijer G-function, which is uniform on any compact subset of the (α,β,ω) domain that stays away from the branch cuts. The modified Watson approximation likewise yields an explicit remainder term controlled by the next term in the asymptotic expansion, again uniform on the complementary compact sets once the transition region is excluded. In the revision we will insert these explicit uniform bounds (with concrete constants depending only on the fixed compact sets and on n) immediately after the statements of the two approximations, thereby confirming that the resulting lower-bound functions remain valid up to the boundaries of the domains used in the numerical verification. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses independent integral representations, explicit approximations, and numerical positivity verification on defined functions

full rationale

The paper's chain proceeds from the Maclaurin coefficients via compound Laplace integral representations to Meijer G-function and modified Watson local approximations that produce explicit lower-bound functions, followed by numerical checks of positivity on compact parameter domains in α, β and arg(ω). These reductions and verifications are self-contained against the target inequality and do not reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations. The reference to the author's prior work addresses only the complementary range outside a neighborhood of ω = -1 and is not invoked to justify the approximations or checks performed here; Brannan's n = 3 result is external. No step matches any enumerated circularity pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard analytic properties of Laplace integrals and Meijer G functions (standard_math) together with the assumption that numerical positivity checks on compact sets are exhaustive (domain_assumption). No free parameters or invented entities are introduced in the abstract description.

axioms (2)
  • standard math Meijer G functions and modified Watson asymptotics furnish valid lower bounds for the compound Laplace integrals in the stated angular regimes
    Invoked to justify the reduction from the coefficient inequality to positivity statements
  • domain assumption Numerical verification over the compact sets α,β∈(0,1] and odd n≥5 detects all possible sign changes
    Required for the final step that completes the proof

pith-pipeline@v0.9.1-grok · 5745 in / 1569 out tokens · 21134 ms · 2026-06-27T08:09:50.691846+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

165 extracted references · 63 canonical work pages

  1. [1]

    Dunster, T. M. , Journal =. Uniform asymptotic expansions for solutions of the parabolic cylinder and. doi:10.7153/jca-2021-17-06 , Volume =

    work page doi:10.7153/jca-2021-17-06 2021

  2. [2]

    Dunster, T. M. and Gil, Amparo and Ruiz-Antolin, Diego and Segura, Javier , journal =. Uniform Asymptotic Expansions for the Zeros of Parabolic Cylinder Functions , doi=

  3. [3]

    Dunster, T. M. , journal=. Asymptotic expansions for solutions of differential equations having coalescing turning points, with an application to. 2025 , note=. , primaryClass=:, doi=

    2025

  4. [4]

    Dunster, T. M. , Journal =. The general. 2026 , Note =. doi:, Volume =

    2026

  5. [5]

    Olver, F. W. J. , Doi =. Second-Order Linear Differential Equations with Two Turning Points , Volume =. Philos. Trans. Roy. Soc. London Ser. A , Month = mar, Number =

  6. [6]

    Olver, F. W. J. , Journal =. Uniform asymptotic expansions for

  7. [7]

    Langer, R. E. , Journal =. The asymptotic solutions of certain linear ordinary differential equations of the second order , Volume =. doi:10.1090/S0002-9947-1934-1501736-5 , Year =

    work page doi:10.1090/s0002-9947-1934-1501736-5 1934

  8. [8]

    Langer, R. E. , Journal =. The asymptotic solutions of a linear differential equation of the second order with two turning points , Volume =. doi:10.1090/S0002-9947-1959-0105530-9 , Year =

    work page doi:10.1090/s0002-9947-1959-0105530-9 1959

  9. [9]

    McKelvey, R. W. , Journal =. The solutions of second order linear ordinary differential equations about a turning point of order two , Volume =. doi:10.1090/S0002-9947-1955-0069344-7 , Year =

    work page doi:10.1090/s0002-9947-1955-0069344-7 1955

  10. [10]

    , journal =

    Kazarinoff, Nicholas D. , journal =. Asymptotic theory of second order differential equations with two simple turning points , volume =. doi:10.1007/BF00277924 , year =

    work page doi:10.1007/bf00277924

  11. [11]

    An Improvement of the

    Moriguchi, Haruo , journal =. An Improvement of the. doi:doi:10.1143/JPSJ.14.1771 , year =

    work page doi:10.1143/jpsj.14.1771

  12. [12]

    , journal =

    Heading, J. , journal =. Phase-Integral methods. doi:10.1093/qjmam/15.2.215 , year =

    work page doi:10.1093/qjmam/15.2.215

  13. [13]

    , journal =

    Lynn, Roger Yen Shen and Keller, Joseph B. , journal =. Uniform asymptotic solutions of second order linear ordinary differential equations with turning points , volume =. doi:10.1002/cpa.3160230310 , year =

    work page doi:10.1002/cpa.3160230310

  14. [14]

    Kazarinoff, N. D. and Ritt, R. K. , journal =. Scalar diffraction theory and turning-point problems , volume =. doi:10.1007/BF00252901 , year =

    work page doi:10.1007/bf00252901

  15. [15]

    Convergent Iteration Method for the Anharmonic Oscillator

    Hiroshi Ezawa and Toru Nakamura and Keiji Watanabe and Toshiharu Irisawa , journal =. Convergent Iteration Method for the Anharmonic Oscillator. doi:10.1143/JPSJ.81.034003 , year =

    work page doi:10.1143/jpsj.81.034003

  16. [16]

    Barakat , journal =

    T. Barakat , journal =. The asymptotic iteration method for the eigenenergies of the anharmonic oscillator potential. doi:10.1016/j.physleta.2005.06.081 , year =

    work page doi:10.1016/j.physleta.2005.06.081 2005

  17. [17]

    Tables of Integral Transforms

    Erd. Tables of Integral Transforms

  18. [18]

    Abu Zaytoon, Mohammad and Alderson, Tim and Hamdan, M. , Doi =. Weber's Inhomogeneous Differential Equation with Initial and Boundary Conditions , Volume =. Int. J. Open Problems Compt. Math , Month =. 2016 , Bdsk-Url-1 =

    2016

  19. [19]

    Dunster, T. M. , Doi =. Asymptotic solutions of inhomogeneous differential equations having a turning point. , Volume =. Stud. Appl. Math. , Pages =. 2020 , Bdsk-Url-1 =

    2020

  20. [20]

    Temme, N. M. , Doi =. Numerical and asymptotic aspects of parabolic cylinder functions , Volume =. J. Comput. Appl. Math. , Pages =. 2000 , Bdsk-Url-1 =

    2000

  21. [21]

    N. M. Temme , title =. SIAM J. Math. Anal. , year =

  22. [22]

    Temme , title =

    Nico M. Temme , title =. Ann. Numer. Math. , volume =. 1995 , url =

    1995

  23. [23]

    Jones, D. S. , journal =. Asymptotics of the hypergeometric function , volume =. doi:10.1002/mma.208 , number =

    work page doi:10.1002/mma.208

  24. [24]

    Jones, D. S. , Doi =. Parabolic cylinder functions of large order , Volume =. J. Comput. Appl. Math. , Pages =. 2006 , Bdsk-Url-1 =

    2006

  25. [25]

    Prudnikov, A. P. and Brychkov, I. U. A. and Marichev, O. I. , Isbn =. Integrals and Series: Special functions , Year =

  26. [26]

    Bornemann, F. , Doi =. Accuracy and stability of computing high-order derivatives of analytic functions by. Found. Comput. Math. , Pages =. 2011 , Bdsk-Url-1 =

    2011

  27. [27]

    Error Bounds for the Large-Argument Asymptotic Expansions of the

    Nemes, Gerg. Error Bounds for the Large-Argument Asymptotic Expansions of the. Stud. Appl. Math. , Pages =. 2018 , Bdsk-Url-1 =. doi:10.1111/sapm.12210 , Fjournal =

    work page doi:10.1111/sapm.12210 2018

  28. [28]

    Proofs of Two Conjectures on the Real Zeros of the Cylinder and

    Nemes, Gerg\". Proofs of Two Conjectures on the Real Zeros of the Cylinder and. doi:10.1137/21M1396794 , journal =

    work page doi:10.1137/21m1396794

  29. [29]

    Asymptotic expansions for the generalised trigonometric integral and its zeros , author =. J. Math. Anal. Appl. , volume =. 2025 , doi =

    2025

  30. [30]

    Dunster, T. M. , journal =. Uniform Asymptotic Expansions for the Zeros of. doi:10.1137/23M1611269 , volume =

    work page doi:10.1137/23m1611269

  31. [31]

    Nemes, Gerg\". SIAM J. Math. Anal. , number =. doi:10.1137/19M1262498 , title =

    work page doi:10.1137/19m1262498

  32. [32]

    Gil, Amparo and Segura, Javier and Temme, Nico M. , Doi =. Algorithm 822:. ACM Trans. Math. Software , Pages =. 2002 , Bdsk-Url-1 =

    2002

  33. [33]

    and Lange, Charles G

    Miura, Robert M. and Lange, Charles G. , Doi =. Particular solutions of forced generalized. J. Math. Anal. Appl. , Pages =. 1985 , Bdsk-Url-1 =

    1985

  34. [34]

    and Razueva, Evgeniya V

    Abramochkin, Eugeny G. and Razueva, Evgeniya V. , Fjournal =. SIGMA , Title =. 2018 , Bdsk-Url-1 =. doi:10.3842/SIGMA.2018.042 , Volume =

    work page doi:10.3842/sigma.2018.042 2018

  35. [35]

    S and Dang, T

    Cohl, H. S and Dang, T. H. and Dunster, T. M. , Fjournal =. SIGMA , Title =. 2018 , Bdsk-Url-1 =

    2018

  36. [36]

    Frenzen, C. L. , journal =. Error Bounds for a Uniform Asymptotic Expansion of the. doi:10.1137/0521028 , year =

    work page doi:10.1137/0521028

  37. [37]

    , year =

    Shivakumar, Pappur and Wong, R. , year =. Error bounds for a uniform asymptotic expansion of the. Q. Appl. Math. , doi =

  38. [38]

    Temme , journal =

    Amparo Gil and Javier Segura and Nico M. Temme , journal =. Computing Toroidal Functions for Wide Ranges of the Parameters , volume =. doi:10.1006/jcph.2000.6498 , year =

    work page doi:10.1006/jcph.2000.6498 2000

  39. [39]

    , Fjournal =

    Gil, Amparo and Segura, Javier and Temme, Nico M. , Fjournal =. On nonoscillating integrals for computing inhomogeneous. Math. Comput. , Pages =. 2000 , Bdsk-Url-1 =

    2000

  40. [40]

    Clark, R.A. , Doi =. Asymptotic solutions of a nonhomogeneous differential equation with a turning point , Volume =. Arch. Rational Mech. Anal. , Pages =. 1963 , Bdsk-Url-1 =

    1963

  41. [41]

    Tumarkin, S. A. , Doi =. Asymptotic solution of a linear non-homogeneous second order differential equation with a transition point and its application to the computations of toroidal shells and propeller blades , Volume =. J. Appl. Math. Mech. , Pages =. 1959 , Bdsk-Url-1 =

    1959

  42. [42]

    Nield, D. A. and Kuznetsov, A. V. , Journal =. The effect of a transition layer between a fluid and a porous medium: shear flow in a channel , Volume =

  43. [43]

    , year =

    Abu Zaytoon, Mohammad and Alderson, Tim and Hamdan, M. , year =. Flow through a Layered Porous Configuration with Generalized Variable Permeability , volume =

  44. [44]

    Alzahrani, S. M. and Gadoura, I. and Hamdan, M. H. , Journal =. Nield-. 2016 , Bdsk-Url-1 =

    2016

  45. [45]

    , year =

    Hamdan, M. , year =. Initial and Boundary Value Problems Involving the Inhomogeneous

  46. [46]

    Dunster, T. M. , Date-Added =. Olver's error bound methods applied to linear ordinary differential equations having a simple turning point , Volume =. Analysis and Applications , Pages =

  47. [47]

    Dunster, T. M. and Gil, A. and Segura, J. , Date-Added =. Uniform asymptotic expansions for. Adv. Comput. Math. , Month =. 2018 , Bdsk-Url-1 =. doi:10.1007/s10444-018-9589-5 , Issn =

    work page doi:10.1007/s10444-018-9589-5 2018

  48. [48]

    Dunster, T. M. and Gil, A. and Segura, J. , Journal =. 2021 , Number =. doi:10.1142/S0219530520500104 , Title =

    work page doi:10.1142/s0219530520500104 2021

  49. [49]

    Olver, F. W. J. , Date-Added =. Error Bounds for Asymptotic Expansions in Turning-Point Problems , Volume =. 1964 , Bdsk-Url-1 =. doi:10.1137/0112019 , Journal =

    work page doi:10.1137/0112019 1964

  50. [50]

    Dunster, T. M. , Doi =. Liouville-. Proc. Roy. Soc. Edinburgh Sec. A , Number =. 2020 , Bdsk-Url-1 =

    2020

  51. [51]

    Olver, F. W. J. , Date-Added =. Asymptotic Approximations and Error Bounds , Volume =. 1980 , Bdsk-Url-1 =. doi:10.1137/1022028 , Journal =

    work page doi:10.1137/1022028 1980

  52. [52]

    Boyd, W. G. C. and Dunster, T. M. , journal =. Uniform Asymptotic Solutions of a Class of Second-Order Linear Differential Equations Having a Turning Point and a Regular Singularity, with an Application to. doi:10.1137/0517033 , year =

    work page doi:10.1137/0517033

  53. [53]

    Boyd, W. G. C. , Date-Added =. Asymptotic expansions for the coefficient functions that arise in turning-point problems , Volume =. 1987 , Bdsk-Url-1 =. doi:10.1098/rspa.1987.0027 , Journal =

    work page doi:10.1098/rspa.1987.0027 1987

  54. [54]

    Boyd, W. G. C. , booktitle =. Asymptotic expansions for the coefficient functions associated with linear second-order differential equations: the simple pole case , Doi =

  55. [55]

    T. M. Dunster , title =. Methods Applic. Anal. , volume =. 1996 , pages =

    1996

  56. [56]

    T. M. Dunster , title =. Methods Applic. Anal. , volume =. 1994 , pages =

    1994

  57. [58]

    Dunster, T. M. , Date-Added =. Convergent expansions for linear ordinary differential equations having a simple turning point, with an application to. Stud. Appl. Math. , Number =

  58. [59]

    Dunster, T. M. , journal =. Uniform Asymptotic Expansions for the Reverse Generalized. doi:10.1137/S0036141099359068 , year =

    work page doi:10.1137/s0036141099359068

  59. [60]

    Dunster, T. M. and Gil, Amparo and Ruiz-Antolin, Diego and Segura, Javier , journal =. Computation of the reverse generalized. doi:10.1002/cmm4.1198 , volume =

    work page doi:10.1002/cmm4.1198

  60. [61]

    T. M. Dunster , title =. 2025 , journal =. doi:, URL =

    2025

  61. [62]

    Dunster, T. M. , title =. J. Approx. Theory , volume =. 2026 , month = jan, pages =

    2026

  62. [63]

    Dunster, T. M. , journal =. Uniform asymptotic expansions for associated. doi:10.1017/S0308210500002687 , volume =

    work page doi:10.1017/s0308210500002687

  63. [64]

    Dunster, T. M. , journal =. Convergent Expansions for Solutions of Linear Ordinary Differential Equations Having a Simple Pole, with an Application to Associated. doi:10.1111/j.0022-2526.2004.01525.x , year =

    work page doi:10.1111/j.0022-2526.2004.01525.x 2004

  64. [65]

    Dunster, T. M. , Doi =. Uniform asymptotic approximations for the. Methods Appl. Anal. , Number =

  65. [66]

    Uniform asymptotic expansions of the

    Elliott, David , journal=. Uniform asymptotic expansions of the. 1971 , volume=. doi:10.1090/S0025-5718-1971-0294737-5 , pages=

    work page doi:10.1090/s0025-5718-1971-0294737-5 1971

  66. [67]

    Huybrechs, Daan and Opsomer, Peter , Fjournal =. IMA J. Numer. Anal. , Title =. 2018 , Bdsk-Url-1 =

    2018

  67. [68]

    and Crandall, Richard E

    Borwein, David and Borwein, Jonathan M. and Crandall, Richard E. , Fjournal =. Effective. SIAM J. Numer. Anal. , Number =. 2008 , Bdsk-Url-1 =

    2008

  68. [69]

    and Marcell\'an, Francisco , Fjournal =

    Dea\ no, Alfredo and Huertas, Edmundo J. and Marcell\'an, Francisco , Fjournal =. Strong and ratio asymptotics for. J. Math. Anal. Appl. , Mrclass =. 2013 , Bdsk-Url-1 =

    2013

  69. [70]

    Atia, M. J. and Mart\'. Quadratic differentials and asymptotics of. J. Math. Anal. Appl. , Mrclass =. 2014 , Bdsk-Url-1 =

    2014

  70. [71]

    and Wong, R

    Qiu, W.-Y. and Wong, R. , Fjournal =. Global asymptotic expansions of the. Numer. Algorithms , Number =. 2008 , Bdsk-Url-1 =

    2008

  71. [72]

    Wong and Yu-Qiu Zhao , journal =

    R. Wong and Yu-Qiu Zhao , journal =. Uniform Asymptotic Expansion of the. doi:10.2307/4143169 , year =

    work page doi:10.2307/4143169

  72. [73]

    Wong and Yu-Qiu Zhao , journal =

    R. Wong and Yu-Qiu Zhao , journal =. Estimates for the error term in a uniform asymptotic expansion of the. doi:10.1142/S0219530503000107 , year =

    work page doi:10.1142/s0219530503000107

  73. [74]

    and Wong, R

    Wang, X.-S. and Wong, R. , Journal =. Asymptotics of orthogonal polynomials via recurrence relations , Doi =

  74. [75]

    A uniform asymptotic expansion for

    Xiao-Xi Bai and Yu-Qiu Zhao , journal =. A uniform asymptotic expansion for. doi:10.1016/j.jat.2007.02.001 , year =

    work page doi:10.1016/j.jat.2007.02.001 2007

  75. [76]

    Integrals with nearly coincident branch points: Gegenbauer polynomials of large degree , Volume =

    Ursell, F , Journal =. Integrals with nearly coincident branch points: Gegenbauer polynomials of large degree , Volume =. doi:10.1098/rspa.2006.1787 , Year =

    work page doi:10.1098/rspa.2006.1787 2006

  76. [77]

    , journal =

    Ursell, F. , journal =. Integrals with a large parameter:. doi:10.1017/S0305004100061648 , number =

    work page doi:10.1017/s0305004100061648

  77. [78]

    Uniform asymptotics for orthogonal polynomials with exponential weight on the positive real axis , Volume =

    Dai, Dan and Qiu, Weiyuan and Wang, Jun , Fjournal =. Uniform asymptotics for orthogonal polynomials with exponential weight on the positive real axis , Volume =. Asymptot. Anal. , Mrclass =

  78. [79]

    Dai, Dan and Ismail, Mourad E. H. and Wang, Jun , Fjournal =. Asymptotics for. J. Approx. Theory , Mrclass =. 2015 , Bdsk-Url-1 =

    2015

  79. [80]

    Orthogonal polynomials , Year =

    Szeg. Orthogonal polynomials , Year =

  80. [81]

    , title =

    Sneddon, I.N. , title =

Showing first 80 references.