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Computing ${}_{q+1}{F}_q$ via rational conformal maps

John Lopez, Richard Mikael Slevinsky

arXiv:2610.35577v1math.NAcs.NA

TL;DR

Möbius transformations leave challenging fixed points for Gauss-function computation. The paper uses rational conformal maps to form transformed Maclaurin series for generalized hypergeometric functions, reporting faster convergence than mapped Chebyshev series and optimal complexity.

  • Problem

    Möbius transformations do not bring e±iπ/3 closer to the origin, making numerical computation challenging near those points.

    Source: §1 · p. 1

  • Method

    The paper applies rational conformal maps to transformed Maclaurin series and exploits the factored differential equation to compute coefficients in O(qmn) flops and O(qm) storage.

    Source: §1 · p. 2, §2.1 · p. 7

  • Results

    Schwarz–Christoffel mapping converges at the same exponential rate as approximate mapped Chebyshev series, while rational maps with m = 3 and 4 converge faster and attain O(qmn) flops and O(qm) storage.

    Source: §6 · p. 15

  • Takeaways & Limitations

    The approach provides pointwise convergence for q+1Fq throughout C \ [1, ∞), including regions challenging for the stated Möbius transformations.

    Source: §1 · pp. 1–2

    The modulus-minimizing γopt map supports global exponential convergence only through m = 3, while r(z; −1, m) does so through m = 4.

    Source: §2 · p. 4

Abstract

from arXiv · show

It is well-known that certain M\"obius (linear fractional) transformations provide analytic continuation of the Gauss hypergeometric function. These transformations work by exchanging the roles of the three regular singular points and $0$, $1$, and $\infty$ of the Gauss hypergeometric differential equation. Not one of these transformations, however, brings the points $e^{\pm{\rm i}\pi/3}$ closer to the origin, and convergence in a neighbourhood of these is also troublesome. In this work, we produce a Maclaurin series in optimal complexity after rational conformal mapping that induces pointwise convergence to any ${}_{q+1}F_q$ throughout the cut complex plane $\mathbb{C}\setminus[1,\infty)$. We show that for the optimal parameter values this series converges faster than an approximate mapped Chebyshev series obtained by the ultraspherical spectral method.

1 Introduction

Möbius transformations for the Gauss function leave e±iπ/3 fixed, complicating computation there; the paper introduces a rational conformal-map strategy for generalized hypergeometric functions on the cut plane.

1.2 Analytic continuation by rational transformations

  • Prior continuation approaches include the Meijer G-function, Bühring, Nørlund, and López-group formulas, with convergence regions or expansions varying by method.The introduction also notes that most numerical algorithms focus on special cases with p + q ≤ 3.Source: §1 · p. 1, §1.2 · pp. 2–3

1 Introduction

  • The paper targets difficult computation near the Möbius-fixed points e±iπ/3 with a rational conformal map whose transformed Maclaurin series converges exponentially throughout C \ [1, ∞).The proposed factored-equation approach gives O(qmn) flops, and the paper also generalizes the Euler transformation for positively balanced functions near unity.Source: §1 · pp. 1–2

1.1 The polynomial conformal map of Jacobs, Piret, and Fornberg

  • Jacobs, Piret, and Fornberg use a polynomial conformal map that yields an (m+1)-term recurrence costing O(mn), with m = 4 reported as the best accuracy-efficiency balance.Its transformed series can converge beyond the unit disk, although the convergence set is bounded.Source: §1.1 · p. 2

1.3 The rational conformal maps of Skorokhodov

  • Skorokhodov studied linear- and quadratic-fractional transformations; the quadratic-fractional map is a Schwarz–Christoffel map from the unit disk onto C \ [1, ∞).An optimal parameter minimizes |w(z)| and gives convergence throughout the cut plane, although each fixed parameter's convergence region is only a half-plane.Source: §1.3 · p. 3

2 Our rational conformal map

The section develops a rational conformal map for Maclaurin-series continuation and transforms the generalized hypergeometric differential equation into a factored form. A compound recurrence preserves that factorization to compute coefficients efficiently, subject to restrictions imposed by the transformed equation’s singularities.

2 Our rational conformal map

  • The rational map r(z; γ, m) is used as the argument of a Maclaurin series and draws on polynomial and fractional maps introduced in earlier work.The paper identifies its antecedents as Jacobs, Piret, and Fornberg and Skorokhodov.Source: §2 · p. 4
  • For m = 2, |r(z; γ, 2)| ≤ 1 throughout the complex plane, with equality only on the cut, and each m has an optimal parameter maximizing the denominator magnitude.The map can also be expressed as a difference of powers to avoid subtractive cancellation near the origin.Source: §2 · p. 4
  • The rational map’s modulus is uniformly bounded over the complex plane, while both γ and m strongly affect its value.Figure 1 compares the algebraic map and rational maps with γ = −1 and γopt.Source: §2 · p. 4, Figure 1 · p. 5
  • The passage identifies r(z; −1, 4) as the conformal map with the smallest modulus; r(z; −1, m) provides global exponential convergence up to m = 4, while r(z; γopt, m) may do so only up to m = 3.It states |r(z; −1, 4)| ≤ |r(z; γopt, 3)| and notes that using γopt may make a Maclaurin series in the transformed variable represent an analytic function as a series of non-analytic ones.Source: §2 · p. 4

2.1 The transformed generalized hypergeometric differential equation

  • The transformed equation’s singular points restrict the rational-algebraic transformation to 1 ≤ m ≤ 4; for m > 4, roots of p(w; m) enter the unit disk.The roots are purely imaginary, and the singular points are regular.Source: §2.1 · p. 5
  • The transformed differential equation is factored into first-order polynomial-coefficient operators, whose discretization matrices combine to give lower bandwidth O(qm).The resulting truncated system can be solved by forward substitution, but explicitly assembling the matrices is more costly than the compound recurrence.Source: §2.1 · pp. 6–7

2.2 Compound recurrence relation

  • The compound recurrence computes the first n coefficients in O(qmn) flops and O(qm) storage, avoiding O(q^2m^2n) matrix-assembly work.It applies the factors successively rather than expanding them into one long recurrence.Source: §2.1 · p. 7, §2.2 · p. 10

3 An approximate Chebyshev series solution to the generalized hypergeometric differential equation

The section compares an approximate Chebyshev solution of the generalized hypergeometric differential equation with mapped Maclaurin-series approaches. Its convergence rate matches that of the Schwarz–Christoffel-mapped Maclaurin series, while the proposed recurrence is substantially faster to compute.

3 An approximate Chebyshev series solution to the generalized hypergeometric differential equation

  • The mapped Chebyshev approach builds on analytic continuation for the Gauss function, for which earlier whole-plane continuation required substantial storage and precomputation.The cited procedure stitches together real-line series and solves a Laplace problem on ellipses.Source: §3 · p. 10
  • The method approximates the solution at the right endpoint of a mapped interval by a Chebyshev series for the differential equation.The interval is linearly mapped from [−1, 1] to [0, z], and the analytic function on this segment supports exponential Chebyshev convergence.Source: §3 · pp. 10–11
  • The ultraspherical spectral method computes Chebyshev coefficients through an infinite-dimensional banded-plus-rank-one linear system, and its approximation is within a constant factor of the true truncated series.The comparison uses a Sobolev norm.Source: §3 · p. 11
  • The Chebyshev series converges at the same exponential rate as the Maclaurin series after Schwarz–Christoffel mapping with prevertices at ±1.The comparison follows from matching the Bernstein ellipse parameter to the mapped-series convergence parameter.Source: §3 · p. 11
  • Near the origin in w, the Chebyshev series in w converges at most four times faster than the original-variable Chebyshev expansion or the rationally mapped Maclaurin series.The estimated rate is smallest as w approaches the singular points ±1 and largest near the origin.Source: §3 · p. 12
  • The Chebyshev differential-equation solution is a few orders of magnitude slower than the compound recurrence, although its exponential convergence rate equals that of the quadratic-fractional transformation.The authors also report that rational conformal maps with m = 3 and 4 can outperform the Chebyshev method; the Chebyshev system requires O(qmn) storage.Source: §3 · p. 12

4 A generalized Euler transformation

A generalized Euler transformation factors parameter-dependent algebraic behavior from the solution before applying the rational conformal map. Its transformed equation retains the factorized recurrence, so the coefficients can be computed with the same O(qmn) procedure while removing the associated algebraic singular behavior.

4 A generalized Euler transformation

  • Factoring the singular part can change Maclaurin coefficients from large algebraic growth to algebraic decay when the Gauss function is singular at z = 1.When the function instead tends to zero there, origin-centered basis functions lead to subtractive cancellation near z = 1.Source: §4 · p. 13
  • The transformation sets u(z) = (1 − z)^δv(z), with δ measuring the imbalance in parameters, then solves the resulting differential equation using the same conformal map.The equation is multiplied by a polynomial factor to obtain polynomial coefficients and written in factored operator form.Source: §4 · p. 13
  • The factorized recurrence avoids expanding its operator products into one long recurrence and uses the same auxiliary-sequence procedure as the earlier method.The transformed equation’s operators are given in factored form.Source: §4 · pp. 13–14
  • The generalized Euler transformation preserves the factorized recurrence for computing the transformed solution’s coefficients and retains O(qmn) complexity while removing the algebraic singular behavior associated with (1 − z)^δ.Once v is computed, the original solution is recovered using the transformation relation.Source: §4 · p. 14

5 Numerical results

The experiments compare convergence rates and computational cost across conformal maps and the ultraspherical spectral method. They report faster convergence for the rational map than for same-degree polynomial maps, with m = 4 optimal, and demonstrate O(qmn) computation.

5 Numerical results

  • Doubling the number of denominator parameters approximately doubles computation time, consistent with O(qmn) complexity; subnormal numbers cause a timing bump near n = 300.The timing experiment averages 100 trials for the rational map with m = 4; the bump does not affect asymptotic linear complexity.Source: §5 · p. 14
  • Figures 3–8 show 2F1((1, 7/2), (−7/4, ), z) approximations on a square complex-plane subset, with phase portraits above relative-error modulus contours, comparing left-panel 30-term truncations against right-panel stopping-criterion results.Source: §5 · pp. 14–15
  • The rational conformal map converges faster than same-degree polynomial maps, while the Schwarz–Christoffel map matches the approximate Chebyshev series’ rate; m = 4 is optimal.Figure 2 compares the rates for 2F1 at z = e^(iπ/3).Source: §5 · p. 14, Figure 2 · p. 15
  • For the polynomial map with m = 4, the Maclaurin series converges beyond the theoretical extremal level sets, while the experiments characterize rational transformations as more powerful analytic-continuation tools.The comparison figures vary the polynomial and rational map degrees from m = 2 to m = 4.Source: §5 · p. 15

6 Conclusion & future directions

The paper concludes that rational conformal maps yield faster convergence than Schwarz–Christoffel mapping for m = 3 and 4, with optimal recurrence costs. It identifies multiply-cut domains and formally divergent hypergeometric series as directions for further work.

6 Conclusion & future directions

  • Schwarz–Christoffel mapping matches the approximate mapped Chebyshev series’ exponential rate, while rational maps are faster for m = 3 and 4 and achieve O(qmn) flops with O(qm) storage.These costs improve on the product approach’s O(q^2m^2n) flops and O(qmn) storage.Source: §6 · p. 15
  • For multiply-cut planes, extensions to Fuchsian equations and Appell or Heun functions are proposed, but computing the rational maps remains an open challenge.Exact integration avoids quadrature for the Schwarz–Christoffel parameter problem, while finite slit vertices introduce logarithms that densify the transformed-series recurrences.Source: §6 · p. 15
  • Figures 3–8 illustrate polynomial-map and rational-map approximations of the same 2F1 function for m = 2, 3, and 4.The polynomial-map cases are shown in Figures 3–5 and rational-map cases in Figures 6–8.Source: Figure 3 · p. 16, Figure 4 · p. 17, Figure 5 · p. 18
  • For formally divergent pFq series with p > q + 1, the proposed direction maps the cut plane to the unit disk but places the initial condition at its boundary, leaving a numerical approach for future study.The origin is an irregular singular point, manifesting as an essential but removable singularity.Source: §6 · p. 22

A Proof of theorem 2.1

The proof of Theorem 2.1 establishes a bound for the rational map and reduces the key comparison to inequalities over a specified angular range. It handles the m = 1 case directly and treats m ≥ 2 through auxiliary inequalities.

A Proof of theorem 2.1

  • The proof begins with |r(z; −1, 2)| ≤ 1 and invokes Skorokhodov’s optimization argument to establish the next bound.The supplied proof passage does not include the displayed bound itself.Source: A Proof of theorem 2.1 · p. 23
  • For the main inequality, it suffices to consider n = m + 1, with 0 ≤ b ≤ π, and the proof reduces it to two separate inequalities.One resulting inequality is tan(b/(2m + 2)) ≤ sin(b/2m).Source: A Proof of theorem 2.1 · pp. 23–24
  • The m = 1 case reduces to tan(b/4) ≤ sin(b/2), equivalent to 2 cos(b/4)^2 ≥ 1.The proof states this holds.Source: A Proof of theorem 2.1 · p. 24
  • For m ≥ 2, the proof introduces x = b/2m and y = rx with r = m/(m + 1), and establishes the inequality for b^2 ≤ 96/7, a range extending beyond π^2.The argument also invokes a comparison involving sinh^2(a/2m) and cos(b/n).Source: A Proof of theorem 2.1 · pp. 24–25

B On the ultraspherical spectral method

The appendix sets notation and collects matrix relations used by the ultraspherical spectral method. These relations represent multiplication and differentiation for Chebyshev and ultraspherical polynomial bases.

B On the ultraspherical spectral method

  • The ultraspherical spectral method represents differentiation, multiplication by x, and basis conversion as sparse operators on coefficient sequences.The appendix states standard recurrence and differentiation relations in matrix form to fix notation and indexing conventions.Source: B On the ultraspherical spectral method · p. 25
  • Second-kind Chebyshev polynomials are identified with ultraspherical polynomials of order 1.The appendix denotes the ultraspherical polynomial quasimatrix for order λ > −1/2, λ ≠ 0.Source: B On the ultraspherical spectral method · p. 25
  • The appendix lists matrix relations for multiplication by x in first-kind Chebyshev and ultraspherical polynomial bases.These are identified as Propositions B.1 and B.2.Source: B On the ultraspherical spectral method · p. 25
  • The appendix also lists conversion relations for first-kind Chebyshev and ultraspherical polynomials, followed by differentiation relations for both bases.These are identified as Propositions B.3–B.6.Source: B On the ultraspherical spectral method · pp. 25–26
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