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arxiv: 1907.04597 · v1 · pith:TFMNOJADnew · submitted 2019-07-10 · 🧮 math.CA

The Fox-Wright function near the singularity and branch cut

Dmitrii Karp , Elena Prilepkina This is my paper

Pith reviewed 2026-05-24 23:33 UTC · model grok-4.3

classification 🧮 math.CA
keywords Fox-Wright functiongeneralized hypergeometric seriessingularitybranch cutanalytic continuationrecursive coefficientsconvergent expansionfractional calculus
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The pith

The Fox-Wright function admits a convergent expansion with recursive coefficients near its positive singularity when upper and lower scaling sums match, plus explicit jump and average values across the branch cut.

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

The Fox-Wright function extends the generalized hypergeometric series by inserting arbitrary positive scaling factors into the gamma-function arguments of each term. When the sum of the upper scaling factors equals the sum of the lower ones, the defining power series converges inside a disk of finite positive radius and admits analytic continuation to the cut plane. The paper establishes that, under further restrictions on the parameters, a convergent series expansion around the positive boundary point of that disk fully characterizes the local singular behavior, with all coefficients obtainable recursively. It additionally supplies explicit formulas for the jump discontinuity and the average value of the function when the branch cut is crossed.

Core claim

The Fox-Wright function can be extended to a holomorphic function in the complex plane cut along a ray from the positive point on the boundary of the disk of convergence to infinity. Under certain restrictions a convergent expansion with recursively computed coefficients completely characterizes the behavior near this positive singular point. The jump and the average value of the Fox-Wright function on the two banks of the branch cut are computed explicitly.

What carries the argument

Convergent expansion with recursively computed coefficients that describes the local behavior near the positive singular point on the circle of convergence.

If this is right

  • The singular behavior near the positive boundary point is completely described by the recursive series.
  • The function values on opposite sides of the branch cut differ by the computed jump.
  • The average value supplies a canonical principal-value interpretation when the cut is crossed.
  • The recursive coefficients allow direct numerical computation of the function arbitrarily close to the singularity.

Where Pith is reading between the lines

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

  • The expansion supplies a practical tool for evaluating the function in regions relevant to fractional differential equations.
  • Analogous recursive expansions may exist for other families of scaled hypergeometric series.
  • The explicit jump formula permits residue calculations or contour deformations that avoid the cut.

Load-bearing premise

The sums of the scaling factors in the top and bottom parameters are equal together with additional restrictions on the parameters that are required for the convergent expansion to hold.

What would settle it

Numerical evaluation of the Fox-Wright function at a sequence of points approaching the positive singular point from inside the disk, compared against partial sums of the proposed recursive expansion at those same points.

read the original abstract

The Fox-Wright function is a further extension of the generalized hypergeometric function obtained by introducing arbitrary positive scaling factors into the arguments of the gamma functions in the summand. Its importance comes mostly from its role in fractional calculus although other interesting applications also exist. If the sums of the scaling factors in the top and bottom parameters are equal, the series defining the Fox-Wright function has a finite non-zero radius of convergence. It was demonstrated by Braaksma in 1964 that the Fox-Wright function can then be extended to a holomorphic function in the complex plane cut along a ray from the positive point on the boundary of the disk of convergence to the point at infinity. In this paper we study the behavior of the Fox-Wright function in the neighborhood of this positive singular point. Under certain restrictions we give a convergent expansion with recursively computed coefficients completely characterizing this behavior. We further compute the jump and the average value of the Fox-Wright function on the banks of the branch cut.

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

1 major / 3 minor

Summary. The paper analyzes the Fox-Wright function _pΨ_q when the sums of the scaling factors of the gamma functions in the numerator and denominator are equal, yielding a series with finite nonzero radius of convergence. Building on Braaksma's 1964 global continuation, it derives under explicit non-resonance and pole-avoidance restrictions a convergent local expansion near the positive real singularity on the circle of convergence, with coefficients obtained recursively by direct substitution of a suitable ansatz into the defining series. It also computes the jump across the branch cut and the average value on the two banks.

Significance. If the local expansion and jump formulas hold, the work supplies an explicit, recursively computable characterization of the singular behavior that is directly usable in fractional-calculus applications. The recursive construction and the explicit difference of limiting values constitute concrete, verifiable additions to the existing global theory; the absence of fitted parameters or hidden constants in the recursion is a methodological strength.

major comments (1)
  1. [§3] §3 (local expansion): the recursion for the coefficients is obtained by formal substitution, but the manuscript must verify that the resulting power series has a positive radius of convergence under the stated non-resonance conditions; without an explicit estimate or comparison test this step remains formal and load-bearing for the central claim.
minor comments (3)
  1. [Abstract, §1] Abstract and §1: the phrase 'positive point on the boundary of the disk of convergence' should be replaced by an explicit reference to the point z = R (R the radius) to avoid ambiguity.
  2. [§2] §2: the non-resonance conditions on the scaling factors are listed but their precise relation to the avoidance of pole alignments in the local ansatz should be cross-referenced to the convergence argument in §3.
  3. Notation: the symbol used for the Fox-Wright function should be consistently typeset as _pΨ_q throughout, including in displayed equations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comment on the local expansion. We address the single major point below.

read point-by-point responses

  1. Referee: [§3] §3 (local expansion): the recursion for the coefficients is obtained by formal substitution, but the manuscript must verify that the resulting power series has a positive radius of convergence under the stated non-resonance conditions; without an explicit estimate or comparison test this step remains formal and load-bearing for the central claim.

    Authors: We agree that the argument for convergence of the recursively defined series is currently formal and requires an explicit verification. In the revised version we will insert a direct comparison or ratio test (leveraging the non-resonance hypothesis to control the growth of the recursive coefficients) that establishes a strictly positive radius of convergence for the local expansion. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation begins from the defining series of the Fox-Wright function (with equal top/bottom scaling sums yielding finite radius) and invokes Braaksma's 1964 global continuation as an external result. The local expansion near the positive singularity is obtained by substituting a power-series ansatz directly into that series and equating coefficients to produce a recursion; the branch-cut jump is the difference of the two one-sided limits. Both steps are explicit constructions from the input series and the cited continuation theorem; no parameter is fitted to data and then relabeled a prediction, no self-citation supplies a uniqueness theorem or ansatz, and the cited prior work is independent (1964, non-overlapping authors). The manuscript therefore remains self-contained against external analytic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified from the given text.

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Works this paper leans on

58 extracted references · 58 canonical work pages · 2 internal anchors

  1. [1]

    Alzer, On some inequalities for the gamma and psi functions, Ma thematics of Computation, Volume 66, Number 217 (1997), 373–389

    H. Alzer, On some inequalities for the gamma and psi functions, Ma thematics of Computation, Volume 66, Number 217 (1997), 373–389. 13

    work page 1997

  2. [2]

    Anderson, R

    G.D. Anderson, R. W.Barnard, K.C. Ricards, M.K. Vamanamurthy, and M. Vuorinen, Inequalities for Zero-Balanced Hypergeometric Functions, Tran. Amer. Math. So c. 347(5)(1993) 1713–1723. 17

    work page 1993

  3. [3]

    Andrews, R

    G.E. Andrews, R. Askey and R. Roy, Special functions, Cambridg e University Press, 1999. 9

    work page 1999

  4. [4]

    Benedetto, W

    J.J. Benedetto, W. Czaja, Integration and Modern Analysis, Bir kh¨ auser, 2009. 5, 9

    work page 2009

  5. [5]

    Braaksma, Asymptotic expansions and analytic continuatio n for a class of Barnes integrals, Composito Math

    B.L.J. Braaksma, Asymptotic expansions and analytic continuatio n for a class of Barnes integrals, Composito Math. 15, 3(1962–64), 239–341. http://eudml.org/doc/88877 2, 3, 8 17

    work page 1962

  6. [6]

    B¨ uhring, The behavior at unit argument of the hypergeomet ric function 3F2, SIAM J

    W. B¨ uhring, The behavior at unit argument of the hypergeomet ric function 3F2, SIAM J. Math. Anal., Vol. 18, 5(1987), 1227–1234. http://dx.doi.org/10.1137/0518089 4

    work page doi:10.1137/0518089 1987

  7. [7]

    B¨ uhring, Generalized hypergeometric functions at unit argu ment, Proceedings of American Math- ematical Society, Vol.114, 1(1992), 145–153

    W. B¨ uhring, Generalized hypergeometric functions at unit argu ment, Proceedings of American Math- ematical Society, Vol.114, 1(1992), 145–153. http://dx.doi.org/10.1090/S0002-9939-19 92-1068116-2 4, 11

    work page doi:10.1090/s0002-9939-19 1992

  8. [8]

    B¨ uhring and H.M

    W. B¨ uhring and H.M. Srivastava, Analytic continuation of the gen eralized hypergeometric series near unit argument with emphasis on the zero-balanced series, pages 17 –35 in: Approximation Theory and Applications, Hadronic Press, 1998. 4

    work page 1998

  9. [9]

    Carlitz, Weighted Stirling numbers of the first and second kind – II, The Fibonacci Quarterly 18(1980), 242–257

    L. Carlitz, Weighted Stirling numbers of the first and second kind – II, The Fibonacci Quarterly 18(1980), 242–257. 6

    work page 1980

  10. [10]

    Charalambides, Enumerative Combinatorics, Chapman and H all/CRC, 2002

    C.A. Charalambides, Enumerative Combinatorics, Chapman and H all/CRC, 2002. 6, 7

    work page 2002

  11. [11]

    Collins, The role of Bell polynomials in integration, Journal of C omputational and Applied Math- ematics 131 (2001), 195–222

    C.B. Collins, The role of Bell polynomials in integration, Journal of C omputational and Applied Math- ematics 131 (2001), 195–222. 7

    work page 2001

  12. [12]

    Erd´ elyi, W

    A. Erd´ elyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Bate man Manuscript Project: Higher Transcendental Functions, Vol. I, McGraw-Hill, New York-Toront o-London, 1953. 2, 12

    work page 1953

  13. [13]

    Flajolet, A

    Ph. Flajolet, A. Odlyzko, Singularity analysis of generating func tions, SIAM J. Disc. Math., 3, 2(1990), 216–240. 8

    work page 1990

  14. [14]

    Flajolet and R

    Ph. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridg e University Press, 2009. 8

    work page 2009

  15. [15]

    Fox, The asymptotic expansion of generalized hypergeomet ric functions, Proc

    C. Fox, The asymptotic expansion of generalized hypergeomet ric functions, Proc. London Math. Soc. (2), 27 (1928), 389–400. 2

    work page 1928

  16. [16]

    Chu and X

    W. Chu and X. Wang, Summation formulae on Fox-Wright Ψ-funct ions, Integral Transforms and Spe- cial Functions, 19:8(2008), 545–561. 2

    work page 2008

  17. [17]

    Gorenflo, Y

    R. Gorenflo, Y. Luchko, F. Mainardi, Analytical properties and applications of the Wright function, Fractional Calculus and Applied Analysis, Vol.2 No.4 (1999), 383–414. 2

    work page 1999

  18. [18]

    A note on Fox's H function in the light of Braaksma's results

    D.B. Karp, A note on Fox’s H function in the light of Braaksma’s res ults, submitted to ”Special Func- tions and Analysis of Differential Equations” (edited by P. Agarwal, R .V. Agarwal and M. Ruzhansky), Taylor & Francis, 2019. Preprint arXiv:1904.10651 5, 7, 8, 11, 16

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  19. [19]

    Karp, Representations and inequalities for generalized hype rgeometric functions, Journal of Math- ematical Sciences, Volume 207, Issue 6(2015), 885–897

    D. Karp, Representations and inequalities for generalized hype rgeometric functions, Journal of Math- ematical Sciences, Volume 207, Issue 6(2015), 885–897. 2

    work page 2015

  20. [20]

    Karp and J.L

    D. Karp and J.L. L´ opez, Representations of hypergeometric functions for arbitrary values of the pa- rameters and their use, Journal of Approximation Theory, Volume 218(2017), 42–70. 2

    work page 2017

  21. [21]

    Karp and E

    D. Karp and E. Prilepkina, Hypergeometric functions as genera lized Stieltjes transforms, Journal of Mathematical Analysis and Applications, Volume 393, Issue 2 (2012) , 348–359. 2

    work page 2012

  22. [22]

    Karp and E

    D. Karp and E. Prilepkina, Completely monotonic gamma ratio and in finitely divisible H function of Fox, Computational Methods and Function Theory, 16:1(2016), 1 35–153. 2, 4, 5

    work page 2016

  23. [23]

    Karp and E

    D. Karp and E. Prilepkina, Some new facts around the delta neut ral H function of Fox, Computational Methods and Function Theory, June 2017, Volume 17, Issue 2, 343 –367. 2, 5, 7

    work page 2017

  24. [24]

    Karp and E

    D. Karp and E. Prilepkina, Applications of the Stieltjes and Laplac e transform representations of the hypergeometric functions, Integral Transforms and Special Fu nctions, volume 28, no.10 (2017), 710–

    work page 2017

  25. [25]

    Karp and E.G

    D.B. Karp and E.G. Prilepkina, An inverse factorial series for a ge neral gamma ratio and related prop- erties of the Nørlund-Bernoulli polynomials, Journal of Mathematic al Sciences, Vol. 234, No. 5 (2018), 680–696. 10

    work page 2018

  26. [26]

    Kilbas, Fractional calculus of the generalized Wright functio n, Fractional Calculus and Applied Analysis 8(2005), 114–126

    A.A. Kilbas, Fractional calculus of the generalized Wright functio n, Fractional Calculus and Applied Analysis 8(2005), 114–126. 2 18

    work page 2005

  27. [27]

    Kilbas, M

    A.A. Kilbas, M. Saigo, J.J. Trujillo, On the generalized Wright funct ion, Fractional Calculus and Ap- plied Analysis 5(2002), 437–460. 2, 4

    work page 2002

  28. [28]

    Kilbas, H.M

    A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equa- tions, North-Holland Mathematics Studies 204, Elsevier, 2006. 2

    work page 2006

  29. [29]

    Kilbas, R.K

    A.A. Kilbas, R.K. Saxena, M. Saigo and J.J. Trujillo, Generalized Wrig ht function as the H-function, in A.A. Kilbas and S.V. Rogosin (Eds.), Analytic Methods of Analysis and D ifferential Equations, AMADE 2003, Camb. Sci. Publ., Cambridge, 2006, 115–132. 2, 4

    work page 2003

  30. [30]

    Kilbas, M

    A.A. Kilbas, M. Saigo, H-transforms and applications, Analytical Methods and Special Fun ctions, Vol- ume 9, Chapman & Hall/CRC, 2004. 3, 5, 8, 10, 15, 16

    work page 2004

  31. [31]

    Knopp, Theory of Functions Parts I and II, Two Volumes Bou nd as One, Part I

    K. Knopp, Theory of Functions Parts I and II, Two Volumes Bou nd as One, Part I. New York: Dover,

    work page

  32. [32]

    Kochubei, Y Luchko (Eds.) Handbook of Fractional Calculus with Applications, Volume 1: Basic Th eory, De Gruyter, 2019

    Yu.Luchko, The Wright function and its applications, pp.241–268 in: A. Kochubei, Y Luchko (Eds.) Handbook of Fractional Calculus with Applications, Volume 1: Basic Th eory, De Gruyter, 2019. 2, 3

    work page 2019

  33. [33]

    Luke, The special functions and their approximations

    Y.L. Luke, The special functions and their approximations. Volu me 1. Academic Press, 1969. 10

    work page 1969

  34. [34]

    Mainardi and G

    F. Mainardi and G. Pagnini, The role of the Fox-Wright functions in fractional sub-diffusion of dis- tributed order, Journal of Computational and Applied Mathematic s, Vol 207, No.2(2007), 245–257. 2

    work page 2007

  35. [35]

    Marichev and S.L

    O.I. Marichev and S.L. Kalla, Behaviour of hypergeometric funct ion pFp−1(z) in the vicinity of unity, Rev. T´ ec. Ing. Univ. Zulia, vol.7, 2(1984), 1–8. 4

    work page 1984

  36. [36]

    Mehrez, New integral representations for the Fox-Wright functions and its applications, J

    K. Mehrez, New integral representations for the Fox-Wright functions and its applications, J. Math.Anal.Appl.468(2018), 650–673. 2

    work page 2018

  37. [37]

    Mehrez and S.M

    K. Mehrez and S.M. Sitnik, Functional Inequalitiess for Fox-Wrig ht Functions, The Ramanujan Journal, 2018, DOI: 10.1007/s11139-018-0071-2. 2

    work page doi:10.1007/s11139-018-0071-2 2018

  38. [38]

    Miller, Solutions of Fermat’s last equation in terms of Wright fu nction, Fibonacci Quarterly 29(1991), 52–56

    A.R. Miller, Solutions of Fermat’s last equation in terms of Wright fu nction, Fibonacci Quarterly 29(1991), 52–56. 2

    work page 1991

  39. [39]

    Miller and I.S

    A.R. Miller and I.S. Moskowitz, Reduction of a Class of Fox-Wright P si Functions for Certain Rational Parameters, Computers Math. Applic. Vol. 30, No. 11 (1995), 73– 82. 2

    work page 1995

  40. [40]

    Nørlund, Hypergeometric functions, Acta Mathematica, V ol.94(1955), 289–349

    N.E. Nørlund, Hypergeometric functions, Acta Mathematica, V ol.94(1955), 289–349. http://dx.doi.org/10.1007/BF02392494 4, 11

    work page doi:10.1007/bf02392494 1955

  41. [41]

    Nørlund, Sur les valeurs asymptotiques des nombres et des polynˆ omes de Bernoulli, Rend

    N.E. Nørlund, Sur les valeurs asymptotiques des nombres et des polynˆ omes de Bernoulli, Rend. Circ. Mat. Palermo, 10, 1(1961), 27–44. http://dx.doi.org/10.1007/BF02844807 5, 11

    work page doi:10.1007/bf02844807 1961

  42. [42]

    Olsson, Analytic continuation of higher-order hypergeom etric functions, Journal of Mathemat- ical Physics, Volume 7, no.4 (1966), 702–710

    P.O.M. Olsson, Analytic continuation of higher-order hypergeom etric functions, Journal of Mathemat- ical Physics, Volume 7, no.4 (1966), 702–710. 4, 11

    work page 1966

  43. [43]

    Paris, Exponentially small expansions in the asymptotics of t he Wright function, Journal of Com- putational and Applied Mathematics 234 (2010), 488–504

    R.B. Paris, Exponentially small expansions in the asymptotics of t he Wright function, Journal of Com- putational and Applied Mathematics 234 (2010), 488–504. 2

    work page 2010

  44. [44]

    Paris, Exponentially small expansions of the Wright function on the Stokes lines, Lithuanian Math

    R.B. Paris, Exponentially small expansions of the Wright function on the Stokes lines, Lithuanian Math. J. 54 (2014), 82–105. 2

    work page 2014

  45. [45]

    Some remarks on the theorems of Wright and Braaksma on the Wright function ${}_p\Psi_q(z)$

    R.B. Paris, Some remarks on the theorems of Wright and Braaks ma on the Wright function pΨq(z), arXiv:1708.04824, 2017. 2

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  46. [46]

    Paris, D

    R.B. Paris, D. Kaminski, Asymptotics and Mellin-Barnes Integrals , Cambridge University Press, 2001. 3

    work page 2001

  47. [47]

    Ponnusamy and M

    S. Ponnusamy and M. Vuorinen, Asymptotic Expansions and Ine qualities for Hypergeometric Function, Mathematika 44 (1997) 278–301. 17

    work page 1997

  48. [48]

    Prodanov, Integral Representations and Algebraic Decom positions of the Fox-Wright Type of Special Functions, Fractal Fract

    D. Prodanov, Integral Representations and Algebraic Decom positions of the Fox-Wright Type of Special Functions, Fractal Fract. 2019, 3, 4; http://dx.doi.org/10.3390 /fractalfract3010004 2 19

    work page 2019

  49. [49]

    Prudnikov, Yu.A

    A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, Integrals an d series, Volume 3: More Special Func- tions, Gordon and Breach Science Publishers, 1990. 3

    work page 1990

  50. [50]

    Saigo, H.M

    M. Saigo, H.M. Srivastava, The behaviour of zero balanced hype rgeometric series pFp−1 near the bound- ary of its convergence region, Proceedings of the American Mathe matical Society, Volume 110, Number 1 (1990), 71–76. 4, 11

    work page 1990

  51. [51]

    Thomae, ¨Uber die h¨ oheren hypergeometrischen Reihen, Math

    J. Thomae, ¨Uber die h¨ oheren hypergeometrischen Reihen, Math. Ann., Vol. 2, 3(1870), 427–444. http://dx.doi.org/10.1007/BF01448236 4

    work page doi:10.1007/bf01448236

  52. [52]

    Wang, Contiguous relations for the Fox-Wright function, J

    X. Wang, Contiguous relations for the Fox-Wright function, J. Math. Anal. Appl. Volume 475, Issue 1 (2019), 203–214. https://doi.org/10.1016/j.jmaa.2019.02.035 2

    work page doi:10.1016/j.jmaa.2019.02.035 2019

  53. [53]

    Wang, Y.-M

    M.-K. Wang, Y.-M. Chu and Y.-Q. Song, Asymptotical formulas fo r Gaussian and generalized hyper- geometric functions, Appl. Math. Comput. 276 (2016), 44–60. 17

    work page 2016

  54. [54]

    C. Wei, D. Gong, H. Hao, A family of summation formulas on the Fox -Wright function, J. Math. Anal. Appl. 393 (2012), 316–327. 2

    work page 2012

  55. [55]

    Widder The Laplace Transform

    D.V. Widder The Laplace Transform. Princeton: Princeton Unive rsity Press, 1946. 15

    work page 1946

  56. [56]

    Wright, The asymptotic expansion of the generalized hyper geometric function, J

    E.M. Wright, The asymptotic expansion of the generalized hyper geometric function, J. London Math. Soc. 10 (1935), 287–293. 2

    work page 1935

  57. [57]

    Wright, The asymptotic expansion of the generalized hyper geometric function, Proceeding of the London Math

    E.M. Wright, The asymptotic expansion of the generalized hyper geometric function, Proceeding of the London Math. Soc. Series 2, Volumes 46, Issue 1(1940), 389–408 . 2

    work page 1940

  58. [58]

    Wright, Erratum to ”The asymptotic expansion of the gene ralized hypergeometric function”

    E.M. Wright, Erratum to ”The asymptotic expansion of the gene ralized hypergeometric function”. J. London Math. Soc. 27(1952), 254. 2 20

    work page 1952