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REVIEW 2 major objections 7 minor 45 references

Covariant reggeization framework for diffraction. Part I: Hadronic tensors in Minkovsky space-time of any dimension

T0 review · 2 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read All diffractive hadronic tensors reduce to irreducible TST tensors in any spacetime dimension, with closed-form coefficients for arbitrary spin.

desk verdict A serious, dense formal extension of covariant reggeization to arbitrary spacetime dimension, with a correct V-tensor core but Y/H coefficient formulas that need independent verification; worth refereeing with a request for symbolic checks. read the letter →

arxiv 2507.16019 v6 pith:PVJMPPP2 submitted 2025-07-21 hep-ph

Roman Ryutin This is my paper · ORCID
classification hep-ph MSC 81V0581-0181-0881Q9981T9981U9981V2581V99
keywords covariantreggeizationirreducibletensorsTSTRarita-Schwingerconditionshadronicdiffractionreggeongeneratingfunctionsarbitraryspacetimedimension
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper develops a covariant reggeization framework in which the amplitudes of all major diffractive processes are built from irreducible transverse-symmetric-traceless (TST) tensor structures. The central claim is that the relevant hadronic tensors, for any integer reggeon spin and any spacetime dimension D, can be expanded in these irreducible tensors, and that all expansion coefficients can be obtained in closed form. A sympathetic reader would care because this reduces the calculation of diffractive cross sections to contracting a small set of universal tensor building blocks, with no need to repeat the Lorentz-algebra work process by process. The paper explicitly constructs the vertex and amplitude tensors V, W, F, Y, and H, and gives their coefficient solutions for elastic scattering, central production, single and double dissociation, and inclusive variants.

What carries the argument

The central object is the irreducible transverse-symmetric-traceless tensor built from transverse projectors $G_{(rr)}$, $G_{(rr')}$, $\hat{G}_{(rs)}$, the orthogonal momenta $P_{(i)}$, and the metric $g$, with the Rarita-Schwinger conditions (transversality, symmetry, and tracelessness in every index group) imposed on the reggeon currents and vertex functions. These TST structures carry the argument: applying the trace operator to a generating function in auxiliary vectors produces linear recurrent equations for the expansion coefficients, and the generating-function method converts those recurrences into closed-form coefficient solutions. For the two-reggeon tensors $F$ and $W$ the solutions reduce to sums of binomial and hypergeometric terms; for the three- and four-reggeon tensors $Y$ and $H$ the solutions are given by an operator algorithm that builds higher-occupancy coefficients step by step from the boundary coefficients.

What would settle it

Take a specific model of a spin-J reggeon current with a non-conserved component, compute its trace in momentum space, and check whether it vanishes; if the trace is nonzero, the TST recurrence (46) is violated and the given coefficients cannot describe that model. Alternatively, measure the single-diffraction dissociation cross section at very small momentum transfer: the conserved-current TST solution predicts a minimum there, so data showing a monotonic rise would falsify the TST assumption for the pomeron current.

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Extended reading notes

Core claim

The paper claims that all hadronic irreducible tensors representing the amplitudes of the most important diffractive processes can be expanded in irreducible transverse-symmetric-traceless (TST) tensors, and provides explicit closed-form solutions for the expansion coefficients for arbitrary spin and arbitrary spacetime dimension D. The tensors V (scalar-scalar-spin-J vertex), W (forward reggeon-hadron amplitude), F (reggeon-reggeon fusion vertex), Y (three-reggeon vertex), and H (four-reggeon forward amplitude) are each decomposed into TST structures built from transverse projectors, moments orthogonal to the momentum transfers, and the metric. The coefficients obey recurrent equations derived from the tracelessness condition in each index group, and the solutions are expressed in terms of Pochhammer symbols, binomial sums, and hypergeometric functions. Generalization to non-conserved currents is also given, with the non-irreducible tensor written as a sum of irreducible tensors times transferred momenta.

Load-bearing premise

The entire decomposition rests on imposing the Rarita-Schwinger conditions, that the reggeon current is transverse, symmetric, and traceless; if real reggeons violate any of these, the closed-form coefficient solutions do not apply to physical diffraction.

Editorial extensions

If this is right

  • If the TST expansion is correct, every diffractive cross section listed in the paper, elastic, central exclusive and inclusive production, single dissociation, double dissociation, and their combinations, follows from contracting the corresponding tensors and taking the imaginary part through the optical theorem.
  • The framework supplies a universal Lego set of tensor structures, so switching between processes only requires choosing the appropriate vertex tensors and form factors rather than re-deriving Lorentz structures.
  • The explicit dependence on spacetime dimension D allows the theory to predict how diffractive observables, such as the shape of the single-dissociation cross section at small momentum transfer, would change if extra dimensions exist.
  • For non-conserved currents, the expansion into irreducible tensors plus transferred momenta provides a systematic way to parametrize models in which the pomeron current is effectively non-conserved in four dimensions but may be conserved in higher dimensions.
  • The closed-form coefficients for arbitrary spin make the framework directly applicable to effective models for higher-spin particles beyond diffraction, since the mathematical construction is general.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's recurrence solutions effectively give an algorithm that could be turned into a computer routine for generating TST expansion coefficients for arbitrary spin and D, something the paper does not explicitly provide as code.
  • If the conserved-current TST condition is relaxed, the paper's formalism predicts definite signatures, such as a possible minimum in the single-dissociation cross section at very small momentum transfer; this prediction could be tested against existing low-t data as a way to infer whether the physical pomeron current is conserved.
  • The same generating-function technology could be extended to continuous-spin representations or to curved backgrounds, directions the paper mentions but does not pursue; a natural next step is to test whether the D-dependence of the coefficients survives the projection onto four-dimensional observables.
  • Because the amplitudes are written as contractions of TST tensors, the framework is well suited for later unitarization studies, since each tensor structure carries a definite spin and transverse structure that can be resummed in the complex-J plane.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 7 minor

Summary. This paper develops a covariant framework for reggeized hadronic amplitudes in arbitrary spacetime dimension D. The author introduces irreducible transverse-symmetric-traceless (TST) tensors V, W, F, Y, H that serve as building blocks for the amplitudes of elastic, central exclusive/inclusive, and single/double dissociative processes. The main technical results are explicit solutions for the expansion coefficients: the V coefficients are given by a Pochhammer formula (48), the F and W coefficients by the closed forms (57)-(58) obtained from generating functions, and the Y and H coefficients by the operator-based expressions (65) and (71). The paper also extends the construction to non-conserved currents by expanding general symmetric tensors in TST structures and momenta (Section 4, Appendix C). Examples for spin-2 F and W are given in Appendix E. The cross-sections themselves are deferred to a subsequent publication.

Significance. If the coefficient solutions are correct, the paper provides a systematic, dimension-independent method to construct all diffractive hadronic tensors, which could be useful for forward physics and effective higher-spin models. The V recurrence is solved correctly, and the spin-2 F example in Eq. (197) matches the general formula (57) for k=0, giving some evidence for the F/W sector. The framework is self-contained, does not fit parameters (the hadronic form-factors are arbitrary), and explicitly covers non-conserved currents. However, the solutions for the three- and four-reggeon tensors Y and H are given through a complex operator algorithm whose correctness is not independently demonstrated, and the notation is so dense that verification is difficult. The paper is a mathematical construction paper; its utility will depend on the promised Part II.

major comments (2)
  1. [§5.4–5.5, Appendices B.6–B.7] Equations (65) and (71) are claimed to be the general solutions of the recurrences (64) and (70) for the coefficients of the Y and H tensors. The derivation in Appendices B.6 and B.7 constructs the solution by iterating the recurrence from a single non-zero component of the vector index and then averaging over permutations of the index groups. This presupposes that the system of recurrences for different index groups is consistent and that the iterative result is independent of the order in which the components are reduced. Neither property is proved in the text, and the final averaged expressions are not shown to satisfy the original recurrences. Because these coefficients are the core of the claimed closed-form TST expansions for the three- and four-reggeon tensors, this is a load-bearing gap. Please provide a proof of consistency of the recurrence system or an explicit verification for a low-spin case (e.g., all J_i=2 or J_i=1) covering all allowed indices.
  2. [§4, Eqs. (31)–(35)] The orthogonality relations (31)–(35) are asserted without proof. These relations are stated to justify that the expansions (26)–(30) are possible and that the coefficients are uniquely defined. While the coefficients are in fact solved from the tracelessness constraints, the paper does not demonstrate that the set of structures is linearly independent or that the stated orthogonality holds under the defined contraction. A proof or an explicit reference for these relations should be provided.
minor comments (7)
  1. [§B.3, Eq. (101)] The definition of k'_2 repeats k'_{12} instead of including k'_{1'2}; the correct expression should be k'_2 = k'_{22'} + k'_{12} + k'_{1'2}.
  2. [§5.2, Eq. (52)] In the definition of W, the second sum is over n_{1,1'} but the tensor S^{W;J_1,J_1'}_{k', n_1 n_2} carries a subscript n_2; this should be n_1 n_{1'}.
  3. [§5.2, Eq. (120)] The change of variables used to solve the F and W recurrences is not clearly written: the relation between f^{k-l}_{n_1 n_2}, f^l_{n_1 n_2}, and f^l_{i;n_1 n_2} is ambiguous and the displayed equation is incomplete. It should be rewritten with explicit definitions of all auxiliary quantities.
  4. [Throughout] The paper consistently misspells 'Minkowski' as 'Minkovsky' (title, abstract, body); this should be corrected.
  5. [§5.4–5.5] The word 'numder' appears in the description of the regions \bar{\Omega}^Y and \bar{\Omega}^H; it should read 'number'.
  6. [§B.6–B.7, Eqs. (65) and (71)] The notation for the operators \hat{S} and the subscripts such as {10,11,12} is introduced only in the appendices; a forward reference in the main text would help the reader understand the final formulas.
  7. [Appendix D, Eq. (196)] The definition of the Stirling-number identity is correct, but the surrounding notation is cramped; a more standard notation, such as Stirling brace notation, would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: tensor expansion coefficients are solved from tracelessness recurrences, not fitted; self-citations are motivational only.

full rationale

The paper's central result is the explicit TST decomposition of hadronic tensors. The load-bearing conditions are the Rarita-Schwinger TST constraints (20)-(25), stated as the defining property of reggeon currents and fields. From these, the recurrence relations (46), (53), (55), (64), and (70) are derived via trace extraction or generating functions, and the coefficient solutions (48), (57), (58), (65), and (71) are obtained algebraically in Appendices B.4, B.6, and B.7. No coefficient is fitted to data: the form-factors are introduced as independent arbitrary scalar functions (hat upsilon_0, hat f, hat w, hat y, hat h) and are never used to determine the expansion coefficients. The orthogonality relations (31)-(35) justify the decomposition, and the non-conserved current case is handled by the same trace-solving algorithm in Appendix C, with factorized solutions (187)-(188). Self-citations to Refs. [1]-[3] are contextual and motivational; the statement in Section 6 that a conserved-current cross-section minimum 'was shown in Refs. [2,3]' is a self-citation but is not load-bearing for the algebraic derivations presented here. There is no step in which a target quantity is defined in terms of itself or a fitted parameter is renamed as a prediction. A non-circular caveat: the closed-form coefficient formulas are lengthy, and Eq. (101) contains an evident typo (k'_12 repeated in the definition of k'_2), so independent symbolic checking is advisable; this is a correctness risk, not circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No new particles or physical entities are introduced. The reggeon is an effective field from prior Regge theory, and the TST tensors are mathematical constructs. The derivation is parameter-free apart from the arbitrary hadronic form-factors, which are left for future phenomenological fitting.

free parameters (1)
  • Hadronic form-factors (hat v_0, hat f_k, hat w_k, hat y_k, hat h_k)
    Arbitrary scalar functions of momentum transfers, declared as independent external inputs in Eqs. (45), (49), (51), (60), (66). They are not fitted in this paper and do not enter the coefficient derivations.
assumptions (4)
  • domain assumption Reggeon fields obey Rarita-Schwinger TST conditions: transverse, symmetric, traceless, Eqs. (20)-(25).
    The entire framework assumes the reggeon is an irreducible Poincare tensor field satisfying these conditions; the assumption is stated in Section 2 rather than derived.
  • domain assumption The reggeization prescription Eq. (18) replaces a sum over integer spin J with a Regge trajectory expression.
    Quoted from previous Regge theory literature and used to connect the tensor formalism to physical amplitudes; not derived in this paper.
  • standard math Any symmetric tensor has a unique orthogonal decomposition into traceless symmetric pieces (ST structures).
    Used implicitly in Section 4 and Appendix C; standard representation theory of symmetric tensors.
  • standard math Generating function manipulations and multinomial expansions are valid for the recurrences.
    Used in Appendices B.2-B.7; standard combinatorial methods, assumed valid.

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Cite this review

Pith. "Pith review of Covariant reggeization framework for diffraction. Part I: Hadronic tensors in Minkovsky space-time of any dimension." pith.science (2026). https://pith.science/paper/PVJMPPP2

@misc{pith2026250716019,
  author       = {Pith},
  title        = {Pith review of: Covariant reggeization framework for diffraction. Part I: Hadronic tensors in Minkovsky space-time of any dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PVJMPPP2}},
  note         = {Machine review of arXiv:2507.16019}
}
abstract

In this paper we consider the general structure of irreducible tensor representations of the Poincar\'e group of arbitrary space-time dimension $D$ with multiple sets of Lorentz indices and different ways to construct them from basic elements (Lorentz vectors and the metric tensor). Then we apply the same methods to obtain the expansion of general hadronic tensors in terms of these irreducible tensors. We propose to use an effective approach in hadronic diffraction, which was usually called covariant reggeization, and obtain basic functions and tensors to calculate all the diffractive cross-sections.

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Works this paper leans on

45 extracted references · 43 canonical work pages

  1. [1]

    Petrov, R.A

    V.A. Petrov, R.A. Ryutin, A.E. Sobol and J.-P. Guillaud,Azimuthal Angular Dis- tributions in EDDE as Spin-Parity Analyser and Glueball Filter for LHC, JHEP06, 007 (2005)

    work page 2005

  2. [2]

    Petrov and R.A

    V.A. Petrov and R.A. Ryutin,Single and double diffractive dissociation and the problem of extraction of the proton–Pomeron cross-section, Int. J. Mod. Phys. 31, No. 10, 1650049 (2016)

    work page 2016

  3. [3]

    Ryutin,Visualizations of exclusive central diffraction, Eur

    R.A. Ryutin,Visualizations of exclusive central diffraction, Eur. Phys. J. C 74, 3162 (2014)

    work page 2014

  4. [4]

    Gribov,The Theory of Complex Angular Momenta

    V.N. Gribov,The Theory of Complex Angular Momenta. Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2003

    work page 2003

  5. [5]

    P. D. B. Collins,An Introduction to Regge Theory and High Energy Physics (Cambridge Monographs on Mathematical Physics), Cambridge University Press, 1977

    work page 1977

  6. [6]

    Jones, M.D

    H.F. Jones, M.D. Scadron & F.D. Gault,Gauge invariance and covariant Reggeization, Nuovo Cimento A 66, 424–436 (1970)

    work page 1970

  7. [7]

    Scadron,Covariant Propagators and Vertex Functions for Any Spin, Phys

    Michael D. Scadron,Covariant Propagators and Vertex Functions for Any Spin, Phys. Rev. 165, 1640 (1968)

    work page 1968

  8. [8]

    Gault, H.F

    F.D. Gault, H.F. Jones,Covariant reggeization and t-channel dip mechanisms in s-channel helicity amplitudes, Nuclear Physics B V. 30, Issue 1, 68-76 (1971). 44

    work page 1971

Show all 45 references
  1. [9]

    Aye,Reggeized differential cross sections for fermion-exchange processes in the covariant formalism, J

    T.M. Aye,Reggeized differential cross sections for fermion-exchange processes in the covariant formalism, J. Phys. A, V. 4, Number 6, 846 (1971)

    1971

  2. [10]

    Jones, M.D

    H.F. Jones, M.D. Scadron,Reggeization of invariant amplitudes, Nuclear Physics B, V. 4, Issue 2, 267-276 (1968)

    1968

  3. [11]

    Wigner,On unitary representations of the inhomogeneous Lorentz group, Ann

    E.P. Wigner,On unitary representations of the inhomogeneous Lorentz group, Ann. Math., 40, 149–204 (1939)

    1939

  4. [12]

    Mojtaba Najafizadeh,Modified Wigner equations and continuous spin gauge field, Phys. Rev. D 97, 065009 (2018)

    2018

  5. [13]

    Rivelles,Gauge theory formulations for continuous and higher spin fields, Phys

    Victor O. Rivelles,Gauge theory formulations for continuous and higher spin fields, Phys. Rev. D 91, 125035 (2015)

    2015

  6. [14]

    Metsaev,Cubic interaction vertices for fermionic and bosonic arbitrary spin fields, Nuclear Physics B, V

    R.R. Metsaev,Cubic interaction vertices for fermionic and bosonic arbitrary spin fields, Nuclear Physics B, V. 859, Issue 1, 13-69 (2012)

    2012

  7. [15]

    Metsaev,Extended hamiltonian action for arbitrary spin fields in flat and AdS spaces, arXiv:1112.0976 [hep-th]

    R.R. Metsaev,Extended hamiltonian action for arbitrary spin fields in flat and AdS spaces, arXiv:1112.0976 [hep-th]

    arXiv

  8. [16]

    2006, JHEP01, 115 (2006)

    Xavier Bekaert and Jihad Mourad,The continuous spin limit of higher spin field equations, Journal of High Energy Physics, V. 2006, JHEP01, 115 (2006)

    2006

  9. [17]

    Bekaert, J

    X. Bekaert, J. Mourad & M. Najafizadeh,Continuous-spin field propagator and interaction with matter, JHEP 2017, 113 (2017)

    2017

  10. [18]

    Sagnotti,Higher Spins and Current Exchanges, arXiv:1002.3388 [hep-th]

    A. Sagnotti,Higher Spins and Current Exchanges, arXiv:1002.3388 [hep-th]

    arXiv

  11. [19]

    Lindwasser,Covariant actions and propagators for all spins, masses, and dimensions, Phys

    Lukas W. Lindwasser,Covariant actions and propagators for all spins, masses, and dimensions, Phys. Rev. D 109, 085010 (2024)

    2024

  12. [20]

    Buchbinder and Benjamin J

    Evgeny I. Buchbinder and Benjamin J. Stone,Three-point functions of conserved currents in 3D CFT: general formalism for arbitrary spins, Phys. Rev. D 107, 046007 (2023)

    2023

  13. [21]

    Buchbinder and Benjamin J

    Evgeny I. Buchbinder and Benjamin J. Stone,Three-point functions of conserved currents in 4D CFT: General formalism for arbitrary spins, Phys. Rev. D 108, 086017 (2023)

    2023

  14. [22]

    Chung (Brookhaven),A General formulation of covariant helicity coupling amplitudes, Phys.Rev.D 57, 431-442 (1998)

    S.U. Chung (Brookhaven),A General formulation of covariant helicity coupling amplitudes, Phys.Rev.D 57, 431-442 (1998)

    1998

  15. [23]

    Seong Youl Choi and Jae Hoon Jeong,Weaving the covariant three-point vertices efficiently, Phys. Rev. D 104, 055046 (2021)

    2021

  16. [24]

    P. R. Auvil and J. J. Brehm,Wave Functions for Particles of Higher Spin, Phys. Rev. 145, 1152 (1966). 45

    1966

  17. [25]

    Caudrey, I.J

    P.J. Caudrey, I.J. Ketley, R.C. King,Covariant arbitrary-spin wave functions and helicity couplings, Nuclear Physics B, V. 6, Issue 6, 671-686 (1968)

    1968

  18. [26]

    Chung (Mainz U., Inst

    S.U. Chung (Mainz U., Inst. Phys.),Helicity coupling amplitudes in tensor for- malism, Phys.Rev.D 48, 1225-1239 (1993); Phys.Rev.D 56, 4419 (1997) (erratum)

    1993

  19. [27]

    19, 1767-1770 (2002)

    Shi-Zhong Huang, Tu-Nan Ruan, Ning Wu, Zhi-Peng Zheng,Projection operator and propagator for an arbitrary integral spin, Chin.Phys.Lett. 19, 1767-1770 (2002)

    2002

  20. [28]

    Shi-Zhong, Z

    H. Shi-Zhong, Z. Peng-Fei, R. Tu-Nan et al.,Feynman propagator for a particle with arbitrary spin, Eur. Phys. J. C 42, 375–389 (2005)

    2005

  21. [29]

    Huang, P.-F

    S.-Z. Huang, P.-F. Zhang, Tingitng Ruan, Neil wu, Zhiquan Zheng,Propagator for an arbitrary integral spin, Acta Physica Sinica 52(8) (2003)

    2003

  22. [30]

    Rakibur Rahman,Higher Spin Theory - Part I, arXiv:1307.3199v2 [hep-th]

    arXiv

  23. [31]

    Isaev, M.A

    A.P. Isaev, M.A. Podoinitsyn,Polarization Tensors for Massive Arbitrary-Spin Particles and the Behrends–Fronsdal Projection Operator, Theor. Math. Phys. 198, 89–99 (2019)

    2019

  24. [32]

    Isaev, M.A

    A.P. Isaev, M.A. Podoinitsyn,Behrends–Fronsdal Spin Projection Operator in Space-Time with Arbitrary Dimension. In: Dobrev, V. (eds) Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1 . LT-XII/QTS-X 2017. Springer Proceedings in Mathematics ...

    2017

  25. [33]

    Bedn´ aˇ r, P

    M. Bedn´ aˇ r, P. Kol´ aˇ r,On the relativistic spin projection operators, Czech. J. Phys. 43, 777–782 (1993)

    1993

  26. [34]

    Frishman, E

    Y. Frishman, E. Gotsman,Explicit Wave Functions for Any Spin, Phys. Rev. 140, B1151 (1965)

    1965

  27. [35]

    Bouatta, G

    N. Bouatta, G. Compere, A. Sagnotti,An Introduction to Free Higher-Spin Fields, arXiv:hep-th/0409068

    arXiv

  28. [36]

    Close, G.A

    F.E. Close, G.A. Schuller,Evidence that the pomeron transforms as a noncon- served vector current, Phys. Lett. B 464, 279 (1999)

    1999

  29. [37]

    Close, G.A

    F.E. Close, G.A. Schuller,Central production of mesons: Exotic states versus pomeron structure, Phys. Lett. B 458, 127 (1999)

    1999

  30. [38]

    Kruchinin, V

    D. Kruchinin, V. Kruchinin, Y. Shablya,Method for Obtaining Coefficients of Powers of Multivariate Generating Functions, Mathematics 2023, 11, 2859 (2023)

    2023

  31. [39]

    Stanley (1972),On the founda- tions of combinatorial theory (VI): The idea of generating function

    Peter Doubilet, Gian-Carlo Rota, and Richard P. Stanley (1972),On the founda- tions of combinatorial theory (VI): The idea of generating function. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability 46 (Berkeley, Calif., 1970/71), Vol. II...

    1972

  32. [40]

    Melczer,Generating Functions and Analytic Combinatorics

    S. Melczer,Generating Functions and Analytic Combinatorics. In: An Invita- tion to Analytic Combinatorics. Texts & Monographs in Symbolic Computation. Springer, Cham, 2021

    2021

  33. [41]

    Stanley,Enumerative Combinatorics: Volume 1,2, Cambridge Univer- sity Press, 2011

    Richard P. Stanley,Enumerative Combinatorics: Volume 1,2, Cambridge Univer- sity Press, 2011

    2011

  34. [42]

    Wilf,Generatingfunctionology, 2nd edition, Academic Press, INC., 1994

    Herbert S. Wilf,Generatingfunctionology, 2nd edition, Academic Press, INC., 1994

    1994

  35. [43]

    Schlosser,Multiple Hypergeometric Series: Appell Series and Beyond

    M.J. Schlosser,Multiple Hypergeometric Series: Appell Series and Beyond. In: Schneider, C., Bl¨ umlein, J. (eds) Computer Algebra in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Vienna, 2013

    2013

  36. [44]

    312, Issue 24, 3553-3560 (2012)

    Herwig Hauser, Christoph Koutschan, Discrete Mathematics,Multivariate linear recurrences and power series division, V. 312, Issue 24, 3553-3560 (2012)

    2012

  37. [45]

    Bousquet-M´ elou, M

    M. Bousquet-M´ elou, M. Petkovˇ sek,Linear recurrences with constant coefficients: the multivariate case, Discrete Mathematics, 225 (1), pp. 51-75 (2000). 47

    2000

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