An exactly solvable self-convolutive recurrence

Martin, R. J.; Kearney, M. J.

doi:10.1007/s00010-010-0051-0

Cited by 17 publications

(23 citation statements)

References 18 publications

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“…Thus for example one cannot immediately recover the (u n ) from G via integral formulae. On the other hand, by treating it as the asymptotic expansion of a recognisable function that, on account of the recurrence, obeys a differential equation, one can obtain analytical results: that is the key insight of [6]. Of course it may be that the asymptotic series is in fact convergent; it then produces the same results as do standard techniques.…”

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confidence: 62%

“…In a recent paper [6] we studied in some detail the sequence defined by the self-convolutive recurrence (1) u n = (α 1 n + α 2 )u n−1 + α 3 n−1 j=1 u j u n−j , u 1 = 1, and showed how to derive a closed-form solution which, under mild assumptions discussed in [6], is…”

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confidence: 99%

“…In [6] we found several examples from combinatorics that pertain to the enumeration of permutations and Feynman diagrams. A trivial example is the factorial numbers, u n = n!, generated 1 by φ(z) = ze z E 1 (z).…”

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confidence: 99%

“…In consequence, combinatorially-related sequences are treated together. Here, all are related to the exponential integral function, and the suggestion is that they are algebraically distinct from other combinatorial recurrences such as those in quantum electrodynamics that invoke the modified Bessel function (see [6]), to which the transformation methods that we use below could doubtless be applied.…”

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confidence: 99%

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Integral representation of certain combinatorial recurrences

Martin

Kearney

2014

Combinatorica

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Many recurrences that occur in combinatorics incorporate linear and self-convolutive terms. The generating function associated to these is usually not well defined because it has zero radius of convergence. However, the sequence may be identifiable as the asymptotic expansion of a function, and then contour integration can be applied to obtain an expression as the moment sequence of a (possibly signed) measure. We find examples that in combinatorics are all connected with permutations, and whose generating functions are related to the exponential integral function.

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confidence: 62%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Integral representation of certain combinatorial recurrences

Martin

Kearney

2014

Combinatorica

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“…Thus This is a recurrence relation for the number of rooted triangulations on n darts (equivalently, n/3 triangles). The corresponding number sequence can be also expressed as the S(6, −8, 1) sequence from [22]. Example 5.3.…”

Section: The Riccati Hierarchy and Recurrence Relationsmentioning

confidence: 99%

Free subgroups of free products and combinatorial hypermaps

Ciobanu

Kolpakov

2019

Discrete Mathematics

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We derive a generating series for the number of free subgroups of finite index in ∆ + = Z p * Z q by using a connection between free subgroups of ∆ + and certain hypermaps (also known as ribbon graphs or "fat" graphs), and show that this generating series is transcendental. We provide non-linear recurrence relations for the above numbers based on differential equations that are part of the Riccati hierarchy.We also study the generating series for conjugacy classes of free subgroups of finite index in ∆ + , which correspond to isomorphism classes of hypermaps. Asymptotic formulas are provided for the numbers of free subgroups of given finite index, conjugacy classes of such subgroups, or, equivalently, various types of hypermaps and their isomorphism classes.

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Double logarithms in $$ \mathcal{N} $$ ≥ 4 supergravity: weak gravity and Shapiro’s time delay

Vera

2020

J. High Energ. Phys.

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A study of the double logarithmic in the center-of-mass energy, s, contributions to the four-graviton scattering amplitude is presented for four-dimensional N ≥ 4 supergravities. This includes a novel representation for the coefficients of the perturbative expansion based on exactly solvable recurrences. A review is given of the structure in the complex angular momentum plane for the t-channel partial wave singularities of the different amplitudes. Working in impact parameter representation, ρ, it is shown that the resummation of double logarithms makes gravity weaker in regions of small ρ and large s. This screening of the gravitational interaction at short distances in the double logarithmic sector of the amplitudes is more acute as the number of gravitinos in the theory increases. It brings corrections to the eikonal phase which can change the sign of the graviton's deflection angle and generate regions with repulsive interaction. For very small impact parameters there appears a constant negative shift in both the eikonal phase and Shapiro's time delay which is not large enough to generate causality violation.The AdS/CFT correspondence [1-3] and the physics program at the Large Hadron Collider (LHC) have revamped the interest in the calculation of scattering amplitudes. Being fundamental for the former and an interesting simplified model for the latter, N = 4 supersymmetric Yang-Mills theory has played a pivotal role in recent developments. A very interesting finding [4] has been the double-copy structure linking this theory to N = 8 supergravity [5,6].An interesting limit where to investigate scattering amplitudes in gravitational theories, of relevance in the work here presented, is that of multi-Regge kinematics. In this case the center-of-mass energy squared, s, is asymptotically larger than all other Mandelstam invariants and amplitudes factorize in terms of impact factors and reggeized gravitons exchanged in the t-channel [7,8]. For gravity, eikonal contributions are also relevant together with double-logarithmic in s (DL) terms [9][10][11]. Both can be studied using the high energy effective action derived by Lipatov [12]. In this framework the leading contributions to scattering amplitudes are generated by bunches of produced gravitons well separated in rapidity from each other. The regions in rapidity without radiation have their origin in the exchange of reggeized gravitons. These are created (annihilated) in the t-channel by A ++ (A −− ) fields, which are subject to the constraints ∂ ± A ±± = 0 in order to generate the rapidity gaps. The local and non-local (in rapidity) effective interactions which appear in this context are described by the action given in [12]. It contains the Einstein-Hilbert action together with a kinetic term for the reggeon fields plus induced contributions. Within this setup it is possible to calculate the graviton Regge trajectory and those effective vertices needed to produce inelastic amplitudes.Connecting with the double-copy structure of gravity [13], it is remarkable t...

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An exactly solvable self-convolutive recurrence

Cited by 17 publications

References 18 publications

Integral representation of certain combinatorial recurrences

Integral representation of certain combinatorial recurrences

Free subgroups of free products and combinatorial hypermaps

Double logarithms in $$ \mathcal{N} $$ ≥ 4 supergravity: weak gravity and Shapiro’s time delay

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