On a generalization of the generating function for Gegenbauer polynomials

In this paper we continue to develop further our prescription [arXiv:1602.02962] to holographically compute the conformal partial waves of CFT correlation functions using the gravitational open Wilson network operators in the bulk. In particular, we demonstrate how to implement it to compute four-point scalar partial waves in general dimension. In the process we introduce the concept of OPE modules, that helps us simplify the computations. Our result for scalar partial waves is naturally given in terms of the Gegenbauer polynomials. We also provide a simpler proof of a previously known recursion relation for the even dimensional CFT partial waves, which naturally leads us to an odd dimensional counterpart.

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“…ρ → ∞ Figure 6: Another OPE module. Now amputating the in-going leg (2.44) starting from y, we obtain 34 2 l+s…”

Section: Computing the 4-point Cpwmentioning

confidence: 99%

Scalar blocks as gravitational Wilson networks

Bhatta

Raman

Suryanarayana

2018

J. High Energ. Phys.

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“…and substituting these into (13) and (16) determines the coefficients u ijk (r), respectively. Combining (18) and (20) gives a series expansion for the Hadamard parametrix in terms of the expansion parameters w, s and ∆r.…”

Section: The Singular Propagatormentioning

confidence: 99%

Mode-sum prescription for vacuum polarization in black hole spacetimes in even dimensions

Taylor

Breen²

2017

Phys. Rev. D

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We present a mode-sum regularization prescription for computing the vacuum polarization of a scalar field in static spherically-symmetric black hole spacetimes in even dimensions. This is the first general and systematic approach to regularized vacuum polarization in higher even dimensions, building upon a previous scheme we developed for odd dimensions. Things are more complicated here since the even-dimensional propagator possesses logarithmic singularities which must be regularized. However, in spite of this complication, the regularization parameters can be computed in closed form in arbitrary even dimensions and for arbitrary metric function f (r). As an explicit example of our method, we show plots for vacuum polarization of a massless scalar field in the Schwarzschild-Tangherlini spacetime for even d = 4, ..., 10. However, the method presented applies straightforwardly to massive fields or to nonvacuum spacetimes.

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“…Finally, it should be mentioned that Brafman's extension procedure, leading to identities parametrized by u, is not the only one that can be applied to Gegenbauer generating functions. By exploiting the connection formula for Gegenbauer polynomials, Cohl and collaborators 28,29 have obtained novel extensions of the defining relation (1), as Eq.…”

Section: Final Remarksmentioning

confidence: 99%

Algebraic Generating Functions for Gegenbauer Polynomials

Maier¹

2018

Frontiers in Orthogonal Polynomials and q-Series

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It is shown that several of Brafman's generating functions for the Gegenbauer polynomials are algebraic functions of their arguments, if the Gegenbauer parameter differs from an integer by one-fourth or one-sixth. Two examples are given, which come from recently derived expressions for associated Legendre functions with octahedral or tetrahedral monodromy. It is also shown that if the Gegenbauer parameter is restricted as stated, the Poisson kernel for the Gegenbauer polynomials can be expressed in terms of complete elliptic integrals. An example is given.

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On a generalization of the generating function for Gegenbauer polynomials

Cited by 29 publications

References 19 publications

Scalar blocks as gravitational Wilson networks

Scalar blocks as gravitational Wilson networks

Mode-sum prescription for vacuum polarization in black hole spacetimes in even dimensions

Algebraic Generating Functions for Gegenbauer Polynomials

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