Wolter Groenevelt

Bio: Wolter Groenevelt is an academic researcher from Delft University of Technology. The author has contributed to research in topics: Orthogonal polynomials & Tensor product. The author has an hindex of 14, co-authored 46 publications receiving 560 citations. Previous affiliations of Wolter Groenevelt include Chalmers University of Technology & University of Lisbon.

The Wilson function transform

Abstract: Two unitary integral transforms with a very-well poised $_7F_6$-function as a kernel are given. For both integral transforms the inverse is the same as the original transform after an involution on the parameters. The $_7F_6$-function involved can be considered as a non-polynomial extension of the Wilson polynomial, and is therefore called a Wilson function. The two integral transforms are called a Wilson function transform of type I and type II. Furthermore, a few explicit transformations of hypergeometric functions are calculated, and it is shown that the Wilson function transform of type I maps a basis of orthogonal polynomials onto a similar basis of polynomials.

Wilson Function Transforms Related to Racah Coefficients

Abstract: The irreducible $*$ -representations of the Lie algebra $${\mathfrak {\rm u}}(1,1)$$ consist of discrete series representations, principal unitary series and complementary series. We calculate Racah coefficients for tensor product representations that consist of at least two discrete series representations. We use the explicit expressions for the Clebsch–Gordan coefficients as hypergeometric functions to find explicit expressions for the Racah coefficients. The Racah coefficients are Wilson polynomials and Wilson functions. This leads to natural interpretations of the Wilson function transforms. As an application several sum and integral identities are obtained involving Wilson polynomials and Wilson functions. We also compute Racah coefficients for $${\mathcal U}_q({\mathfrak{\rm u}}(1,1))$$ , which turn out to be Askey–Wilson functions and Askey–Wilson polynomials.

The Wilson function transform

Abstract: Two unitary integral transforms with a very-well poised $_7F_6$-function as a kernel are given. For both integral transforms the inverse is the same as the original transform after an involution on the parameters. The $_7F_6$-function involved can be considered as a non-polynomial extension of the Wilson polynomial, and is therefore called a Wilson function. The two integral transforms are called a Wilson function transform of type I and type II. Furthermore, a few explicit transformations of hypergeometric functions are calculated, and it is shown that the Wilson function transform of type I maps a basis of orthogonal polynomials onto a similar basis of polynomials.

Meixner functions and polynomials related to lie algebra representations

Abstract: The decomposition of the tensor product of a positive and a negative discrete series representation of the Lie algebra (1,1) is a direct integral over the principal unitary series representations. In the decomposition discrete terms can occur, and these are a finite number of discrete series representations or one complementary series representation. The interpretation of Meixner functions and polynomials as overlap coefficients in the four classes of representations and the Clebsch–Gordan decomposition, lead to a general bilinear generating function for the Meixner polynomials. Finally, realizing the positive and negative discrete series representations as operators on the spaces of holomorphic and anti-holomorphic functions, respectively, a non-symmetric type Poisson kernel is found for the Meixner functions.

Bilinear Summation Formulas from Quantum Algebra Representations

Abstract: The tensor product of a positive and a negative discrete series representation of the quantum algebra U q (su(1,1)) decomposes as a direct integral over the principal unitary series representations. Discrete terms can appear, and these terms are a finite number of discrete series representations, or one complementary series representation. From the interpretation as overlap coefficients of little q-Jacobi functions and Al-Salam and Chihara polynomials in base q and base q−1, two closely related bilinear summation formulas for the Al-Salam and Chihara polynomials are derived. The formulas involve Askey-Wilson polynomials, continuous dual q-Hahn polynomials and little q-Jacobi functions. The realization of the discrete series as q-difference operators on the spaces of holomorphic and anti-holomorphic functions, leads to a bilinear generating function for a certain type of 2φ1-series, which can be considered as a special case of the dual transmutation kernel for little q-Jacobi functions.

A guide to quantum groups

Abstract: By Vyjayanthi Chari and Andrew Pressley: 651 pp., £22.95 (US$34.95), isbn 0 521 55884 0 (Cambridge University Press, 1994).

Solving the Schwarzian via the conformal bootstrap

Abstract: We obtain exact expressions for a general class of correlation functions in the 1D quantum mechanical model described by the Schwarzian action, that arises as the low energy limit of the SYK model. The answer takes the form of an integral of a momentum space amplitude obtained via a simple set of diagrammatic rules. The derivation relies on the precise equivalence between the 1D Schwarzian theory and a suitable large c limit of 2D Virasoro CFT. The mapping from the 1D to the 2D theory is similar to the construction of kinematic space. We also compute the out-of-time ordered four point function. The momentum space amplitude in this case contains an extra factor in the form of a crossing kernel, or R-matrix, given by a 6j-symbol of SU(1,1). We argue that the R-matrix describes the gravitational scattering amplitude near the horizon of an AdS2 black hole. Finally, we discuss the generalization of some of our results to $$ \mathcal{N}=1 $$ and $$ \mathcal{N}=2 $$ supersymmetric Schwarzian QM.

Confluent Hypergeometric Functions

Abstract: Confluent Hypergeometric Functions By Dr L J Slater Pp ix + 247 (Cambridge: At the University Press, 1960) 65s net

Shockwave S-matrix from Schwarzian Quantum Mechanics

Abstract: Schwarzian quantum mechanics describes the collective IR mode of the SYK model and captures key features of 2D black hole dynamics. Exact results for its correlation functions were obtained in [1]. We compare these results with bulk gravity expectations. We find that the semi-classical limit of the OTO four-point function exactly matches with the scattering amplitude obtained from the Dray-’t Hooft shockwave $$ \mathcal{S} $$ -matrix. We show that the two point function of heavy operators reduces to the semi-classical saddle-point of the Schwarzian action. We also explain a previously noted match between the OTO four point functions and 2D conformal blocks. Generalizations to higher-point functions are discussed.

Crossing symmetry in alpha space

Abstract: We initiate the study of the conformal bootstrap using Sturm-Liouville theory, specializing to four-point functions in one-dimensional CFTs. We do so by decomposing conformal correlators using a basis of eigenfunctions of the Casimir which are labeled by a complex number α. This leads to a systematic method for computing conformal block decompositions. Analyzing bootstrap equations in alpha space turns crossing symmetry into an eigenvalue problem for an integral operator K. The operator K is closely related to the Wilson transform, and some of its eigenfunctions can be found in closed form.