Discrete Darboux transformation for discrete polynomials of hypergeometric type

Darboux Transformation, well known in second order differential operator theory, is applied here to the difference equation satisfied by the discrete hypergeometric polynomials(Charlier, Meixner-Krawchuk, Hahn)

On a modular property of N=2 superconformal theories in four dimensions

In this note we discuss several properties of the Schur index of N=2 superconformal theories in four dimensions. In particular, we study modular properties of this index under SL(2,Z) transformations of its parameters.Comment: 23 page, 2 figure

Modeling of pipe-drawing tool for drawing the multifaceted pipes of nonferrous metals on an immediate arbor

A method of mathematical modeling of a pipe-drawing tool for drawing the multifaceted pipes of nonferrous metals and alloys using the vector-matrix apparatus, which can be applied for the analytical description of the bulk deformation region, is presented. Arbors with various geometries of the reduction zone are considered. As a result of modeling the deformation region, which appears when manufacturing the profiled multifaceted pipes by arbor drawing using all types of considered arbors, it is established that the best result with the smallest rounding radii is attained for arbors with a pyramidal input into the reduction zone. © 2013 Allerton Press, Inc

Superconformal indices of three-dimensional theories related by mirror symmetry

Recently, Kim and Imamura and Yokoyama derived an exact formula for superconformal indices in three-dimensional field theories. Using their results, we prove analytically the equality of superconformal indices in some U(1)-gauge group theories related by the mirror symmetry. The proofs are based on the well known identities of the theory of $q$-special functions. We also suggest the general index formula taking into account the $U(1)_J$ global symmetry present for abelian theories.Comment: 17 pages; minor change

q-Ultraspherical polynomials for q a root of unity

Properties of the $q$-ultraspherical polynomials for $q$ being a primitive root of unity are derived using a formalism of the $so_q(3)$ algebra. The orthogonality condition for these polynomials provides a new class of trigonometric identities representing discrete finite-dimensional analogs of $q$-beta integrals of Ramanujan.Comment: 7 pages, LATE

Raising and lowering operators, factorization and differential/difference operators of hypergeometric type

Starting from Rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variable on uniform lattice. In order to have these operators mutually adjoint we introduce orthonormal functions with respect to the scalar product of unit weight. Using the Infeld-Hull factorization method, we generate from the raising and lowering operators the second order self-adjoint differential/difference operator of hypergeometric type.Comment: LaTeX, 24 pages, iopart style (late submission

S-duality and 2d Topological QFT

We study the superconformal index for the class of N=2 4d superconformal field theories recently introduced by Gaiotto. These theories are defined by compactifying the (2,0) 6d theory on a Riemann surface with punctures. We interpret the index of the 4d theory associated to an n-punctured Riemann surface as the n-point correlation function of a 2d topological QFT living on the surface. Invariance of the index under generalized S-duality transformations (the mapping class group of the Riemann surface) translates into associativity of the operator algebra of the 2d TQFT. In the A_1 case, for which the 4d SCFTs have a Lagrangian realization, the structure constants and metric of the 2d TQFT can be calculated explicitly in terms of elliptic gamma functions. Associativity then holds thanks to a remarkable symmetry of an elliptic hypergeometric beta integral, proved very recently by van de Bult.Comment: 25 pages, 11 figure

Quasi-exactly solvable problems and the dual (q-)Hahn polynomials

A second-order differential (q-difference) eigenvalue equation is constructed whose solutions are generating functions of the dual (q-)Hahn polynomials. The fact is noticed that these generating functions are reduced to the (little q-)Jacobi polynomials, and implications of this for quasi-exactly solvable problems are studied. A connection with the Azbel-Hofstadter problem is indicated.Comment: 15 pages, LaTex; final version, presentation improved, title changed, to appear in J.Math.Phy