The number of points on certain families of hypersurfaces over finite fields
The number of representations by sums of squares and triangular numbers.
The Numerical Computation of the Confluent Hypergeometric Function U(a,b,z)
The numerical radius of a weighted shift operator with geometric weights.
The order-independence of the polylogarithmic ladder structure - implications for a new category of functional equations.
The permutation group method for the dilogarithm
We give qualitative and quantitative improvements on all the best previously known irrationality results for dilogarithms of positive rational numbers. We obtain such improvements by applying our permutation group method to the diophantine study of double integrals of rational functions related to the dilogarithm.
The Poisson integral for a ball in spaces of constant curvature
We present explicit expressions of the Poisson kernels for geodesic balls in the higher dimensional spheres and real hyperbolic spaces. As a consequence, the Dirichlet problem for the projective space is explicitly solved. Comparison of different expressions for the same Poisson kernel lead to interesting identities concerning special functions.
The Poisson Kernel for Heisenberg Polynomials on the Disk.
The polylogarithm in the field of two irreducible quintics.
The Principles of Elliptic and Hyperbolic Analysis [Book]
The properties, inequalities and numerical approximation of modified Bessel functions.
The -analogue of Hölder’s theorem for the gamma function
The q-gamma function for q > 1.
The q-gamma function for x<0.
The quantized Jacobi polynomials
The author studies a system of polynomials orthogonal at a finite set of points its weight approximating that of the orthogonal system of classical Jacobi polynomials.
The Rahman polynomials are bispectral.
The rate of convergence for spectra of GUE and LUE matrix ensembles
We obtain optimal bounds of order O(n −1) for the rate of convergence to the semicircle law and to the Marchenko-Pastur law for the expected spectral distribution functions of random matrices from the GUE and LUE, respectively.
The rational qKZ equation and shifted non-symmetric Jack polynomials.
The reciprocal of the beta function and Whittaker functions
In this paper we derive, using the Gauss summation theorem for hypergeometric series, a simple integral expression for the reciprocal of Euler’s beta function. This expression is similar in form to several well-known integrals for the beta function itself.We then apply our new formula to the study of Whittaker functions, which are special functions that arise in the Fourier theory for automorphic forms on the general linear group. Specifically, we deduce explicit integral representations of “fundamental”...
The relational translators of the hyperspherical functional matrix.