Related papers: The Extended Zeilberger's Algorithm with Parameter…
We present a complexity reduction algorithm for a family of parameter-dependent linear systems when the system parameters belong to a compact semi-algebraic set. This algorithm potentially describes the underlying dynamical system with…
Periodicity is often studied in timeseries modelling with autoregressive methods but is less popular in the kernel literature, particularly for higher dimensional problems such as in textures, crystallography, and quantum mechanics. Large…
Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and…
In previous articles, we showed that, based on large-order asymptotic behavior, one can approximate a divergent series via the parametrization of a specific hypergeometric approximant. The analytical continuation is then carried out through…
We prove a priori bounds for Feigenbaum quadratic polynomials, i.e., infinitely renormalizable polynomials $f_c: z\mapsto z^2+c$ of bounded type. It implies local connectivity of the corresponding Julia sets $J(f_c)$ and MLC (local…
In some applications, one is interested in reconstructing a function $f$ from its Fourier series coefficients. The problem is that the Fourier series is slowly convergent if the function is non-periodic, or is non-smooth. In this paper, we…
In many applications (hupergeometric-type) special functions like orthogonal polynomials are needed. For example in more than 50% of the published solutions for the (application-oriented) questions in the "Problems Section" of SIAM Review…
In a previous paper, we presented conjectures of the recurrence relations with constant coefficients for the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. In this paper we present a proof for the…
Orthogonal polynomials of a continuous variable in the Askey scheme satisfying second order difference equations, such as the Askey-Wilson polynomial, can be studied by the quantum mechanical formulation, idQM (discrete quantum mechanics…
In 2010, Kh. Hessami Pilehrood and T. Hessami Pilehrood introduced generating function identities used to obtain series accelerations for values of Dirichlet's $\beta$ function, via the Markov--Wilf--Zeilberger method. Inspired by these…
By using contiguous relations for basic hypergeometric series, we give simple proofs of Bailey's $_4\phi_3$ summation, Carlitz's $_5\phi_4$ summation, Sears' $_3\phi_2$ to $_5\phi_4$ transformation, Sears' ${}_4\phi_3$ transformations,…
We prove the following theorems: 1) The Laurent expansions in epsilon of the Gauss hypergeometric functions 2F1(I_1+a*epsilon, I_2+b*epsilon; I_3+p/q + c epsilon; z), 2F1(I_1+p/q+a*epsilon, I_2+p/q+b*epsilon; I_3+ p/q+c*epsilon;z),…
Usually creative telescoping is used to derive recurrences for sums. In this article we show that the non-existence of a creative telescoping solution, and more generally, of a parameterized telescoping solution, proves algebraic…
Given a matrix $A$ and $k\geq 0$, we study the problem of finding the $k\times k$ submatrix of $A$ with the maximum determinant in absolute value. This problem is motivated by the question of computing the determinant-based lower bound of…
New explicit as well as manifestly symmetric three-term summationformulas are derived for the Clausenian hypergeometric series $_3F_2(1)$ with negative integral parameter differences. Our results generalize and naturally extend several…
The number of linear independent algebraic relations among elementary symmetric polynomial functions over finite fields is computed. An algorithm able to find all such relations is described. It is proved that the basis of the ideal of…
We present a method to prove hypergeometric double summation identities. Given a hypergeometric term $F(n,i,j)$, we aim to find a difference operator $ L=a_0(n) N^0 + a_1(n) N^1 +...+a_r(n) N^r $ and rational functions…
In this article three expansion formulas for a generalized hypergeometric function $_4F_3$ are derived, when its upper parameters differ by integers. Though the results are special cases of a general continuation formula for $_pF_q$, they…
Several problems on Fourier series and trigonometric approximation on a hexagon and a triangle are studied. The results include Abel and Ces\`aro summability of Fourier series, degree of approximation and best approximation by trigonometric…
We define a parametric variant of generalized Euler sums and construct contour integration to give some explicit evaluations of these parametric Euler sums. In particular, we establish several explicit formulas of (Hurwitz) zeta functions,…