Related papers: Factorial Basis Method for q-Series Applications

A decomposition of a higher order linear differential operator with polynomial coefficients into a direct sum of two factor operators is obtained. This leads to a lower echelon matrix representation for operators of the above mentioned type…

In this paper new descent line search iterative schemes for unconstrained as well as constrained optimization problems are developed using q-derivative. At every iteration of the scheme, a positive definite matrix is provided which is…

We describe a simple method that produces automatically closed forms for the coefficients of continued fractions expansions of a large number of special functions. The function is specified by a non-linear differential equation and initial…

Using the simple case of Blasius similarity solution, we illustrate a recently developed general method that reduces a strongly nonlinear problem into a weakly nonlinear analysis. The basic idea is to find a quasi-solution $F_0$ that…

For the quantum integer [n]_q = 1+q+q^2+... + q^{n-1} there is a natural polynomial multiplication such that [mn]_q = [m]_q \otimes_q [n]_q. This multiplication is given by the functional equation f_{mn}(q) = f_m(q) f_n(q^m), defined on a…

Here we consider the $q$-series coming from the Hall-Littlewood polynomials, \begin{equation*} R_\nu(a,b;q)=\sum_{\substack{\lambda \\[1pt] \lambda_1\leq a}} q^{c|\lambda|} P_{2\lambda}\big(1,q,q^2,\dots;q^{2b+d}\big). \end{equation*} These…

We prove two new summation formulae of Hall-Littlewood polynomials over partitions into bounded parts and derive some new multiple $q$-identities of Rogers-Ramanujan type.

Let the symmetric functions be defined for the pair of integers $\left( n,r\right) $, $n\geq r\geq 1$, by $p_{n}^{\left( r\right) }=\sum m_{\lambda }$ where $m_{\lambda }$ are the monomial symmetric functions, the sum being over the…

Using the theory of functions of several variables and $q$-calculus, we prove an expansion theorem for the analytic function in several variables which satisfies a system of $q$-partial differential equations. Some curious applications of…

We propose a new method for simulating QCD at finite density. The method is based on a general factorization property of distribution functions of observables, and it is therefore applicable to any system with a complex action. The…

The q-binomial formula in the limit q->1 is shown to be equivalent to the Rogers five term dilogarithm identity.

We propose a randomized polynomial time algorithm for computing nontrivial zeros of quadratic forms in 4 or more variables over $\mathbb{F}_q(t)$, where $\mathbb{F}_q$ is a finite field of odd characteristic. The algorithm is based on a…

We propose a fast and stable method for constructing matrix approximations to fractional integral operators applied to series in the Chebyshev fractional polynomials. This method utilizes a recurrence relation satisfied by the fractional…

A Lagrangian method is introduced recently for deriving indefinite integrals of special functions that satisfy homogeneous (nonhomogeneous) second-order linear differential equations. This paper extends this method to include indefinite…

Finding shape correspondences can be formulated as an NP-hard quadratic assignment problem (QAP) that becomes infeasible for shapes with high sampling density. A promising research direction is to tackle such quadratic optimization problems…

Let $\mathbb{A}=\mathbb{F}_{q}[T]$ be the polynomial ring over finite field $\mathbb{F}_{q}$, and $\mathbb{A}_{+}$ be the set of monic polynomials in $\mathbb{A}$. In this paper, we show that a large class of arithmetic functions in…

We evaluate $q$-Bessel functions at an infinite sequence of points and introduce a generalization of the Ramanujan function and give an extension of the $m$-version of the Rogers-Ramanujan identities. We also prove several generating…

One method for deriving a factorization for QCD processes is to use successive integration over fields in the functional integral. In this approach, we separate the fields into two categories: dynamical fields with momenta above a relevant…

Bayesian probabilistic numerical methods are a set of tools providing posterior distributions on the output of numerical methods. The use of these methods is usually motivated by the fact that they can represent our uncertainty due to…

Strict partitions are enumerated with respect to the weight, the number of parts, and the number of sequences of odd length. We write this trivariate generating function as a double sum $q$-series. Equipped with such a combinatorial set-up,…