Related papers: Recurrence relations for harmonic and derangement …
Explicit formulas for the solutions of the recurrence relations for 3--loop vacuum integrals are suggested. This formulas can be used for direct calculations and demonstrate a high efficiency. They also produce a new type of recurrence…
We formalize the general principle of significance with respect to binary relations which is a universal tool for description and analysis of various situations in and apart from mathematics. We derive the basic properties and focus on a…
Lanczos-type algorithms are efficient and easy to implement. Unfortunately they breakdown frequently and well before convergence has been achieved. These algorithms are typically based on recurrence relations which involve formal orthogonal…
The notion of a mesh pattern was introduced recently, but it has already proved to be a useful tool for description purposes related to sets of permutations. In this paper we study eight mesh patterns of small lengths. In particular, we…
In this short note, we provide an elementary complex analytic method for converting known real integrals into numerous strange and interesting looking real integrals.
We show some of the mathematics that is being developed for the computation of deep inelastic structure functions to three loops. These include harmonic sums, harmonic polylogarithms and a class of difference equations that can be solved…
By means of the generating function method, a linear recurrence relation is explicitly resolved. The solution is expressed in terms of the Stirling numbers of both the first and the second kind. Two remarkable pairs of combinatorial…
We present new proofs for some summation identities involving Stirling numbers of both first and second kind. The two main identities show a connection between Stirling numbers and Bessel numbers. Our method is based on solving a particular…
Two-term recurrence relations are supplied for indefinite integrals of functions that involve factors of the types ${P_2}^n$, ${P_3}^n$, ${P_4}^n$, ${P_1}^m {Q_1}^n$, $E_1 {P_1}^n$, ${P_1}^m {Q_2}^n$, $E_1 {P_2}^n$, ${P_2}^m {Q_2}^n$,…
Number sequences defined by a linear recursion relation are studied by means of generating functions. Indices of the terms in the recursion relation have arbitrary differenses. In addition to formulas for the nth term an algorithm is…
We give some results and conjectures about recurrence relations for certain sequences of binomial sums.
The degenerate Stirling numbers of the second kind and of the first kind, which are respectively degenerate versions of the Stirling numbers of the second kind and of the first kind, appear frequently when we study various degenerate…
We establish three-term recurrence relations for the ${}_1\phi_1$ and ${}_0\phi_1$ basic hypergeometric series involving multiplicative shifts of the parameters and the variable by integer powers of q. The coefficients of these recurrence…
In this work we develop an algebraic theory of linear recurrence equations and systems with constant coefficients and reflection. We obtain explicit solutions and the Green's functions associated to different problems under general linear…
We establish supercongruences for two kinds of Ap\'ery-like numbers, which involve Bernoulli numbers and Bernoulli polynomials. Conjectural supercongruences of the same type for another four kinds of Ap\'ery-like numbers are also proposed.
Metric embedding is a powerful tool used extensively in mathematics and computer science. We devise a new method of using metric embeddings recursively, which turns out to be particularly effective in $\ell_p$ spaces, $p>2$, yielding…
In this paper, we study the combinatorial structures of straight and ordinary m\'enage permutations. Based on these structures, we prove four formulas. The first two formulas define a relationship between the m\'enage numbers and the…
We develop a new approach to recurrence and the existence of non-constant harmonic functions on infinite weighted graphs. The approach is based on the capacity of subsets of metric boundaries with respect to intrinsic metrics. The main tool…
In this paper we introduce and experimentally compare alternative algorithms to join uncertain relations. Different algorithms are based on specific principles, e.g., sorting, indexing, or building intermediate relational tables to apply…
This short course offers a new perspective on randomized algorithms for matrix computations. It explores the distinct ways in which probability can be used to design algorithms for numerical linear algebra. Each design template is…