Related papers: Another proof of Bailey's 6-psi-6 summation
We continue the analysis of higher and multiple Mahler measures using log-sine integrals as started in "Log-sine evaluations of Mahler measures" and "Special values of generalized log-sine integrals" by two of the authors. This motivates a…
We augment the method of Wooley (2015) by some new ideas and in a series of results, improve his metric bounds on the Weyl sums and the discrepancy of fractional parts of real polynomials with partially prescribed coefficients. We also…
The equation commonly known as Sury's identity is a deceptively simple summation formula that connects the Lucas numbers, Fibonacci numbers, and powers of two. Many authors have given extensions and generalizations over the years; in this…
A generalised summation method is considered based on the Fourier series of periodic distributions. It is shown that $$ e^{it}-2e^{2it}+3e^{3it}-4e^{4it}+-\cdots = {\mathrm P\mathrm f} {\displaystyle \frac{e^{it}}{(1+e^{it})^2}} +i\pi…
We use Bailey pairs to prove $q$-series identities at roots of unity due to Cohen and Bryson-Ono-Pitman-Rhoades. The proofs use Bailey pairs with quadratic forms developed in the study of mock theta functions. In addition to the standard…
We sketch several proofs of F\'ary--Milnor theorem.
The main results extend to sums over primes in a short interval earlier estimates by the author for "long" Weyl sums over primes.
In this paper we prove some interpolation theorems for the multipliers of the Cauchy- Stiltjes type integrals
New lower bounds involving sum, difference, product, and ratio sets for a set $A\subset \C$ are given. The estimates involving the sum set match, up to constants, the one obtained by Solymosi for the reals and are obtained by generalising…
Following the Mellin and inverse Mellin transform techniques presented in our paper arXiv:1606.02150 (NT), we have established close forms of Laurent series expansions of products of bi- and trigamma functions /psi(z)*/psi(-z) and…
A flexible representation of uncertainty that remains within the standard framework of probabilistic measure theory is presented along with a study of its properties. This representation relies on a specific type of outer measure that is…
Some new sufficient conditions for the weighted Chebyshev's inequality for real numbers to hold are provided.
We generalize the method of combinatorial telescoping to the case of multiple summations. We shall demonstrate this idea by giving combinatorial proofs for two identities of Andrews on parity indices of partitions.
Several new identities for elliptic hypergeometric series are proved. Remarkably, some of these are elliptic analogues of identities for basic hypergeometric series that are balanced but not very-well-poised.
We give a new proof of Lucas' Theorem in elementary number theory.
In this article, a short combinatorial proof of the Capelli's identity is given. It also leads to an easy proof of the Capelli--Cauchy--Binet identity, a more general form of Capelli's identity. With the technique introduced, the Turnbull's…
The paper presents several combinatorial properties of the boolean cumulants. A corollary is a new proof of the multiplicative property of the boolean cumulant series that can be easily adapted for the case of boolean independence with…
We propose a novel approach to Bayesian analysis that is provably robust to outliers in the data and often has computational advantages over standard methods. Our technique is based on splitting the data into non-overlapping subgroups,…
We prove some new results related to Tanaka's formula.
First we introduce the Bailleul-Hoshino's result [4], which links the theory of regularity structures and the paracontrolled calculus. As an application of their result, we give another algebraic proof of the multicomponent commutator…