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We develop a compositional approach for automatic and symbolic differentiation based on categorical constructions in functional analysis where derivatives are linear functions on abstract vectors rather than being limited to scalars,…

In this paper, we discuss sums of powers of the positive integers and compute both the exponential and ordinary generating functions for these sums. We express these generating functions in terms of exponential and geometric polynomials and…

We present the transformation of several sums of positive integer powers of the sine and cosine into non-trigonometric combinatorial forms. The results are applied to the derivation of generating functions and to the number of the closed…

In this article, we give a formula for the generalization of the binomial coefficient to the complex numbers as a linear combination of $\sinc$ functions. We then give a general formula to compute the integral on the real line of the…

This paper develops a generalized cotangent-type series, extending classical expansions to higher-order lattice sums. By introducing a new family of series indexed by integer powers, we derive closed form representations that combine…

Permutations can be represented as linear combinations of natural numbers with different powers. In this paper, its coefficient matrix and inverse matrix is derived, and the results show the coefficient matrix is a lower triangular matrix…

This is the first in a set of three papers providing an introduction to generalised Cesaro convergence. We start with traditional Cesaro methods for extending classical convergence and further generalise these to allow the calculation of…

A big class of Feynman integrals, in particular, the coefficients of their Laurent series expansion w.r.t.\ the dimension parameter $\ep$ can be transformed to multi-sums over hypergeometric terms and harmonic sums. In this article, we…

We define base-extension semantics (Bes) using atomic systems based on sequent calculus rather than natural deduction. While traditional Bes aligns naturally with intuitionistic logic due to its constructive foundations, we show that…

We develop a theory of sesquilinear forms over finite fields, investigating their representations via polynomials and coefficient matrices, along with classification results for these forms. Through their connection to quadratic forms, we…

Regular expressions are widely used in software. Various regular expression engines support different combinations of extensions to classical regular constructs such as Kleene star, concatenation, nondeterministic choice (union in terms of…

Spectral decomposition of matrices is a recurring and important task in applied mathematics, physics and engineering. Many application problems require the consideration of matrices of size three with spectral decomposition over the real…

This paper presents a pure neural solver for arithmetic expression calculation (AEC) problem. Previous work utilizes the powerful capabilities of deep neural networks and attempts to build an end-to-end model to solve this problem. However,…

We consider here a particular quadratic equation linking two elements of a C-Algebra. By analysing powers of the unknowns, it appears a double sequence of polynomials related to classical Bernoulli polynomials. We get the generating…

Diagrams as a graphic expresion of derivatives is proposed for calculation of derivatives for composed function. The concret diagram is understood as a virtual derivative in contrast of concret derivative. In polynomial expression of…

This is a collection of definitions, notations and proofs for the Bernoulli numbers $B_n$ appearing in formulas for the sum of integer powers, some of which can be found scattered in the large related historical literature in French,…

The Euler-Maclaurin summation formula is generalized to a modified form by expanding the periodic Bernoulli polynomials as its Fourier series and taking cuts, which includes both the Euler-Maclaurin summation formula and the Poission…

We investigate the representation of symmetric polynomials as a sum of squares. Since this task is solved using semidefinite programming tools we explore the geometric, algebraic, and computational implications of the presence of discrete…

An overlooked formula of E. Lucas for the generalized Bernoulli numbers is proved using generating functions. This is then used to provide a new proof and a new form of a sum involving classical Bernoulli numbers studied by K. Dilcher. The…

We describe a neural-based method for generating exact or approximate solutions to differential equations in the form of mathematical expressions. Unlike other neural methods, our system returns symbolic expressions that can be interpreted…