Related papers: Fa\`a di Bruno's note on eponymous formula, trilin…
In this note we explore the relationship between the operation of convolution of functions and the Eulerian integrals. This approach allow us to obtain some expressions for the convolution of a certain class of functions in terms of the…
The Fibonacci polynomials are defined recursively as $f_{n}(x)=xf_{n-1}(x)+f_{n-2}(x)$, where $f_0(x) = 0$ and $f_1(x)= 1$. We generalize these polynomials to an arbitrary number of variables with the $r$-Fibonacci polynomial. We extend…
Roman logarithmic binomial formula analogue has been found . It is presented here also for the case of fibonomial coefficients which recently have been given a combinatorial interpretation by the present author.
The aim of this note is to survey the factorizations of the Fibonacci infinite word that make use of the Fibonacci words and other related words, and to show that all these factorizations can be easily derived in sequence starting from…
A sequence of rational numbers as a generalization of the sequence of Bernoulli numbers is introduced. Sums of products involving the terms of this generalized sequence are then obtained using an application of the Fa\`a di Bruno's formula.…
Using both fractional derivatives, defined in the Riemann-Liouville and Caputo senses, and classical derivatives of the integer order we examine different numerical approaches to ordinary differential equations. Generally we formulate some…
A lemniscate is a curve defined by two foci, F1 and F2. If the distance between the focal points of F1 - F2 is 2a (a: constant), then any point P on the lemniscate curve satisfy the equation PF1 PF2 = a^2. Jacob Bernoulli first described…
In this paper we use the viewpoint of the formal calculus underlying vertex operator algebra theory to study certain aspects of the classical umbral calculus and we introduce and study certain operators generalizing the classical umbral…
We introduce a new numerical method, based on Bernoulli polynomials, for solving multiterm variable-order fractional differential equations. The variable-order fractional derivative was considered in the Caputo sense, while the…
The concept of fractional order derivative can be found in extensive range of many different subject areas. For this reason, the concept of fractional order derivative should be examined. After giving different methods mostly used in…
The trial and condemnation of Giordano Bruno was mainly based on arguments of philosophical and theological nature, and therefore different from Galilei's. Such elements contribute to unfairly devalue the scientific contribution of Bruno…
In this paper I discuss Enrico Fermi's view of identical particles, taking a lecture that he gave in 1933 as a starting point. Fermi used his lecture as a basis for a paper that was published in 1934: the paper is in italian and is not…
This is an English translation and digitisation of Frobenius' and Stickelberger's "On the theory of elliptic functions" first published in Journal fur die reine und angewandte Mathematik (Crelle's journal), 83, 175-179 (1877) with the title…
In this paper, we show that a partitioned formula \phi is dependent if and only if \phi has uniform definability of types over finite partial order indiscernibles. This generalizes our result from a previous paper [1]. We show this by…
Double Fibonacci sequences are introduced and they are related to operations with Fibonacci modules. Generalizations and examples are also discussed.
The details for the construction of an explicit formula for the divisors function d(n) = #{d | n} are formalized in this article. This formula facilitates a unified approach to the investigation of the error terms of the divisor problem and…
Connected the generalized Goncharov polynomials associated to a pair ($\partial,\mathcal{Z}$) if a delta operator $\partial$ and an interpolation grid $\mathcal{Z}$, introduced by Lorentz, Tringali and Yan in [7], with the theory of…
The purpose of the present paper is to introduce a new subclasses of the function class of bi-univalent functions defined in the open unit disc. Furthermore, we obtain estimates on the coefficients $|a_{2}|$ and $|a_{3}|$ for functions of…
We outline a proof of a remarkable formula for Hodge integrals conjectured by Marino and Vafa in hep-th/0108064 base on large N duality.
In this paper, we apply high level versions of Jacobi's derivative formula to number theory such as quarternary quadratic forms and convolution sums of some arithmetical functions.