Roudy El Haddad

Sorbonne Université | UPMC · Campus UPMC

Repeated integration is a major topic of integral calculus. In this article, we study repeated integration. In particular, we study repeated integrals and recurrent integrals. For each of these integrals, we develop reduction formulae for both the definite as well as indefinite form. These reduction formulae express these repetitive integrals in te...

This manuscript presents and details all stages of developing, simulating, and controlling a mobile robot using ROS and Gazebo. The codes and documentation are done on ROS melodic and Gazebo version 1.9.

Multiple zeta values have become of great interest due to their numerous applications in mathematics and physics. In this article, we present a generalization, which we will refer to as multiple sums, where the reciprocals are replaced with arbitrary sequences. We develop formulae to help with manipulating such sums. We develop variation formulae t...

Multiple zeta star values have become a central concept in number theory with a wide variety of applications. In this article, we propose a generalization, which we will refer to as recurrent sums, where the reciprocals are replaced by arbitrary sequences. We introduce a toolbox of formulas for the manipulation of such sums. We begin by developing...

All OEIS sequence added or edited by Roudy El Haddad.

Binomial coefficients have been used for centuries in a variety of fields and have accumulated numerous definitions. In this paper, we introduce a new way of defining binomial coefficients as repeated sums of ones. A multitude of binomial coefficient identities will be shown in order to prove this definition. Using this new definition, we simplify...

Leibniz's rule for the $n$-th derivative of a product is a very well known and extremely useful formula. Unfortunately, currently, there does not exist a similar explicit formula for the $n$-th derivative of a quotient. In this article, we introduce an explicit formula for the $n$-th derivative of a quotient of two functions. Later, we use this for...

Repeated integration is a major topic of multivariable calculus. In this article, we study repeated integration. In particular, we study repeated integrals and recurrent integrals. For each of these integrals, we develop reduction formulae for both the definite as well as indefinite form. These reduction formulae express these repetitive integrals...

Binomial coefficients have been used for centuries in a variety of fields and have accumulated numerous definitions. In this paper, we introduce a new way of defining binomial coefficients as repeated sums of ones. A multitude of binomial coefficient identities will be shown in order to prove this definition. Using this new definition, we simplify...

Sums of the form $\sum_{q \leq N_1 < \cdots < N_m \leq n}{a_{(m);N_m}\cdots a_{(2);N_2}a_{(1);N_1}}$ date back to the sixteen century when Vi\`ete illustrated that the relation linking the roots and coefficients of a polynomial had this form. In more recent years, such sums have become increasingly used with a diversity of applications. In this pap...

Sums of the form $\sum_{N_m=q}^{n}{\cdots \sum_{N_1=q}^{N_2}{a_{(m);N_m}\cdots a_{(1);N_1}}}$ where the $a_{(k);N_k}$'s are same or distinct sequences appear quite often in mathematics. We will refer to them as recurrent sums. In this paper, we introduce a variety of formulas to help manipulate and work with this type of sums. We begin by developin...