Purpose: The purpose of this paper is to develop a set of identities for Euler type sums of products of harmonic numbers and reciprocal binomial coefficients.Method: We use analytical methods to obtain our results.Results: We obtain identities for variant Euler sums of the type ¥Sn=1H2n/n (n+k)k, and its finite counterpart, which generalize some results obtained by other authors.Conclusions: Identities are successfully achieved for the sums under investigation. Some published results have been successfully generalized.

In the paper, the authors view some ordinary differential equations and their solutions from the angle of (generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius– Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds.

Exponential partial Bell polynomials are a main tool for computing explicit formula for a large family of numbers and polynomials. Using these polynomials, we review most of the polynomials already studied in the book of Djordjevic-Milovanovic (2014). This study allows to unify the writing of these polynomials and obtain other extensions. Consequently the obtained results find their application in Number Theory,notably for the arithmetical properties of power products, Fermat numbers, Mersenne numbers and Fermat Last Theorem.

Zernike circle polynomials use for wavefront analysis because of their orthogonality over a circular. ‏‎ So far, different representations have been presented for these polynomials. In this paper, ‎ ‎‎we introduce Zernike polynomials and their important properties. Also, we present some other orthogonal polynomials such ‎as‎ Bessel ‎functions, ‎ Legendre polynomials and ‎ Jaccobi polynomials ‎and ‎give some important results of them‏. ‎‎ Finally, we express the relationship between Zernike orthogonal polynomials and these well-known orthogonal polynomials. ‏‎ For different cases, the ‎radial‎ polynomials in the Zernike functions are obtained using the well-known orthogonal polynomials‏. These relations frequently used for representing‎ wavefront ‎aberrations.

We study the calculus of variations problem in the presence of a system of differential-integral (D-I) equations. In order to identify the necessary optimality conditions for this problem, we derive the so-called D-I Euler–Lagrange equations. We also generalize this problem to other cases, such as the case of higher orders, the problem of optimal control, and we derive the so-called D-I Pontryagin equations. In special cases, these formulations lead to classical Euler–Lagrange equations. To illustrate our results, we provide simple examples and applications such as obtaining the minimumpower for an RLC circuit.