In this note, the author proves that sums of powers of the first n positive integers can be expressed as finite discrete convolutions.

A finite discrete convolution involving the Jacobi-Stirling numbers of both kinds is expressed in this paper in terms of the Bernoulli polynomials.

The q-binomial coefficients are specializations of the elementary symmetric functions. In this paper, we use this fact to give a new expression for the generating function of the number of divisors. As corollaries, we obtained new connections between partitions and divisors.

The r-Whitney numbers of both kinds are specializations of complete and elementary symmetric functions. In this paper, we use this fact to express a finite discrete convolution involving r-Whitney numbers of both kinds in terms of Bernoulli polynomials.

The complete and elementary symmetric functions are special cases of Schur functions. It is well-known that the Schur functions can be expressed in terms of complete or elementary symmetric functions using two determinant formulas: Jacobi–Trudi identity and Nägelsbach–Kostka identity. In this paper, we study new connections between complete and elementary symmetric functions.

Binomial coefficients can be expressed in terms of multinomial coefficients as sums over integer partitions. This approach allows us to introduce new upper bounds for the number of partitions into a given number of parts.

A very special case of a Ramus’s identity is used in this note to derive an infinite family of inequalities involving finite sums with cosecant function. As a corollary of this result, we obtain the Wallis's formula.