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In this note, the author proves that sums of powers of the first n positive integers can be expressed as finite discrete convolutions.Read more
A finite discrete convolution involving the Jacobi-Stirling numbers of both kinds is expressed in this paper in terms of the Bernoulli polynomials.Read more
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.Read more
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.Read more
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.Read more
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.Read more
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.Read more