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We investigate a connection between generalized Fibonacci numbers and renewal theory for stochastic processes. Using Blackwell's renewal theorem we find an approximation to the generalized Fibonacci numbers. With the help of error estimates…

In this paper, we study various classes of partition functions such as those related to the parity of the number of parts, to differences of partition numbers, and to partitions with a repeated smallest part. We establish identities…

For a given length and a given degree and an arbitrary partition of the positive integers, there always is a cell containing a polynomial progression of that length and that degree; moreover, the coefficients of the generating polynomial…

We establish a new generalized Taylor's formula for power fractional derivatives with nonsingular and nonlocal kernels, which includes many known Taylor's formulas in the literature. Moreover, as a consequence, we obtain a general version…

We derive a formula for $p(n)$ (the number of partitions of $n$) in terms of the partial Bell polynomials using Fa\`{a} di Bruno's formula and Euler's pentagonal number theorem.

We evaluate binomial series with harmonic number coefficients, providing recursion relations, integral representations, and several examples. The results are of interest to analytic number theory, the analysis of algorithms, and…

We formally prove the existence of an enduring incongruence pervading a widespread interpretation of the Bell inequality and explain how to rationally avoid it with a natural assumption justified by explicit reference to a mathematical…

Set partitions and permutations with restrictions on the size of the blocks and cycles are important combinatorial sequences. Counting these objects lead to the sequences generalizing the classical Stirling and Bell numbers. The main focus…

We study a relation between roots of characteristic polynomials and intersection points of line arrangements. Using these results, we obtain a lot of applications for line arrangements. Namely, we give (i) a generalized addition theorem for…

We discuss various universality aspects of numerical computations using standard algorithms. These aspects include empirical observations and rigorous results. We also make various speculations about computation in a broader sense.

The law of large numbers for the empirical density for the pairs of uniformly distributed integers with a given greatest common divisor is a classic result in number theory. In this paper, we study the large deviations of the empirical…

We give new identities for some symmetric polynomials. As applications of these identities, we obtain some formulas for a higher order analogue of Fibonacci and Lucas numbers.

We introduce a novel generalization of deranged Bell numbers by defining the partial deranged Bell numbers $w_{n,r}$, which count the number of set partitions of $\left[ n\right] $ with exactly $r$ fixed blocks, while the remaining blocks…

We study the problem of generalization of Oresme numbers with a new sequence of numbers called Oresme polynomials. Moreover, by using the matrix methods for Oresme polynomials, we obtain the identities including the general bilinear…

The aim of this paper is twofold. Firstly, we investigate a finite sum involving the generalized falling factorial polynomials, in some special cases of which we express it in terms of the degenerate Stirling numbers of the second kind, the…

The present research deals with generalizations of the Salem function with arguments defined in terms of certain alternating expansions of real numbers. The special attention is given to modelling such functions by systems of functional…

I recent years, studying degenerate versions of some special polynomials, which was initiated by Carlitz in an investigation of the degenerate Bernoulli and Euler polynomials, regained lively interest of mant mathematicains. In this paper,…

In this paper we derive congruences expressing Bell numbers and derangement numbers in terms of each other modulo any prime.

In 2008, Spivey found a recurrence relation for the Bell numbers. We consider the probabilistic r-Bell polynomials associated with which are a probabilistic extension of the r-Bell polynomials. Here Y is a random variable whose moment…

In this short note, we establish some identities containing sums of binomials with coefficients satisfying third order linear recursive relations. As a result and in particular, we obtain general forms of earlier identities involving…