A Bijection for the Boolean Numbers of Ferrers Graphs

In this note we prove a new characterization of the derangement sets of Ferrers graphs and present a bijection between the derangement sets and $$F_{\lambda }$$ F λ -Callan sequences. In particular, this connection reveals that the boolean numbers of the complete bipartite graphs are the D-relatives of poly-Bernoulli numbers.

Explicit formulas for Euler polynomials and Bernoulli numbers

In this paper, we give several explicit formulas involving the n-th Euler polynomial E_{n}\left(x\right). For any fixed integer m\geq n, the obtained formulas follow by proving that E_{n}\left(x\right) can be written as a linear combination of the polynomials x^{n}, \left(x+r\right)^{n},\ldots, \left(x+rm\right)^{n}, with r\in \left \{1,-1,\frac{1}{2}\right\}. As consequence, some explicit formulas for Bernoulli numbers may be deduced.

A few remarks on the values of the Bernoulli polynomials at rational arguments and some relations with ζ(2k + 1)

A problem posed by Lehmer in 1938 asks for simple closed formulae for the values of the even Bernoulli polynomials at rational arguments in terms of the Bernoulli numbers. We discuss this problem based on the Fourier expansion of the Bernoulli polynomials. We also give some necessary and sufficient conditions for ζ(2k + 1) be a rational multiple of π2k+1.

A simple proof of Linas’s theorem on Riemann zeta function

Linas Vepštas gives rapidly converging infinite representatives for values of Riemann zeta function at \left(4m-1 \right), where m is a natural number. In this paper, we give a new simple proof. Also, we obtain two equation of values of Bernoulli numbers’ generating function by applying a corollary given in this paper.

Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions

Recent advances in mathematical inequalities suggest that bounds of polynomial-exponential-type are appropriate for evaluating key trigonometric functions. In this paper, we innovate in this sense by establishing new and sharp bounds of the form (1−αx2)eβx2 for the trigonometric sinc and cosine functions. Our main result for the sinc function is a double inequality holding on the interval (0, π), while our main result for the cosine function is a double inequality holding on the interval (0, π/2). Comparable sharp results for hyperbolic functions are also obtained. The proofs are based on series expansions, inequalities on the Bernoulli numbers, and the monotone form of the l’Hospital rule. Some comparable bounds of the literature are improved. Examples of application via integral techniques are given.

Upper bounds for analytic summand functions and related inequalities

The topic of analytic summability of functions was introduced and studied in 2016 by Hooshmand. He presented some inequalities and upper bounds for analytic summand functions by applying Bernoulli polynomials and numbers. In this work we apply upper bounds, represented by Hua-feng, for Bernoulli numbers to improve the inequalities and related results. Then, we observe that the inequalities are sharp and leave a conjecture about them. Also, as some applications, we use them for some special functions and obtain many particular inequalities. Moreover, we arrived at the inequality 1 p + 2 p + 3 p + ⋯ + r p ≤ 1 2 r p + 1 3 r p + 1 ( p + 1 ) + 2 3 p ! π p + 1 sinh ⁡ ( π r ) $1^p + 2^p + 3^p + \dots + r^p \leq \frac{1}{2}r^p + \frac{1}{3}\frac{r^{p+1}}{(p+1)} + \frac{2}{3}\frac{p!}{\pi^{p+1}}\sinh(\pi r)$ , for r sums of power of natural numbers, if p ∈ ℕ e and analogously for the odd case.

Analytical properties of type 2 degenerate poly-Bernoulli polynomials associated with their applications

Recently, Kim et al. (Adv. Differ. Equ. 2020:168, 2020) considered the poly-Bernoulli numbers and polynomials resulting from the moderated version of degenerate polyexponential functions. In this paper, we investigate the degenerate type 2 poly-Bernoulli numbers and polynomials which are derived from the moderated version of degenerate polyexponential functions. Our degenerate type 2 degenerate poly-Bernoulli numbers and polynomials are different from those of Kim et al. (Adv. Differ. Equ. 2020:168, 2020) and Kim and Kim (Russ. J. Math. Phys. 26(1):40–49, 2019). Utilizing the properties of moderated degenerate poly-exponential function, we explore some properties of our type 2 degenerate poly-Bernoulli numbers and polynomials. From our investigation, we derive some explicit expressions for type 2 degenerate poly-Bernoulli numbers and polynomials. In addition, we also scrutinize type 2 degenerate unipoly-Bernoulli polynomials related to an arithmetic function and investigate some identities for those polynomials. In particular, we consider certain new explicit expressions and relations of type 2 degenerate unipoly-Bernoulli polynomials and numbers related to special numbers and polynomials. Further, some related beautiful zeros and graphical representations are displayed with the help of Mathematica.

On a half-discrete Hilbert-type inequality related to hyperbolic functions

By the introduction of a new half-discrete kernel which is composed of several exponent functions, and using the method of weight coefficient, a Hilbert-type inequality and its equivalent forms involving multiple parameters are established. In addition, it is proved that the constant factors of the newly obtained inequalities are the best possible. Furthermore, by the use of the rational fraction expansion of the tangent function and introducing the Bernoulli numbers, some interesting and special half-discrete Hilbert-type inequalities are presented at the end of the paper.

Multi-Lah numbers and multi-Stirling numbers of the first kind

In this paper, we introduce multi-Lah numbers and multi-Stirling numbers of the first kind and recall multi-Bernoulli numbers, all of whose generating functions are given with the help of multiple logarithm. The aim of this paper is to study several relations among those three kinds of numbers. In more detail, we represent the multi-Bernoulli numbers in terms of the multi-Stirling numbers of the first kind and vice versa, and the multi-Lah numbers in terms of multi-Stirling numbers. In addition, we deduce a recurrence relation for multi-Lah numbers.

Fibonacci series from power series

We show how every power series gives rise to a Fibonacci series and a companion series involving Lucas numbers. For illustrative purposes, Fibonacci series arising from trigonometric functions, the gamma function and the digamma function are derived. Infinite series involving Fibonacci and Bernoulli numbers and Fibonacci and Euler numbers are also obtained.