Related papers: The Relationship Between Euler Numbers and Bernoul…

We prove certain identities involving Euler and Bernoulli polynomials that can be treated as recurrences. We use these and also other known identities to indicate connection of Euler and Bernoulli numbers with entries of inverses of certain…

The purpose of this article is to introduce q-deformed Stirling numbers of the first and second kinds. Relations between these numbers, Riemann zeta function and q-Bernoulli numbers of higher order are given. Some relations related to the…

The generalized hyperharmonic numbers $h_n^{(m)}(k)$ are defined by means of the multiple harmonic numbers. We show that the hyperharmonic numbers $h_n^{(m)}(k)$ satisfy certain recurrence relation which allow us to write them in terms of…

In this paper, we introduced some notions on the n-Normed Spaces. Those are bounded k-linear (or multilinear) functionals and k-continuous (or multicontinuous) functions with k \in \mathbb{N}. We defined k-linear functionals under several…

The n-th Delannoy number and the n-th Schr\"oder number given by $D_n=\sum_{k=0}^n\binom{n}{k}\binom{n+k}{k}$ and $S_n=\sum_{k=0}^n\binom{n}{k}\binom{n+k}{k}/(k+1)$ respectively arise naturally from enumerative combinatorics. Let p be an…

Let $B_{n}(t)$ be a $n$-th Stern polynomial and let $e(n)=\op{deg}B_{n}(t)$ be its degree. In this note we continue our study started in \cite{Ul} of the arithmetic properties of the sequence of Stern polynomials and the sequence…

Let $n$ be a nonnegative integer. The $n$-th Ap\'{e}ry number is defined by $$ A_n:=\sum_{k=0}^n\binom{n+k}{k}^2\binom{n}{k}^2. $$ Z.-W. Sun ever investigated the congruence properties of Ap\'{e}ry numbers and posed some conjectures. For…

We relate two fundamental enumerative functions, namely the $I$-functions in the quantum $K$-ring of $G(r,n)$ and of its cotangent bundle, by defining a $K$-theoretic operator on classes, called balancing. This operator lifts the…

Let $\{E_n\}$ be the Euler numbers. In the paper we determine $E_{2^mk+b}-E_b$ modulo $2^{m+7}$, where $k$ and $m$ are positive integers and $b\in{0,2,4,...}$.

For the Stirling numbers of the second kind $S(n,k)$ and the ordered Bell numbers $B(n)$, we prove the identity $\sum_{k=1}^{n/2} S(n,2k)(2k-1)! = B(n-1)$. An analogous identity holds for the sum over odd $k$'s.

The partial Stirling numbers T_n(k) used here are defined as the sum over odd values of i of (n choose i) i^k. Their 2-exponents nu(T_n(k)) are important in algebraic topology. We provide many specific results, applying to all values of n,…

The Euler numbers have been widely studied. A signed version of the Euler numbers of even subscript are given by the coefficients of the exponential generating function 1/(1+x^2/2!+x^4/4!+...). Leeming and MacLeod introduced a…

In this paper, we investigate decompositions of the partition function $p(n)$ from the additive theory of partitions considering the famous M\"{o}bius function $\mu(n)$ from multiplicative number theory. Some combinatorial interpretations…

In this paper we study recurrences concerning the combinatorial sum $[n,r]_m=\sum_{k\equiv r (mod m)}\binom {n}{k}$ and the alternate sum $\sum_{k\equiv r (mod m)}(-1)^{(k-r)/m}\binom{n}{k}$, where m>0, $n\ge 0$ and r are integers. For…

Following the ideas of L. Carlitz we introduce a generalization of the Bernoulli and Eulerian polynomials of higher order to vectorial index and argument. These polynomials are used for computation of the vector partition function $W({\bf…

The $k$-arrangements are permutations whose fixed points are $k$-colored. We prove enumerative results related to statistics and patterns on $k$-arrangements, confirming several conjectures by Blitvi\'c and Steingr\'imsson. In particular,…

We give a complete classification of the factorial functions of Eulerian binomial posets. The factorial function B(n) either coincides with $n!$, the factorial function of the infinite Boolean algebra, or $2^{n-1}$, the factorial function…

Consider generalizations of the boundary singularities B_n of the functions on the real line to the case where the boundary consists of a finite number of l points. These singularities B_n^l could also arise in higher dimensional case, when…

I solve here a question of Vladimir Reshetnikov in Mathoverflow (question 261649) about the values of Fabius function. Namely, I prove that the numbers $R_n:=2^{-\binom{n-1}{2}}(2n)! F(2^{-n})\prod_{m=1}^{\lfloor n/2\rfloor}(2^{2m}-1)$ are…

The main aim of this paper is to investigate and introduce relations between the numbers of k-ary Lyndon words and unified zeta-type functions which was defined by Ozden et al [15, p. 2785]. Finally, we give some identities on generating…