Related papers: Transcendence of generalized Euler-Kronecker const…
In the present article, we study Bell based Euler polynomial of order {\alpha} and investigate some useful correlation formula, summation formula and derivative formula. Also, we introduce some relation of string number of the second kind.…
We define a generalization of the Eulerian polynomials and the Eulerian numbers by considering a descent statistic on segmented permutations coming from the study of 2-species exclusion processes and a change of basis in a Hopf algebra. We…
In this paper, we introduce the hypergeometric Euler number as an analogue of the hypergeometric Bernoulli number and the hypergeometric Cauchy number. We study several expressions and sums of products of hypergeometric Euler numbers. We…
We give an elementary approach to studying whether rings of $S$-integers in complex quadratic fields are Euclidean with respect to the $S$-norm.
A generalization of L{\"u}roth's theorem expresses that every transcendence degree 1 subfield of the rational function field is a simple extension. In this note we show that a classical proof of this theorem also holds to prove this…
By using the elementary symmetric polynomials and some results of number theory, we solve the well known problem of Lehmer on Euler's totient function. As application, we obtain a new characterization of prime numbers.
We study asymptotics of various Euclidean geometric phenomena as the dimension tend to infinity.
We consider a generalized angle in complex normed vector spaces. Its definition corresponds to the definition of the well known Euclidean angle in real inner product spaces. Not surprisingly it yields complex values as `angles'. This…
The Eulerian numbers count permutations according to the number of descents. The two-sided Eulerian numbers count permutations according to number of descents and the number of descents in the inverse permutation. Here we derive some…
The infinite dimensional generalization of the quantum mechanics of extended objects, namely, the quantum field theory of extended objects is presented. The paradigm example studied in this paper is the Euclidean scalar field with a…
In this paper we study a general class of conics starting from a quotient field. We give a group structure over these conics generalizing the construction of a group over the Pell hyperbola. Furthermore, we generalize the definition of…
A general method for proving continuity of the von Neumann entropy on subsets of positive trace-class operators is considered. This makes it possible to re-derive the known conditions for continuity of the entropy in more general forms and…
Recently Raayoni et al. announced various conjectures on continued fractions of fundamental constants automatically generated with machine learning techniques. In this paper we prove some of their stated conjectures for Euler number $e$ and…
This paper, along with E592 and E636, seems to consider the binomial expansion (1+z)^n in the case where z is complex. Euler even gives the sums of divergent series. The paper is translated from Euler's Latin original into German.
We give a logically and mathematically self-consistent procedure of quantization of free scalar field, including quantization on space-like surfaces. A short discussion of possible generalization to interacting fields is added.
We use Euler systems to prove the Gras conjecture for groups generated by Stark units in global function fields. The techniques applied here are classical and go back to Thaine, Kolyvagin and Rubin. We obtain our Euler systems from the…
The paper studies some properties of the ring of integer-valued quasi-polynomials. On this ring, theory of generalized Euclidean division and generalized GCD are presented. Applications to finite simple continued fraction expansion and…
In this note, we study the arithmetic nature of values of modular functions, meromorphic modular forms and meromorphic quasi-modular forms with respect to arbitrary congruence subgroups, that have algebraic Fourier coefficients. This…
Ramanujan studied a general class of Stirling constants that are the resummation of some natural divergent series. These constants include the classical Euler-Mascheroni, Stirling and Glaisher-Kinkelin constants. We find natural integral…
We make some remarks about the relation between the Kronecker limit formula for totally real fields, (normalized) periods of CM abelian varieties, and units in abelian extensions of CM fields.