Using probability theory we derive an expression for the sum of a series of definite integrals involving upper incomplete Gamma functions. In the proof, a normal variance mixture distribution with Beta mixing distributions plays a crucial role. We also give an interesting application of our result, namely, a new summation formula for some derivatives of the Bessel functions of the first kind and the Struve functions with respect to the order.

The exp _q(z) function is the standard q-analogue of the exponential. Since not much is known about this function, our aim is to give a contribution to the knowledge on exp _q . After proving some simpler but new relations for it, we make a complete description of the inverse map of exp _q(z) , including its branch structure and Riemann surface.

Motivated by the work of David Singmaster, we study the number of times an integer can appear among the Stirling numbers of both kinds. We provide an upper bound for the occurrences of all the positive integers, and present certain questions for further study. Some numerical results and conjectures concerning the related diohantine equations are collected.

In this paper, we equip the partition lattice of the positive integers with a metric structure, and study the properties of this metric. We show, that this new space is complete, compact and separable. We also investigate how continuous functions look like in this structure.

The exponential space, which is a Banach function space, can be defined with two very differently looking, but equivalent norms. In this paper, we give estimates for the best constants of the ratio of these two norms. Our result answers a question of C. Bennett, and R. Sharpley.

In this short note, we first give several new infinite product representations for the elliptic theta functions. These will then be used to deduce some special values and estimations for these functions on the real line.

The principal branch of the Lambert W function satisfies a well-known functional equation with respect to the sum of two of its values, but this equation is valid only on a restricted set. In this paper, we extend this equation to all the branches of W and to the arbitrary complex linear combination of two values. The results give an application for the unwinding number, too.

The solution of Kepler's famous equation can be expressed as an infinite sum for which the radius of convergence, lambda, is called the Laplace Limit Constant. So far, no explicit expression has been discovered for this constant. In this note, we point out that lambda can be expressed in closed form, in terms of the r-Lamb ert special function. Based upon this observation, we give a new infinite series representation for lambda in terms of the Laguerre polynomials.

In this paper we study two modifications of the partition lattice, regarding the size of the blocks. After determining the most elementary properties of these lattices, we calculate their Möbius functions, and study the enumeration problems of the chains and maximal length chains in our lattices.

In this paper we study the counting sequences which arise from the enumeration problems of set partitions of type B. These sequences count the total number of set partitions, the number of partitions with a given number of blocks. We thoroughly study the enumeration problem of partitions without singletons.

Extensions of a set partition obtained by imposing bounds on the size of the parts and the coloring of some of the elements are examined. Combinatorial properties and the generating functions of some counting sequences associated with these partitions are established. Connections with Riordan arrays are presented.

The r-Lambert function is a generalization of the classical Lambert W function which has proven to be useful in physics and other disciplines. In this paper we construct the Riemann surface of this function. It turns out that this surface has some peculiar properties, therefore it might be useful for demonstration purposes, for those who would like to see non-standard examples of Riemann surfaces coming from complex function theory.

In this work we study the zeros of the Eulerian and Bell polynomials and their generalizations. More concretely, lower estimates for the leftmost zeros of these polynomials will be given, complementing earlier results where upper estimations were presented.

The logarithm space L p log L can be defined by two equivalent, but very differently looking norms. In this note we give estimates for the best constants for the ratio of these norms.

A hazai termőföld értéke becsült piaci értéken nagyságrendileg néhány ezermilliárd forintra tehető, azonban valós természeti és közgazdasági értékének megállapítása összetett módszertani eljárást igényel. A termőföld értékelésére vonatkozó jelenleg érvényben levő hivatalos, jogszabályban rögzített módszertani irányelv (54/1997 FM rendelet) a hitelfedezeti értékelés céljából került megalkotásra illetve meghirdetésre az akkori jelzálog-intézményi követelményeknek megfelelően. Az azóta eltelt időszakban a társadalmi-gazdasági téren végbement változások szükségszerűvé tették a rendeletben foglalt módszertani alapok újragondolását, pontosítását helyenkénti módosítását. A szerzők az elmúlt két évtized tapasztalatainak felhasználásával a nemzetközi irányelvek és hazai jogszabályi alapok figyelembevételével javaslatokat dolgoztak ki a termőföld-értékelés aktualizálására vonatkozóan a jogalkotók és a vagyonértékelői szakma számára egyaránt.

The Lambert W function is defined by W (a)e(W(a)) - a = 0. One of the many applications of the Lambert W function is in solving delay differential equations (DDEs). In 2003, Asl and Ulsoy provided a solution of some DDEs in terms of the Lambert W functions Asl et al. (2003)[1]. However, the solutions are limited to differential equations with delay in the state variable. Scott et al. (2006)[2] introduced a generalized Lambert function which was further studied by Mezo and Baricz (2017)[3]. In our work, we show that this generalization of the Lambert W function provides an analytical solution to neutral delay differential equations (NDDEs). NDDEs are DDEs with time delay not only in the state variables but also in the derivative terms. This analytical solution is advantageous such that it is similar to the general solutions of linear ODEs. Also, one can identify how the parameters affect the solution of the equation since our proposed solution is written in terms of these parameters. We then propose a new numerical method to solve linear NDDEs using the generalized Lambert W function. We test our method to examples with known solutions. We also provide a real-world application by solving an NDDE model of the population growth of an E. coli culture using our proposed approach. (C) 2020 Elsevier Inc. All rights reserved.

Based on a Problem and its solution published on the pages of SIAM Review, we give an interesting integral representation for the Lambert W function in this short note. In particular, our result yields a new integral representation for the Ω=W(1) constant as well.

The directive of 1666/2015. (IX. 21.) called ’Land for Farmers!’ has changed not only the legal terms and conditions but also the economic basis of land use in the relation of land use and resulting derivative demand. Institutionalized rental fees can be modified to market level only if it is confirmed by qualified expert’s report hired by the new land owner. Setting a fair rental value has quite a few methodological approaches. Due to the lack of a legally recommended calculation process, authors hereby are presenting a method to calculate fair rental value that is beneficial for both renter and owner. Foreign rental conditions related to the topic are also concerned in the article.

Here, the Einstein-Maxwell field equation with plane-symmetry is resolved and a solution involving a generalized Lambert W function is obtained. This generalized Lambert W function is studied and its derivative, Taylor series, Mellin transform and approximation formula are derived and its real branches are determined. Implications of these analytic properties to the solution function are also presented.

The log-sine-polylog integrals were introduced by J. M. Borwein and A. Straub during their studies on special values of the log-sine integrals. In this paper we evaluate some of the log-sine-polylog integrals and show that they are intimately related to alternating Euler sums. These evaluations result in some interesting relations with respect to the first integral moments of the square of the Clausen function Cl-2.