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Assume that [Formula: see text] is a random variable whose moment generating function exists in a neighbourhood of the origin. In this paper, we study the probabilistic degenerate central Bell polynomials associated with [Formula: see text], as probabilistic extension of the degenerate central Bell polynomials. In addition, we investigate the probabilistic degenerate central factorial numbers of the second kind associated with [Formula: see text] and the probabilistic degenerate central Fubini polynomials associated with [Formula: see text]. The aim of this paper is to derive some properties, explicit expressions, certain identities and recurrence relations for those polynomials and numbers.

ABSTRACTRecently, many authors have studied degenerate Bernoulli and degenerate Euler polynomials. Let [Formula: see text] be a random variable whose moment generating function exists in a neighbourhood of the origin. The aim of this paper is to introduce and study the probabilistic extension of degenerate Bernoulli and degenerate Euler polynomials, namely the probabilistic degenerate Bernoulli polynomials associated with [Formula: see text] and the probabilistic degenerate Euler polynomials associated with [Formula: see text]. Also, we intoduce the probabilistic degenerate [Formula: see text]-Stirling numbers of the second associated with [Formula: see text] and the probabilistic degenerate two variable Fubini polynomials associated with [Formula: see text]. We obtain some properties, explicit expressions, recurrence relations and certain identities for those polynomials and numbers. As special cases of [Formula: see text], we treat the gamma random variable with parameters [Formula: see text], the Poisson random variable with parameter [Formula: see text], and the Bernoulli random variable with probability of success [Formula: see text].

ABSTRACTThe aim of this paper is to further study probabilistic versions of the degenerate Stirling polynomials of the second kind, namely the probabilistic degenerate Stirling polynomials of the second kind associated with [Formula: see text], which are also degenerate versions of the probabilistic Stirling polynomials of the second kind investigated earlier. Here [Formula: see text] is a random variable whose moment generating function exists in some neighbourhood of the origin. We derive some properties, explicit expressions and certain identities for those polynomials. In addition, we apply our results to the Bernoulli random variable with probability of success [Formula: see text], the normal random variable with parameters [Formula: see text], and to the uniform random variable on [Formula: see text].

Obesity triggers inflammatory responses in the microenvironment of white adipose tissue, resulting in chronic systemic inflammation and the subsequent development of non-communicable diseases, including type 2 diabetes, coronary heart disease, and breast cancer. Current therapy approaches for obesity-induced non-communicable diseases persist in prioritizing symptom remission while frequently overlooking the criticality of targeting and alleviating inflammation at its source. Accordingly, this review highlights the importance of the microenvironment of obese white adipose tissue and the promising potential of employing immunotherapy to target it as an effective therapeutic approach for non-communicable diseases induced by obesity. Additionally, this review discusses the challenges and offers perspective about the immunotherapy targeting the microenvironment of obese white adipose tissue.

Assume that the moment-generating function of the random variable $ Y $ exists in a neighborhood of the origin. The aim of this paper is to investigate the probabilistic type 2 Bernoulli polynomials associated with $ Y $ and the probabilistic type 2 Euler polynomials associated with $ Y $, along with the probabilistic type 2 cosine-Bernoulli polynomials associated with $ Y $, the probabilistic type 2 sine-Bernoulli polynomials associated with $ Y $, the probabilistic type 2 cosine-Euler polynomials associated with $ Y $, and the probabilistic type 2 sine-Euler polynomials associated with $ Y $. We deal with their properties, related identities and explicit expressions.

In this paper, we study the degenerate unipoly-Dedekind sums rising from the type 2 degenerate unipoly Bernoulli functions and degenerate unipoly-Bernoulli functions of arbitrary indices, respectively.

ABSTRACTRecently, degenerate harmonic numbers and degenerate hyperharmonic numbers are introduced by Kim-Kim. In this paper, we study the series involving the degenerate harmonic numbers and degenerate Stirling numbers and investigate those properties.

ABSTRACTHarmonic numbers and generalized harmonic numbers have been studied in connection with combinatorial problems, many expressions involving special functions in analytic number theory and analysis of algorithms. We introduce Lerch-harmonic numbers which generalize the harmonic numbers and the generalized harmonic numbers. The aim of this note is to derive some identities expressing certain finite sums as the infinite sums involving the Lerch-harmonic numbers. Then, by taking limits of such identities we obtain the corresponding infinite sums of the finite sums as the infinite sums involving the Lerch transcendents.

The aim of this paper is to study the $ \lambda $-Stirling numbers of both kinds, which are $ \lambda $-analogues of Stirling numbers of both kinds. These numbers have nice combinatorial interpretations when $ \lambda $ are positive integers. If $ \lambda = 1 $, then the $ \lambda $-Stirling numbers of both kinds reduce to the Stirling numbers of both kinds. We derive new types of generating functions of the $ \lambda $-Stirling numbers of both kinds which are related to the reciprocals of the generalized rising factorials. Furthermore, some related identities are also derived from those generating functions. In addition, all the corresponding results to the $ \lambda $-Stirling numbers of both kinds are obtained for the $ \lambda $-analogues of $ r $-Stirling numbers of both kinds, which are generalizations of those numbers.

For any positive integer $ r $, the $ r $-truncated (or $ r $-associated) Stirling number of the second kind $ S_{2}^{(r)}(n, k) $ enumerates the number of partitions of the set $ \{1, 2, 3, \dots, n\} $ into $ k $ non-empty disjoint subsets, such that each subset contains at least $ r $ elements. We introduce the degenerate $ r $-truncated Stirling numbers of the second kind and of the first kind. They are degenerate versions of the $ r $-truncated Stirling numbers of the second kind and of the first kind, and reduce to the degenerate Stirling numbers of the second kind and of the first kind for $ r = 1 $. Our aim is to derive recurrence relations for both of those numbers.

The aim of this paper is to introduce several degenerate hyperbolic functions as degenerate versions of the hyperbolic functions, to evaluate Volkenborn and the fermionic $ p $-adic integrals of the degenerate hyperbolic cosine and the degenerate hyperbolic sine functions and to derive from them some identities involving the degenerate Bernoulli numbers, the degenerate Euler numbers and the Cauchy numbers of the first kind.

Abstract Because regional sea-level rise can threaten coastal communities, understanding and quantifying the underlying process contributing to reginal sea-level budget are essential. Here, we assessed whether the regional sea-level rise on the northwestern Pacific marginal seas can be closed with a combination of observations and ocean reanalyses over 1993–2017, as well as with independent observations from in situ profiles including Argo floats and satellite gravity measurements since 2003. The assessment represents that the major contributions come from the land ice melt and sterodynamic components, while the spatial pattern and interannual variability are dominated by sterodynamic effect. The observation-based estimate further shows that along continental shelves, sterodynamic sea-level changes are substantially induced by ocean mass redistribution due to changes in ocean circulation. This result highlights the ocean mass change between the deep ocean and shallow marginal seas, which plays a role in driving regional sea-level rise and variability.

Objectives: Pravastatin sodium is reported to have multiple beneficial effects in cerebral atherosclerosis and neuronal injury; however, the preventive effects on cerebral venous ischemia are still unknown. Herein, we aimed to examine the neuroprotective effects of transoral prior administration of pravastatin sodium against cerebral cortical venous ischemia with suppression of apoptosis. Methods: Thirty 8-week-old male Wistar rats were divided equally into two study groups (n = 15 vs. n = 15); the pravastatin group was fed 1% pravastatin sodium with their usual diet for 2 weeks, while the control group only received the usual diet. Two-vein occlusion (2VO) model was applied for this study, and two adjacent cortical veins in each animal were permanently occluded photochemically with rose bengal dye. During photo-thrombosis, regional changes of the cerebral blood flow (CBF) in area of the venous ischemia were recorded. At 48-h after 2VO, animals were euthanized using perfusion fixation, and we histologically measured ratios of infarcted area to contralateral hemisphere, and counted Bax- and Bcl-2-positive cells in the penumbra to investigate the implications for apoptosis. Results: The ratio of infarcted area was significantly decreased in the pravastatin group compared to the control group (P

Based on the relation between volume transport and sea level difference across the Korea/Tsushima Strait (KTS), in this study, we estimated the volume transport of the Tsushima Warm Current over the past four decades and examined its long-term trend using an ensemble empirical model decomposition (EEMD) method. The estimated geostrophic transports showed very good agreement with those measured by acoustic Doppler current profiler and those estimated from satellite altimetry-based sea level data. This corroborates the reliability of the long-term estimation. EEMD analysis revealed an acceleration in volume transport in the KTS, with a persistent increase in the trend rate occurring after the mid-1980s. Further analyses indicate that the trend shift in the KTS closely coincided with an upper-ocean warming trend in the East/Japan Sea (EJS) after the mid-1980s. Analyzing large-scale wind patterns suggests that long-term trends of wind stress curls over the North Pacific are likely a key driver of the trend shifts in the inflow through the KTS and the upper-ocean warming in the EJS over the past four decades.

Many works have been done in recent years as to explorations for degenerate versions of some special polynomials and numbers, which began with the pioneering work of Carlitz on the degenerate Bernoulli and degenerate Euler polynomials. This paper is focused on the study of some properties, recurrence relations and identities related to the degenerate r-Bell polynomials, the two variable degenerate Fubini polynomials and the degenerate r-Stirling numbers of the second kind. Especially, we express the power series $ \sum _{n=0}^{\infty }\sum _{k=0}^{n}(k+r)_{p,\lambda }\frac {x^n}{n!} $ in terms of the degenerate r-Bell polynomials, of the degenerate r-Stirling numbers of the second kind and of the degenerate Fubini polynomials.

The degenerate exponentials are degenerate versions of the ordinary exponential and the truncated degenerate exponentials are obtained from the Taylor expansions of them by truncating the first finitely many terms. The degenerate exponentials play an important role in recent studies on degenerate versions of many special numbers and polynomials, the degenerate gamma function, the degenerate umbral calculus and the degenerate q-umbral calculus. The aim of this note is to consider infinite series whose terms involve truncated degenerate exponentials together with binomial coefficients, the generalized falling factorials and the Stirling numbers of the second kind, and to find either their values or some other expressions of them as finite sums.

Dedekind-type DC sums and their properties are defined in terms of Euler functions. Ma et al. recently introduced poly-Dedekind-type DC sums and demonstrated that they satisfy a reciprocity relation. In this paper, we introduce the degenerate poly-Euler polynomials and numbers, and we also consider the reciprocity relations of degenerate poly-Dedekind-type DC sums. Equivalently, several properties and identities of degenerate poly-Euler functions are obtained.

We introduce the generalized degenerate Euler–Genocchi polynomials as a degenerate version of the Euler–Genocchi polynomials. In addition, we introduce their higher-order version, namely the generalized degenerate Euler–Genocchi polynomials of order α, as a degenerate version of the generalized Euler–Genocchi polynomials of order α. The aim of this paper is to study certain properties and identities involving those polynomials, the generalized falling factorials, the degenerate Euler polynomials of order α, the degenerate Stirling numbers of the second kind, and the ‘alternating degenerate power sum of integers’.

Recently, Kim-Kim introduced the truncated degenerate Bell polynomials and numbers. In this paper, we introduce the truncated Lah-Bell polynomials and numbers. We obtain some identities, recurrence relations and properties. Furthermore, we also introduce the truncated modified Lah-Bell polynomials and numbers, truncated degenerate Lah-Bell polynomials and numbers and we derive some properties and identities involving with those polynomials and numbers.

The paper has two main objectives. Firstly, it explores the properties of hyperbolic cosine and hyperbolic sine functions by using Volkenborn and the fermionic p-adic integrals, respectively. It also investigates interesting identities involving Bernoulli and degenerate Bernoulli numbers from differential equations with their solutions. Secondly, it introduces a random variable whose mass function is given in terms of Euler polynomials and find some expressions for the expectation of the random variable.

It is remarkable that, in recent years, intensive studies have been done for degenerate versions of many special polynomials and numbers and have yielded many interesting results. The aim of this paper is to study the generalized degenerate $ (r,s) $ -Stirling numbers of the second and their natural extensions to polynomials, namely the generalized degenerate $ (r,s) $ -Bell polynomials, arising from certain ‘degenerate boson normal ordering.’ We derive some properties, explicit expressions and generating functions for those numbers and polynomials. The generalized degenerate $ (r,s) $ -Stirling numbers of the second and the degenerate boson normal ordering are respectively degenerate versions of the generalized $ (r,s) $ -Stirling numbers of the second and the boson normal ordering studied earlier by Blasiak-Person-Solomon.

Our main focus here is a new type of degenerate poly-Euler polynomials and numbers. This focus stems from their nascent importance for applications in combinatorics, number theory and in other aspects of applied mathematics. We construct a new type of degenerate poly-Euler polynomials by using the degenerate polylogarithm functions. We also show several combinatorial identities related to this polynomials and numbers.

Mass loss from ice shelves occurs through ocean-driven melting regulated by dynamic and thermodynamic processes in sub-ice shelf cavities. However, the understanding of these oceanic processes is quite limited because of the scant observations under ice shelves. Here, a regional coupled sea-ice/ocean model that includes physical interactions between the ocean and the ice shelf is used as an alternative tool for exploring ocean-driven melting beneath the Nansen Ice Shelf (NIS) which is a cold-water cavity ice shelf located beside Terra Nova Bay (TNB) in East Antarctica. For the first time, this study identifies the spatiotemporal variability signatures for different modes of ocean-driven melting at the base of NIS. In February (austral summer), basal melting substantially increases where the ice shelf draft is relatively small in the vicinity of the ice shelf front, contributing 78% of the total NIS melting rate. As the dominant source of NIS mass loss, this melting is driven by tide-induced turbulent mixing along the sloping ice shelf base and summer warm surface water intruding beneath and reaching the shallow parts of the ice shelf. In contrast, the NIS has relatively high basal melting rates near the grounding line in September (austral winter) primarily because of the intrusion of high-salinity shelf water produced by polynya activity in TNB that flows into the cavity beneath NIS toward the deep grounding line. Of the total melting rate of NIS in winter, 36% comes from regions near the grounding line. In addition, the contributions of tides and realistic cavity geometry to NIS basal melting are identified by conducting sensitivity experiments. Tidal effects increase the melting of NIS throughout the year, particularly contributing as much as 30% to the areas of ice draft shallower than 200 m in summer. Sensitivity results for uncertainty in cavity geometry show that spurious vertical mixing can be locally induced and enhanced by interaction between tides and the unrealistic topography, resulting in excessive basal melting near the NIS frontal band. The sensitivity experiments have shown that tides and realistic cavity geometry bring a significant improvement in the estimation of basal melt rates through a numerical model.

The normal ordering of an integral power of the number operator in terms of boson operators is expressed with the help of the Stirling numbers of the second kind. As a ‘degenerate version’ of this, we consider the normal ordering of a degenerate integral power of the number operator in terms of boson operators, which is represented by means of the degenerate Stirling numbers of the second kind. As an application of this normal ordering, we derive two equations defining the degenerate Stirling numbers of the second kind and a Dobinski-like formula for the degenerate Bell polynomials.

The nth Fubini number enumerates the number of ordered partitions of a set with n elements and is the number of possible ways to write the Fubini formula for a summation of integration of order n. Further, Fubini polynomials are natural extensions of the Fubini numbers. There are many variants of Fubini numbers and polynomials. Recently, the degenerate Fubini polynomials were introduced by Kim-Kim-Jang as a degenerate version of the Fubini polynomials. The aim of this article is by using generating functions and certain differential operators to further study some identities and properties on the degenerate Fubini polynomials and the higher-order degenerate Fubini polynomials.

Degenerate Dowling and degenerate r-Dowling polynomials were introduced earlier as degenerate versions and further generalizations of Dowling and r-Dowling polynomials. The aim of this paper is to show their connections with Poisson degenerate central moments for a Poisson random variable with a certain parameter and with Charlier polynomials.

In this paper, we derive some identities and recurrence relations for the degenerate Stirling numbers of the first kind and of the second kind, which are degenerate versions of the ordinary Stirling numbers of the first kind and of the second kind. They are deduced from the normal orderings of degenerate integral powers of the number operator and their inversions, certain relations of boson operators and the recurrence relations of the Stirling numbers themselves. Here we note that, while the normal ordering of an integral power of the number operator is expressed with the help of the Stirling numbers of the second kind, that of a degenerate integral power of the number operator is represented by means of the degenerate Stirling numbers of the second kind.

Recently, reactive materials have been developed for penetrative projectiles to improve impact resistance and energy capacity. However, the design of a reactive material structure, involving shape and size, is challenging because of difficulties such as high non-linearity of impact resistance, manufacturing limitations of reactive materials and high expenses of penetration experiments. In this study, a design optimization methodology for the reactive material structure is developed based on the finite element analysis. A finite element model for penetration analysis is introduced to save the expenses of the experiments. Impact resistance is assessed through the analysis, and result is calibrated by comparing with experimental results. Based on the model, topology optimization is introduced to determine shape of the structure. The design variables and constraints of the optimization are proposed considering the manufacturing limitations, and the optimal shape that can be manufactured by cold spraying is determined. Based on the optimal shape, size optimization is introduced to determine the geometric dimensions of the structure. As a result, optimal design of the reactive material structure and steel case of the penetrative projectile, which maximizes the impact resistance, is determined. Using the design process proposed in this study, reactive material structures can be designed considering not only mechanical performances but also manufacturing limitations, with reasonable time and cost.

The n-th Fubini number counts the number of ordered partitions of a set with n elements and is the number of possible ways to write the Fubini formula for a summation of integration of order n. Further, Fubini polynomials are natural extensions of the Fubini numbers. Recently, the generalized Fubini polynomials were introduced by Sebaoui-Laissaoui-Guettai-Rahmani, as one of the variants of the Fubini polynomials. The aim of this paper is to study the degenerate generalized Fubini polynomials, which are a degenerate version of those polynomials, and to find an application of them to probability theory in connection with geometric random variable.

Bell polynomials are widely applied in many problems arising from physics and engineering. The aim of this paper is to introduce new types of special polynomials and numbers, namely Bell polynomials and numbers of the second kind and poly-Bell polynomials and numbers of the second kind, and to derive their explicit expressions, recurrence relations and some identities involving those polynomials and numbers. We also consider degenerate versions of those polynomials and numbers, namely degenerate Bell polynomials and numbers of the second kind and degenerate poly-Bell polynomials and numbers of the second kind, and deduce their similar results.

Recently, Kim-Kim introduced the degenerate Whitney numbers of the first and second kind involving the degenerate Dowling polynomials and numbers. In this paper, we introduce the fully degenerate Dowling polynomials and the fully degenerate Bell polynomials and derive some identities involving these polynomials. We also obtain the generating functions, expressions, and recurrence relations for the fully degenerate Dowling polynomials and the fully degenerate Bell polynomials and other special polynomials and numbers.

Recently, the generalized Euler–Genocchi and generalized degenerate Euler–Genocchi polynomials are introduced. The aim of this note is to study the multi-Euler–Genocchi and degenerate multi-Euler–Genocchi polynomials which are defined by means of the multiple logarithm and generalize, respectively, the generalized Euler–Genocchi and generalized degenerate Euler–Genocchi polynomials. Especially, we express the former by the generalized Euler–Genocchi polynomials, the multi-Stirling numbers of the first kind and Stirling numbers of the second kind, and the latter by the generalized degenerate Euler–Genocchi polynomials, the multi-Stirling numbers of the first kind and Stirling numbers of the second kind.

Abstract In this paper, we introduce poly-central factorial sequences and poly-central Bell polynomials arising from the polyexponential functions, reducing them to central factorials and central Bell polynomials of the second kind respectively when k = 1 $k = 1$ . We also show some relations: between poly-central factorial sequences and power of x; between poly-central Bell polynomials and power of x; between poly-central Bell polynomials and the poly-Bell polynomials; between poly-central Bell polynomials and higher order type 2 Bernoulli polynomials of second kind; recurrence formula of poly-central Bell polynomials.

Abstract The aim of this paper is to determine the λ-linear functionals sending any given polynomial p ( x ) $p(x)$ with coefficients in C p $\mathbb{C}_{p}$ to the p-adic invariant integral of P ( x ) $P(x)$ on Z p $\mathbb{Z}_{p}$ and also to that of P ( x 1 + ⋯ + x r ) $P(x_{1}+\cdots +x_{r})$ on Z p r $\mathbb{Z}_{p}^{r}$ . We show that the former is given by the generating function of degenerate Bernoulli polynomials and the latter by that of degenerate Bernoulli polynomials of order r. For this purpose, we use the λ-umbral algebra which has been recently introduced by Kim and Kim (J. Math. Anal. Appl. 493(1):124521 2021).

Abstract 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.

Abstract The aim of this paper is to introduce the degenerate generalized Laguerre polynomials as the degenerate version of the generalized Laguerre polynomials and to derive some properties related to those polynomials and Lah numbers, including an explicit expression, a Rodrigues type formula, and expressions for the derivatives. The novelty of the present paper is that it is the first paper on degenerate versions of orthogonal polynomials.

In this paper, we consider sums of finite products of the second and third type Chebyshev polynomials, those of the second and fourth type Chebyshev polynomials and those of the third and fourth type Chebyshev polynomials, and represent each of them as linear combinations of Chebyshev polynomials of all types. Here the coefficients involve some terminating hypergeometric functions $ {}_{2}F_{1} $. This problem can be viewed as a generalization of the classical linearization problems and is done by explicit computations.

Abstract The aim of this paper is to study the complete and incomplete degenerate Bell polynomials, which are degenerate versions of the complete and incomplete Bell polynomials, and to derive some properties and identities for those polynomials. In particular, we introduce some new polynomials associated with the incomplete degenerate Bell polynomials. In fact, they are the coefficients of the reciprocal of the power series given by 1 plus the one appearing as the exponent of the generating function of the complete degenerate Bell polynomials.

Abstract Numerous mathematicians have studied ‘poly’ as one of the generalizations to special polynomials, such as Bernoulli, Euler, Cauchy, and Genocchi polynomials. In relation to this, in this paper, we introduce the degenerate poly-Bell polynomials emanating from the degenerate polyexponential functions which are called the poly-Bell polynomials when λ → 0 $\lambda \rightarrow 0$ . Specifically, we demonstrate that they are reduced to the degenerate Bell polynomials if k = 1 $k = 1$ . We also provide explicit representations and combinatorial identities for these polynomials, including Dobinski-like formulas, recurrence relationships, etc.

Abstract In this paper, we study degenerate complete and partial Bell polynomials and establish some new identities for those polynomials. In addition, we investigate the connections between modified degenerate complete and partial Bell polynomials, which are closely related to the degenerate complete and partial Bell polynomials, and the joint distribution of weighted sums of independent degenerate Poisson random variables.

Previous studies have suggested that westward-migrating mesoscale eddies are a dominant factor that modulate the interannual Kuroshio intensity in the East China Sea (ECS), indicating a close positive correlation between them. According to the extended record of altimetry-based sea level anomalies (SLAs) until 2020, however, the interannual variation of the Kuroshio intensity no longer has a strong positive correlation with eddy activity in the subtropical countercurrent (STCC) region since the early 2000s. Our observational analyses showed that the Kuroshio intensity in the ECS can be modulated by the combined effect of westward-migrating mesoscale eddies and westward-propagating oceanic planetary waves from the east. Until the early 2000s, the interannual variability of Kuroshio was mainly affected by eddy migration from the STCC region, associated with oceanic instability driven by large-scale wind patterns over the western North Pacific. Since then, oceanic planetary waves propagating westward across the Pacific basin have largely modulated the interannual variability of the ECS-Kuroshio intensity by superimposing the SLAs related to mesoscale eddies that propagated towards the east of Taiwan.

In this paper, we study the degenerate derangement polynomials and numbers, investigate some properties of those polynomials and numbers and explore their connections with the degenerate gamma distributions. In more detail, we derive their explicit expressions, recurrence relations and some identities involving the degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials and the degenerate Stirling numbers of both kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions.

Abstract A new family of p-Bernoulli numbers and polynomials was introduced by Rahmani (J. Number Theory 157:350–366, 2015) with the help of the Gauss hypergeometric function. Motivated by that paper and in the light of the recent interests in finding degenerate versions, we construct the generalized degenerate Bernoulli numbers and polynomials by means of the Gauss hypergeometric function. In addition, we construct the degenerate type Eulerian numbers as a degenerate version of Eulerian numbers. For the generalized degenerate Bernoulli numbers, we express them in terms of the degenerate Stirling numbers of the second kind, of the degenerate type Eulerian numbers, of the degenerate p-Stirling numbers of the second kind and of an integral on the unit interval. As to the generalized degenerate Bernoulli polynomials, we represent them in terms of the degenerate Stirling polynomials of the second kind.

Abstract As is well known, poly-Bernoulli polynomials are defined in terms of polylogarithm functions. Recently, as degenerate versions of such functions and polynomials, degenerate polylogarithm functions were introduced and degenerate poly-Bernoulli polynomials were defined by means of the degenerate polylogarithm functions, and some of their properties were investigated. The aim of this paper is to further study some properties of the degenerate poly-Bernoulli polynomials by using three formulas coming from the recently developed ‘λ-umbral calculus’. In more detail, among other things, we represent the degenerate poly-Bernoulli polynomials by higher-order degenerate Bernoulli polynomials and by higher-order degenerate derangement polynomials.

A piezoelectric energy harvester (PEH) transduces mechanical energy into electrical energy, which can be utilized as an energy source for self-powered or low-power devices. Therefore, maximizing the power of a PEH is a crucial design objective. It is well known that structural designs are firstly conducted for controlling resonance characteristics, and then circuit designs are pursued through impedance matching for improving power. However, a PEH contains solid mechanics, electrostatics, and even a circuit-coupled multi-physics system. Therefore, this research aims to design a PEH considering a circuit-coupled multi-physics. As a design process, a conceptual design is developed by topology optimization, and a detailed design is developed sequentially by applying size optimization as a post-processing step to refine the conceptual design results for manufacturable design. In the two optimization processes, design optimizations of a structure coupled with circuit resistor are performed to maximize the power, where the electrical and mechanical interactions between PZT, substrate, and circuit resistor are simultaneously considered. Additionally, stress constraints are also added for structural safety to ensure operational life of PEH. As a result of the proposed design methodology, a manufacturable design of PEH having maximum power and operational life is obtained with power density of 6.61 μWg−2mm−3.

Abstract The nth r-extended Lah–Bell number is defined as the number of ways a set with n + r $n+r$ elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks. The aim of this paper is to introduce incomplete r-extended Lah–Bell polynomials and complete r-extended Lah–Bell polynomials respectively as multivariate versions of r-Lah numbers and the r-extended Lah–Bell numbers and to investigate some properties and identities for these polynomials. From these investigations we obtain some expressions for the r-Lah numbers and the r-extended Lah–Bell numbers as finite sums.

Abstract The classical Dedekind sums appear in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. The Dedekind sums and their generalizations are defined in terms of Bernoulli functions and their generalizations, and are shown to satisfy some reciprocity relations. In contrast, Dedekind-type DC (Daehee and Changhee) sums and their generalizations are defined in terms of Euler functions and their generalizations. The purpose of this paper is to introduce the poly-Dedekind-type DC sums, which are obtained from the Dedekind-type DC sums by replacing the Euler function by poly-Euler functions of arbitrary indices, and to show that those sums satisfy, among other things, a reciprocity relation.

A pronounced increase in the intensification of Typhoon Bavi in 2020 was detected when the typhoon passed over the Changjiang plume in the northern East China Sea. Using a coupled atmosphere-ocean modeling system, this study investigates the role of the plume-induced barrier layer (BL) in the air–sea interaction during the intensification of a typhoon. Simple comparative experiments with and without the river plume revealed a strong relationship between BL formation and typhoon intensification as a result of the significant surface freshening discharged from the Changjiang River. The plume-induced BL maintained a warm sea surface before the typhoon approached, thereby influencing the energy transfer at the air–sea interface. The enthalpy and moisture reaching the atmosphere were increased by approximately 20%, leading to the intensification of Typhoon Bavi and providing further support for the results observed in the best-track record. The model comparison also indicates that the salinity-induced BL led to the reduction of the typhoon-induced SST cooling by restricting the vertical diffusion between the surface and the thermocline, and consequently contributed to maintaining the typhoon intensity. This study suggests that the effect of river-induced surface freshening in a coupled atmosphere-ocean model may help in improving typhoon forecasts and may aid in mitigating against the destructive power of typhoons in the future.

Abstract Identities of symmetry in two variables for Bernoulli polynomials and power sums had been investigated by considering suitable symmetric identities. T. Kim used a completely different tool, namely the p-adic Volkenborn integrals, to find the same identities of symmetry in two variables. Not much later, it was observed that this p-adic approach can be generalized to the case of three variables and shown that it gives some new identities of symmetry even in the case of two variables upon specializing one of the three variables. In this paper, we generalize the results in three variables to those in an arbitrary number of variables in a suitable setting and illustrate our results with some examples.

Abstract In this paper, we introduce one discrete random variable, namely the negative λ-binomial random variable. We deduce the expectation of the negative λ-binomial random variable. We also get the variance and explicit expression for the moments of the negative λ-binomial random variable.