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This paper is devoted to reduce the conservativeness of distributionally robust optimization with moments information. Since the optimal solution of distributionally robust optimization is required to be feasible for all uncertain distributions in a given ambiguity distribution set and so the conservativeness of the optimal solution is inevitable. To address this issue, we introduce the globalized distributionally robust counterpart (GDRC) which allows constraint violations controlled by functional distance of the true distribution to the inner distribution set. We obtain the deterministic equivalent forms for several GDRCs under the moment-based framework. To be specific, we show the deterministic equivalent systems of inequalities for GDRCs under second order moment information with a separable convex distance function and a special jointly convex function, respectively. We also obtain the deterministic equivalent inequality for GDRC under first order moment and support information. The computationally tractable examples are presented for these GDRCs. Numerical tests of a portfolio optimization problem are given to show the effectiveness of our method and the results demonstrate that the globalized distributionally robust solution is non-conservative and flexible compared to the distributionally robust solution. [ABSTRACT FROM AUTHOR]

In a recent paper, we used a basic decomposition property of polyanalytic functions of order 2 in one complex variable to characterize solutions of the classical $ \bar {\partial } $ ∂ ¯ -problem for given analytic and polyanalytic data. Our approach suggested the study of a special reproducing kernel Hilbert space that we call the Hörmander–Fock space that will be further investigated in this paper. The main properties of this space are encoded in a specific moment sequence denoted by $ \eta =(\eta _n)_{n\geq ~0} $ η = (η n) n ≥ 0 leading to a special entire function $ \mathsf {E}(z) $ E (z) that is used to express the kernel function of the Hörmander–Fock space. We present also an example of a special function belonging to the class Mittag-Leffler (ML) introduced recently by Alpay et al. and apply a Bochner–Minlos type theorem to this function, thus motivating further connections with the theory of stochastic processes. [ABSTRACT FROM AUTHOR]

AbstractD-allulose, an epimer of D-fructose at C-3 position, is a low-calorie rare sugar with favorable physiochemical properties and special physiological functions, which displays promising perspectives in the food and pharmaceutical industries. Currently, D-allulose is extremely sparse in nature and is predominantly biosynthesized through the isomerization of D-fructose by D-allulose 3-epimerase (DAEase). In recent years, D-allulose 3-epimerase as the key biocatalyst for D-allulose production has received increasing interest. The current review begins by providing a summary of D-allulose regarding its characteristics and applications, as well as different synthesis pathways dominated by biotransformation. Then, the research advances of D-allulose 3-epimerase are systematically reviewed, focusing on heterologous expression and biochemical characterization, crystal structure and molecular modification, and application in D-allulose production. Concerning the constraint of low yield of DAEase for industrial application, this review addresses the various attempts made to promote the production of DAEase in different expression systems. Also, various strategies have been adopted to improve its thermotolerance and catalytic activity, which is mainly based on the structure-function relationship of DAEase. The application of DAEase in D-allulose biosynthesis from D-fructose or low-cost feedstocks through single- or multi-enzymatic cascade reaction has been discussed. Finally, the prospects for related research of D-allulose 3-epimerase are also proposed, facilitating the industrialization of DAEase and more efficient and economical bioproduction of D-allulose. [ABSTRACT FROM AUTHOR]

Special events have gained popularity because they can contribute to the destinations' sustainability. The outbreak of the COVID-19 pandemic resulted in a rise in xenophobia and racism, so the function of special events in society became even more critical. This study explores the potential of special events to promote diversity, equity, and inclusion (DEI) experiences through local festivals during COVID-19. The result of this study is that attendees' perception of the festival was the most salient in predicting personal DEI experience, particularly festival DEI practices and the trust of other festival attendees. In addition, attendees with a high level of festival pride significantly have more positive perceptions than those with a low level. Overall, the results of this study affirm the notion that special events foster diversity, equity, and inclusion (DEI), thereby aiding in the advancement of socially sustainable event planning and management practices. [ABSTRACT FROM AUTHOR]

In this paper, we mainly concern the fixed-time consensus control problem for nonlinear leader-following multi-agent systems(MASs). An edge-based adaptive control protocol is presented with part of parameters updated by relative information of system state. Compared with the common node-based adaptive protocols, the proposed edge-based adaptive control protocol assigning different time-varying weights to different communication edges is more accurate. By selecting special Lyapunov functions and theoretical analysis, the obtained conclusions indicate that both sufficient conditions and the settling time of the fixed-time consensus are independent of the system's global information such as the spectrum of Laplacian matrix, which implies that the given protocol is fully distributed. Finally, numerical simulation is illustrated to support the correctness of the main theoretical results. [ABSTRACT FROM AUTHOR]

For the hierarchical uniform and inverse gamma model, we calculate the Bayes posterior estimator of the parameter of the uniform distribution under Stein's loss function which penalizes gross overestimation and gross underestimation equally and the corresponding Posterior Expected Stein's Loss (PESL). We also obtain the Bayes posterior estimator of the parameter under the squared error loss function and the corresponding PESL. Moreover, we obtain empirical Bayes estimators of the parameter of the uniform distribution by two methods. Note that the estimators of the hyperparameters of the model by the Maximum Likelihood Estimation (MLE) method are summarized in a theorem, whose proof involves the upper incomplete gamma function and a special case of the Meijer G-function. In numerical simulations, we address from four perspectives. First, we exemplify the two inequalities of the Bayes posterior estimators and the PESLs. Second, we illustrate that the moment estimators and the Maximum Likelihood Estimators (MLEs) are consistent estimators of the hyperparameters. Third, we calculate the goodness-of-fit of the model for the simulated data. Fourth, we plot the marginal densities of the model for various hyperparameters. Finally, we utilize the current prices of the 300 component stocks of Shenzhen 300 Index to illustrate our theoretical studies. [ABSTRACT FROM AUTHOR]

When analyzing materials non-isothermally using the Arrhenius equation under linear heating, a temperature integration is necessary. While the frequency factor in this equation is typically assumed to be constant, it can actually vary with temperature for certain solid-state reactions. The resulting temperature integral, known as the Arrhenius or general temperature integrals, usually have no analytical solutions. Therefore, special functions and approximation functions are often used to estimate them. In this particular study, new rational approximations for the Arrhenius and general temperature integrals were derived through the expansion of the incomplete gamma function. Two sets of these rational approximations, which exhibit excellent accuracy, are presented. One set of approximations for the Arrhenius integral matches the widely used Senum and Yang's approximations, while the other set, which offers even greater accuracy, has not been previously reported. An obtained rational approximation has been utilized to simulate the thermal degradation of a commercially available PMMA, illustrating a practical application example. [ABSTRACT FROM AUTHOR]

In this paper we introduce generalized phase retrievable frames or simply g-phase retrievable frames in real or complex n-dimensional Hilbert space H n , which include ordinary phase retrievable frames in H n . Specifically, a g-phase retrievable frame is a λ-phase retrievable frame, where λ is an special function, which is called phase coefficient function. Using the λ-phase retrievable frames, every vector in H n can be reconstructed up to a constant phase coefficient factor from the action of λ on the its frame coefficients, which can be bounded for bounded phase coefficient functions. We obtain some equivalent conditions to g-phase retrievable frames and we study on stability of special g-phase retrievable frames under small perturbation of their vectors and under small perturbation of their phase coefficient functions. [ABSTRACT FROM AUTHOR]

Harmonic 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. [ABSTRACT FROM AUTHOR]

Some new relations for the confluent Horn functions H 10 (c) (a ; c ; w , z) and H 11 (c) (a , c , c ′ ; d ; w , z) are obtained including differentiation and integration formulas, series and reduction formulas. Some generating functions for various special functions are given in terms of these Horn functions. [ABSTRACT FROM AUTHOR]

In this paper, some properties related to the 3 m -parametric Mittag-Leffler (M-L) functions are investigated. More precisely, three families of such functions with different kinds of indices are examined. Appropriate estimates and asymptotic formulae are obtained and which are further used in proving the convergence of series of functions of these families in the whole complex plane, as well as in its compact subsets. Using these auxiliary results, different relations referring to the discussed families are obtained; generalizing the ones already known for particular cases of the parameters. Those are considered different representations of the 3 m -parametric M-L functions pertaining to the families discussed, and with which various equalities are established relating different sums that involve matrix coefficients. Some interesting properties are reached for particular choices of the matrices. Finally, bounds of the 3 m -parametric M-L function are given under a perturbed matrix argument. [ABSTRACT FROM AUTHOR]

The H -function of communication theory plays an important role in the error rate analysis in digital communication with the presence of additive white Gaussian noise (AWGN) and generalized multipath fading conditions. In this paper we investigate several convergent and/or asymptotic expansions of H p (z , b , η) for some limiting values of their variables and parameters: large values of z, large values of p, small values of η, and values of b → 1. We provide explicit and/or recursive algorithms for the computation of the coefficients of the expansions. Some numerical examples illustrate the accuracy of the approximations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a ground-breaking approach to defining fractional calculus for a selected class of analytic functions. Our new definitions, based on a novel and intuitive understanding of fractional derivatives and integrals, offer improved mathematical tractability for a variety of applications, including physics, engineering and finance. Our approach significantly simplifies the complexity of mathematical functions compared to the traditional Riemann-Liouville approach, by using simple functions rather than special functions, while preserving the intrinsic sense of fractional calculus. This article not only presents our proposed definitions but also provides a thorough analysis of their properties and advantages. The conclusion of this paper discusses the potential for future research in the field of fractional calculus. [ABSTRACT FROM AUTHOR]

The rapid expansion of Special Operations Forces (SOF) since the beginning of the twenty-first century is largely seen as a response to the necessity to fight the post–11 September 2001 counterterrorism wars. Strike and intelligence, previously studied as distinct functions for military forces, have been increasingly intertwined in contemporary SOF operations. This study examines the evolution of the strike and intelligence functions of the North Atlantic Treaty Organization and allied SOF in counterterrorism and counterinsurgency campaigns from 2004 to 2017. The coevolution of strike and intelligence functions in SOF operations provides unique insight into the processes of battlefield and doctrinal adaptation in counterterrorism campaigns. Opportunities for further research include links between emerging doctrine and practice in targeting enemy networks. [ABSTRACT FROM AUTHOR]

In a normed space, we study the perturbed minimal time function determined by a bounded closed convex set and a proper lower semicontinuous function. It covers the perturbed distance function as a special case. In particular, we show that the ε-subdifferential and the limiting subdifferential of the perturbed minimal time function are representable by virtue of corresponding subdifferential of the associated function and the support function of the constraint set. [ABSTRACT FROM AUTHOR]

Conway [A Lagrangian method for deriving new indefinite integrals of special functions. Integral Transforms Spec Funct. 2015;26:812–824] introduced a new and simple method named the 'Lagrangian method' for deriving indefinite integrals of both elementary and special functions, provided the function satisfies the second-order linear differential equation. In this paper, different Meijer's G-functions have been used for deriving indefinite integrals, which satisfy second-order differential equations. We have derived recurrence relations and the integration of those recurrence relations by the Lagrangian method. We have discussed the integration of Euler identity, which is given in terms of Meijer's G-function. Different additional relations of Meijer's G-functions have also been discussed. [ABSTRACT FROM AUTHOR]

In this paper, we prove some blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime by using an iteration argument, and we derive upper bound estimates for the lifespan. In particular, we will focus on the critical cases which require the employment of a slicing procedure in the iterative mechanism. Furthermore, in order to deal with the main critical case, we will introduce a non-autonomous and parameter-dependent Cauchy problem for a linear ODE of second-order, whose explicit solution will be determined by applying the theory of special functions. [ABSTRACT FROM AUTHOR]

In this work, we deal with the Hénon and the Lozi map for a choice of parameters where they show transient chaos. Orbits close to the chaotic saddle behave chaotically for a while to eventually escape to an external attractor. Traditionally, to prevent such an escape, the partial control technique has been applied. This method stands out for considering disturbances (noise) affecting the map and for finding a special region of the phase space, called the safe set, where the control required to sustain the orbits is small. However, in this work, we will apply a new approach of the partial control method that has been recently developed. This new approach is based on finding a special function called the safety function which allows to automatically find the minimum control necessary to avoid the escape of the orbits. Furthermore, we will show the strong connection between the safety function and the classical safe set. To illustrate that, we will compute for the first time, safety functions for the two-dimensional Hénon and Lozi maps, where we also show the strong dependence of this function with the magnitude of disturbances affecting the map, and how this change drastically impacts the controlled orbits. [ABSTRACT FROM AUTHOR]

D-Allulose is the C-3 epimer of D-fructose, and widely regarded as a promising substitute for sucrose. It's an excellent low-calorie sweetener, with 70% sweetness of sucrose, 0.4 kcal/g dietary energy, and special physiological functions. It has been approved as GRAS by the U.S. Food and Drug Administration, and is allowed to be excluded from total and added sugar counts on the food labels. Therefore, D-allulose gradually attracts more public attention. Owing to scarcity in nature, the bioproduction of D-allulose by using ketose 3-epimerase (KEase) has become the research hotspot. Herein, we give a summary of the physicochemical properties, physiological function, applications, and the chemical and biochemical synthesis methods of D-allulose. In addition, the recent progress in the D-allulose bioproduction using KEases, and the possible solutions for existing challenges in the D-allulose industrial production are comprehensively discussed, focusing on the molecular modification, immobilization, food-grade expression, utilizing low-cost biomass as feedstock, overcoming thermodynamic limitation, as well as the downstream separation and purification. Finally, Prospects for further development are also proposed. [ABSTRACT FROM AUTHOR]

We study the Korenblum Maximum Principle for various weighted Fock spaces. The main tool relies on applying special cases of Ramanujan's Master Theorem involving the Gamma function and the Mellin transform of the Dirichlet series. It is interesting that with elementary probe functions, we still can obtain closed-form upper bound expressions, in terms of several well-known special functions, for the Korenblum constants of various weighted Fock spaces. For the first time, we obtain upper bounds of the Korenblum constant for the finite and infinite intersections of weighted Fock spaces and proved that the Korenblum Maximum Principle fails for those infinite intersections of weighted Fock spaces. Some open questions are provided. [ABSTRACT FROM AUTHOR]

The aim of this article is to define the bicomplex hypergeometric function. We obtain some useful results from the theory of special functions over a relatively large domain. We discuss the bicomplex holomorphicity of the bicomplex hypergeometric function and establish some significant relations based on it along with some bicomplex geometric representations of conditions of existence. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and non-existence of maximizers for variational problems associated with Caffarelli–Kohn–Nirenberg-type inequalities under the equivalent constraints both in the subcritical case and critical case. Furthermore, if the maximizer exists, we can derive the concrete form of the maximizers. Our method is based on establishing a useful link between the attainability of the supremum in our variational problems and the attainability of the supremum of some special functions defined on (0 , ∞). [ABSTRACT FROM AUTHOR]

We introduce a generic inequality of the form f 2 (x) ≤ f (s) f (2 x − s) and show that many special functions such as gamma and polygamma functions, Riemann zeta function, beta and incomplete beta functions, Gauss and confluent hypergeometric functions, elliptic integrals and error function satisfy such type of the inequality. We also obtain new inequalities of this type for inverse trigonometric functions arcsin ⁡ x and arctan ⁡ x. [ABSTRACT FROM AUTHOR]

The grey pattern problem is to select a pattern of p points in a square that is replicated in adjacent squares so that they are spread out as uniformly as possible. The goal is to cover a large area with many squares of the same pattern of p points. In the original formulation a special objective function is designed. In this paper we suggest the criterion of maximizing the minimum distance between points in the same square and in the eight adjacent squares, four with a common side and four with a common vertex. We prove properties of the proposed objective, and propose alternate formulations of the model. Extensive computational experiments are reported on instances using Euclidean distances and Manhattan distances with good results. [ABSTRACT FROM AUTHOR]

The widely known hermite-hadamard-Fejer type inequalities are so important in the field of mathematical analysis. Many researchers have studied on these inequalities. In this paper, we have obtained several inequalities related to the Hermite-Hadamard inequality for a special class of the functions called uniformly convex functions. We have also presented applications of these obtained inequalities in some error estimates for higher moments of random variables. [ABSTRACT FROM AUTHOR]

Some new relations for the confluent Horn functions H 3 (c) (a , b , c ; d ; w , z) , H 4 (c) (a , c ; d ; w , z) and H 9 (c) (a , b ; c ; w , z) are obtained including differentiation and integration formulas, series and reduction formulas. Some generating functions for various special functions are given in terms of these Horn functions. [ABSTRACT FROM AUTHOR]

Various relations including differentiation formulas, connection with hypergeometric functions, integral representations, formulas of summation and series representations for three types of Neumann polynomials are derived. [ABSTRACT FROM AUTHOR]

We construct Lyapunov functionals for linear neutral delay systems in the Hale's form. Our construction is made for no differentiable initial functions, as required for such a class of systems, but not thus considered in the existing works. The functionals depend on a special matrix-valued function which is the natural extension of the Lyapunov matrix for retarded delay systems. The constructed functionals allow deriving a converse result for exponential stability of the systems and exponential estimates for the solutions. [ABSTRACT FROM AUTHOR]

Over the last two decades, special matrix functions have become a major area of study for mathematicians and physicists. The famous four Appell hypergeometric matrix functions have received considerable attention by many authors from different points of view. The present paper is devoted to provide further investigations on the two variables second Appell hypergeometric matrix functions. Precisely, we derive certain mathematical properties such as limit formulae of Humbert hypergeometric matrix functions, recursion formulae using contiguous matrix relations, some differentiation and summation formulae. We also find general differential operators which induce some differential equations. This enriches the theory of special matrix functions. The obtained results are believed to be newly presented. [ABSTRACT FROM AUTHOR]

In this paper, we study the quantum motions of the non-relativistic particles under the influence of the flux field in the background of a point-like global monopole with Kratzer plus generalised Yukawa potential. The solutions of the quantum system under consideration are obtained using the parametric Nikiforov–Uvarov method by employing the Greene–Aldrich approximation scheme in the centrifugal and reciprocal terms that appear in the radial equation. Afterwards, we use another approximation called the Taylor series expansion up to the first order in the exponential terms and solve the radial equation analytically using the confluent hypergeometric function. It is shown that the topological defect of a point-like global monopole influences the eigenvalue solutions of the non-relativistic particles and gets modified by this defect compared to the flat space result with this superposed potential. We also show that the energy eigenvalue of the particles depends on the quantum flux field and thus gets shifted. Finally, we compare the eigenvalue solutions obtain for these two approximations scheme used here and show graphically that the results are different in these schemes. [ABSTRACT FROM AUTHOR]

Recently, the applications of special functions of matrix arguments have received more attention in many fields, such as theoretical physics, number theory, probability theory, engineering and theory of group representations. Gaining enlightenment from these works, in this paper, we introduce the degenerate gamma matrix function, the degenerate zeta matrix function, the degenerate digamma matrix function, the degenerate polygamma matrix function and the degenerate Gauss hypergeometric matrix function. Basic properties of these functions are discussed. In addition, we derive some interesting formulas related to these functions and special cases. [ABSTRACT FROM AUTHOR]

A modification of Watson's lemma for Laplace transforms ∫ 0 ∞ f (t) e − z t d t was introduced in [Nielsen, 1906], deriving a new asymptotic expansion for large | z | with the extra property of being convergent as well. Inspired in that idea, in this paper we derive asymptotic expansions of two-dimensional Laplace transforms F (x , y) := ∫ 0 ∞ ∫ 0 ∞ f (t , s) e − x t − y s d t d s for large | x | and | y | that are also convergent. The expansions of F (x , y) are accompanied by error bounds. Asymptotic and convergent expansions of some special functions are given as illustration. [ABSTRACT FROM AUTHOR]

q-analogs of special functions, including hypergeometric functions, play a central role in mathematics and have numerous applications in physics. In the theory of probability, q-analogs of various probability distributions have been introduced over the years, including the binomial distribution. Here, I propose a new refinement of the binomial distribution by way of the quantum binomial theorem (also known as the noncommutative q-binomial theorem), where the q is a formal variable in which information related to the sequence of successes and failures in the underlying binomial experiment is encoded in its exponent. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new class of optimization problems, which maximize the terminal wealth and accumulated consumption utility subject to a mean–variance criterion controlling the final risk of the portfolio. The multiple-objective optimization problem is firstly transformed into a single-objective one by introducing the concept of overall 'happiness' of an investor defined as the aggregation of the terminal wealth under the mean–variance criterion and the expected accumulated utility, and then solved under a game-theoretic framework. We have managed to maintain analytical tractability; the closed-form solutions found for a set of special utility functions enable us to discuss some interesting optimal investment strategies that have not been revealed in the literature before. [ABSTRACT FROM AUTHOR]

Some new relations for the Horn function H 7 (a , b , b ′ ; c ; w , z) are obtained including differentiation, integration, reduction and summation formulas. Some generating functions for various special functions are given in terms of these Horn functions. [ABSTRACT FROM AUTHOR]

In this paper, Hyers–Ulam stability of the diffusion equation on the half space R + n + 1 with n ≥ 2 is investigated: ∂ u ∂ t (x , y , t) =△ u (x , y , t) where x ∈ R n , y>0, and t>0. Using integral transforms, precisely, Fourier transform with respect to the tangential space x and Laplace transform for time t, we obtain the generalized Hyers–Ulam stability of diffusion equation in the half space R + n + 1 involving special functions such as gamma and error functions. [ABSTRACT FROM AUTHOR]

Some new relations for the Horn function H6(a, b, b';w, z) and confluent Horn function H(c)8 (a, b;w, z) are obtained including differentiation and integration formulas, series and values for specific choice of parameters and variables. Some generating functions for various special functions are given in terms of these Horn functions. [ABSTRACT FROM AUTHOR]

Some special functions are particularly relevant in applied probability and statistics. For example, the incomplete beta function is the cumulative central beta distribution. In this paper, we consider the inversion of the central Student's-t distribution which is a particular case of the central beta distribution. The inversion of this distribution function is useful in hypothesis testing as well as for generating random samples distributed according to the corresponding probability density function. A new asymptotic representation in terms of the complementary error function will be one of the important ingredients in our analysis. Aswewill show, this asymptotic representation is also useful in the computation of the distribution function. We illustrate the performance of all the obtained approximations with numerical examples. [ABSTRACT FROM AUTHOR]

The theory of Pólya ensembles of positive definite random matrices provides structural formulas for the corresponding biorthogonal pair, and correlation kernel, which are well suited to computing the hard edge large N asymptotics. Such an analysis is carried out for products of Laguerre ensembles, the Laguerre Muttalib–Borodin ensemble, and products of Laguerre ensembles and their inverses. The latter includes, as a special case, the Jacobi unitary ensemble. In each case, the hard edge scaled kernel permits an expansion in powers of 1/N, with the leading term given in a structured form involving the hard-edge scaling of the biorthogonal pair. The Laguerre and Jacobi ensembles have the special feature that their hard edge scaled kernel – the Bessel kernel – is symmetric and this leads to there being a choice of hard edge scaling variables for which the rate of convergence of the correlation functions is O (1 / N 2). [ABSTRACT FROM AUTHOR]

In this article, we compute the generating function of the joint distribution of the first letter and descents statistics on six avoidance classes of permutations corresponding to two patterns of length four thereby demonstrating their equivalence. This distribution is in turn shown to be equivalent to the distribution on a restricted class of inversion sequences for the statistics that record the last letter and number of distinct positive letters, affirming a recent conjecture of Lin and Kim. Members of each avoidance class of permutations and also of the class of inversion sequences are enumerated by the nth large Schröder number, and thus, one obtains a new bivariate refinement of these numbers as a consequence. We make use of auxiliary combinatorial statistics to establish a system of recurrences for the distribution in question in each case and define special generating functions (specific to the class) based on the system. In some cases, we utilize the conjecture itself in a creative way to aid in solving the functional equations satisfied by these associated generating functions and in others use the kernel method. [ABSTRACT FROM AUTHOR]

Some new relations for the Horn function H 5 (a , b ; c ; w , z) and confluent Horn function H 5 (c) (a ; c ; w , z) are obtained including differentiation and integration formulas, series and values for specific choice of parameters and variables. Some generating functions for various special functions are given in terms of these Horn functions. [ABSTRACT FROM AUTHOR]

Many indefinite integrals are derived for Bessel functions and associated Legendre functions from particular transformations of their differential equations which are closely linked to Wronskians. A large portion of the results for Bessel functions is known, but all the results for associated Legendre functions appear to be new. The method can be applied to many other special functions. All results have been checked by differentiation using Mathematica. [ABSTRACT FROM AUTHOR]

Due to the increased knowledge of nutrients and functional ingredients for their positive effects on health, people are tending to choose healthy food products including the snacks rich in these nutrients and ingredients. Healthy meal replacement products, which are calorie-controlled, nutritionally-balanced or/and functionally-fortified "meals," are another food product that meets people's health needs. Novel processing technologies enable the production of new products meeting the expectations of consumers. Among these technologies, freeze drying based processing technologies are especially suitable for producing products containing abundant nutrients and functional ingredients due to the drying environment with lower temperature and oxygen content. In this paper, the characteristics of freeze drying and the novel processing technologies based on freeze drying applied to produce healthy snacks and meal replacement products are critically reviewed, the new freeze dried healthy snacks and meal replacement products enriched with special nutrition and function are also discussed. [ABSTRACT FROM AUTHOR]

This article reports on an exploration into the human relationships practices of school leaders at faith-based schools as viewed through a servant leadership lens. The researchers employed a qualitative research design. Data were collected by means of individual semi-structured interviews and focus group discussions. The interviews were audio-taped and transcribed, and the data were analysed using a process of deductive data analysis. The following themes were employed to understand the human relationship practices of faith-based school leaders: teamwork, collaborative environment, development of others, and enhancing values-based attributes in day-to-day leadership. Overall, the principals were found to be efficacious leaders who were involved in a hands- on manner in both task-orientated and person-orientated activities, such as daily managerial and leadership functions with a special focus on the servant leadership ideal. Among the participants, there was a common understanding that leadership was not practiced as an individual action but was practiced by involving other school members. We strongly recommend that the ongoing professional development of school principals should include components related to servant leadership in their day-to-day leadership practices. [ABSTRACT FROM AUTHOR]

It is shown that an appropriate combination of methods, relevant to operational calculus and to special functions, can be a very useful tool to establish and treat a new modified 2D-Laguerre polynomials. We explore the formal properties of the operational identities to derive a number of properties of the new modified 2D-Laguerre polynomials and discuss the operational links with various known polynomials. [ABSTRACT FROM AUTHOR]

Some new relations for the Horn function H 4 (a , b ; c , c ′ ; w , z) and confluent Horn function H 7 (c) (a ; c , c ′ ; w , z) are obtained including differentiation and integration formulas, series and for special values of parameters and variables. Some generating functions for various special functions are given in terms of these Horn functions. [ABSTRACT FROM AUTHOR]

As of late, the study of Fractional Calculus (FC) and Special Functions (SFs) has been interestingly prompted in various realms of mathematics, engineering and sciences. This is due to the considerable demonstrated potential of their applications. Among these SFs, Gamma function and Mittag-Leffler functions are the most renowned and distinguished. Numerous authors continue to study this line. The current analysis attempts to introduce and further examine new modifications of Gamma and Kummer function in terms of Mittag-Leffler functions, respectively. Several attributes and formulations of this new Kummer-type function that include integral representations, Beta transform, Laplace transform, derivative formulas, and recurrence relation are investigated. Furthermore, outcomes of Riemann-Liouville fractional integral and fractional derivative in relation to this newly established Kummer function are also investigated. [ABSTRACT FROM AUTHOR]

A method was presented recently for deriving integrals of special functions using two kinds of integrating factor for the homogeneous second-order linear differential equations which many special functions obey. The classical orthogonal polynomials are well suited for this method, and results are given here for Jacobi polynomials. Results have been previously presented for Gegenbauer, Hermite and Laguerre polynomials. All the integrals presented here appear to be new and have been checked using Mathematica. [ABSTRACT FROM AUTHOR]

Some new relations for the Horn function H 3 a , b ; c ; w , z , confluent Horn function H 6 (c) a ; c ; w , z and Humbert function Φ 3 b ; c ; w , z are obtained including differentiation and integration formulas, series and values for specific choice of parameters and variables. Some generating fuctions for various special functions are given in terms of these Horn functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the incomplete second Appell hypergeometric matrix functions in terms of the incomplete Pochhammer matrix symbols. We also derive certain properties such as matrix differential equation, integral formula, recurrence relation, recursion formula and differentiation formula of the incomplete second Appell hypergeometric matrix functions. This enriches the theory of special matrix functions. The results listed here are believed to be new. [ABSTRACT FROM AUTHOR]