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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]

The new generalized hypergeometric function (Yang 핐-function) described by the contour-type Mellin–Barnes integral representation serves as the inspiration for this study. The incomplete Yang 핐-functions γ핐 r,sp,q(z) and Γ핐 r,sp,q(z) that we shall introduce here are the appropriate extension of a class of 핐-functions by merit of the gamma functions of incomplete type, γ(σ,x) and Γ(σ,x). In this work, our objective is to study several remarkable characteristics of these newly introduced incomplete 핐-functions viz; reduction formula, derivative, various integral transformations. Additionally, using these newly established incomplete functions, we formulate and solve a generalized fractional kinetic equation in terms of these incomplete 핐-functions. Further, we also propose the incomplete 핎-functions γ핎 p,qr,s and Γ핎 p,qr,s corresponding to a new generalization of the 핐-function. [ABSTRACT FROM AUTHOR]

Hundreds of special functions have been employed in applied mathematics and computing sciences for many centuries due to their outstanding features and wide range of applications. When considering the relevance of these consequences in the evaluation of generalized integrals, applied physics, and many engineering areas, the illustration of image formulas involving one or more variable special functions is significant under various definite integrals. In this paper, it is devoted to study the various integral identities involving incomplete Fox-Wright functions and Srivastava’s polynomials. It is shown that the integrals of the Fox-Wright functions are also the Fox-Wright functions but of greater order. Due to the fact that our results are unified, a substantial number of new results can be constructed as special instances from our leading results. The results obtained in this work are general in nature and very useful in science, engineering and finance. [ABSTRACT FROM AUTHOR]

The authors of this study set out to build a software reliability growth model (SRGM). Software reliability is a crucial attribute that has to be quantified and evaluated. In most cases, software errors happen at unpredictable times. In this article, the failure intensity of the single parameter length-biased exponential class SRGM has been characterized taking into account the Poisson process of the incidence of software faults. The parameters of the proposed SRGM under investigation are the scale parameter (θ1) and the total number of failures (θ0). It is considered that the experimenter may have previous knowledge of the parameters from past or earlier experiences in the form of gamma priors. The posterior probability may be obtained by combining the prior probability with the likelihood of the data, and Bayes estimators can then be suggested. [ABSTRACT FROM AUTHOR]

The current manuscript's goal is to determine the Boros integral with three parameters, which comprises of the multiplication of the incomplete I-function and a family of polynomials. Some interesting corollaries are provided as a specific case of our primary findings. [ABSTRACT FROM AUTHOR]

A Bromwich-Hansen series is extended to derive the infinite sum of the addition of two incomplete gamma functions in terms of the Hurwitz-Lerch Zeta function. This formula is then used to derive infinite sums and products involving elementary and trigonometric functions. In some cases the infinite products are highly-oscillatory and difficult to evaluate. Symmetric plots are produced to showcase interesting features of special cases of infinite product and summation forms. [ABSTRACT FROM AUTHOR]

The topic of this work is the supercritical geometric reproduction of particles in the model of a Markov branching process. The solution to the Kolmogorov equation is expressed by the Wright function. The series expansion of this representation is obtained by the Lagrange inversion method. The asymptotic behavior is described by using two different equivalent forms for the Laplace transform. They include the computation of the limit distribution and its moments. The exact formula for the asymptotic density is written in terms of the reduced Wright function. In particular, when the ultimate extinction probability q = 1/2, the density of the limit random variable is given by the incomplete gamma function. [ABSTRACT FROM AUTHOR]

We introduce a p-adic analogue of the incomplete gamma function. We also introduce quantities (m-values) associated to a function on natural numbers and prove a new characterization of p-adic continuity for functions with p-integral m-values. Combinatorial interpretations for the integral values of the incomplete gamma function and functions with m-values zero or one are obtained, which show that these functions count derangements in generalized symmetric groups and permutations with restricted cycle lengths. [ABSTRACT FROM AUTHOR]

Using a contour integral method, the de Montmort-Prudnikov sum is extended to derive a new series representation involving the incomplete gamma function. The series is uniformly convergent and completely analytical, which can be evaluated for general complex ranges of the parameters involved. Applications and evaluations of this formula are discussed. [ABSTRACT FROM AUTHOR]

The current manuscript's goal is to determine the Boros integral with three parameters, which comprises of the multiplication of the incomplete I-function and a family of polynomials. Some interesting corollaries are provided as a specific case of our primary findings. [ABSTRACT FROM AUTHOR]

In this paper, we construct some new image formulas for the incomplete H-and ...-functions under the Akel's M-transform. We also provide image formulas for the incomplete Meijer's G-functions, incomplete Fox-Wright functions and Fox's H-function, as special cases of our main findings in corollaries. [ABSTRACT FROM AUTHOR]

In this paper, we present new inequalities and bounds involving the incomplete gamma function and related functions. Many of these new inequalities and bounds are based upon results, some well known and some not as well known, from probability theory and reliability theory. When these results are combined with other mathematical techniques, some very good upper and lower bounds are obtained. In particular, improvements of previously discussed bounds are presented. [ABSTRACT FROM AUTHOR]

Nonlinear mixed-effects models allow modeling repeated measures over time. The fixed effects of these models allow incorporating covariates, whereas the random effects reflect the multiple sources of heterogeneity and correlation between and within the units. To estimate the parameters of these models, it is necessary to use iterative processes, which can be done through different approaches, some of which are applied by the statistical software SAS. In this work, through simulation, we studied the performance of the estimators of a Wood's incomplete gamma function, obtained through three different methods: linearization method through a Taylor series expansion on the empirical best linear unbiased predictor of random effects, applied by the NLINMIX macro, and expansion around the expected value of random effects, applied by both the NLINMIX macro and the NLMIXED procedure. We also investigated the impact of an incorrect specification of the covariance matrix for random errors. The linearization method through a Taylor series expansion on the empirical best linear unbiased predictor of random effects, applied by the NLINMIX macro, provided estimators with good performance, approximately normally distributed and with biases lower than those obtained with the other methods, even when the covariance matrix for random errors was incorrectly specified. [ABSTRACT FROM AUTHOR]

Stripe truncation in the normal distribution is introduced such that the variable is truncated when it is located on several intervals like stripes. This truncation includes single, double and elliptical truncation as special cases. Then, the moments and absolute moments of arbitrary orders for the deviation of the truncated variable from an arbitrary reference point are derived using closed-form formulas based on the incomplete gamma function. The corresponding absolute moments of non-integer valued orders are also derived employing the parabolic cylinder distribution. The formulas do not suffer from difficulties associated with the conventional recursive methods for higher-order moments. [ABSTRACT FROM AUTHOR]

We derive a closed-form expression for the infinite sum of the Hurwitz–Lerch zeta function using contour integration. This expression is used to evaluate infinite sum and infinite product formulae involving trigonometric functions expressed in terms of fundamental constants. These types of infinite sums and products have previously been and are currently studied by many mathematicians including Leonhard Euler. The results presented in this paper extend previous work by squaring parameters in the infinite sum of the Hurwitz–Lerch zeta function. This formula allows for new derivations featuring trigonometric functions with angles of powers of 2. The zero distribution of almost all Hurwitz–Lerch zeta functions is asymmetrical. A table of infinite products is produced highlighting the usefulness of this work and for easy reading by researchers interested in such formulae. Mathematica software was used in assisting with the numerical verification of the results in the tables produced. [ABSTRACT FROM AUTHOR]

Many researchers are interested in Lambert's law because of its relevance in light attenuation owing to the characteristics of a material through which the light passes. In this paper, we developed the Lambert's law that involves incomplete -functions. Next, while taking a course in the constraints of incomplete -functions, we give a few special cases of our model, and also mention some known results. [ABSTRACT FROM AUTHOR]

In this paper, we are motivated to establish the generalization of a R q p ⁢ (α , β ; z) {{}_{p}R_{q}(\alpha,\beta;z)} function in terms of incomplete exponential functions and obtain some properties of an incomplete exponential R q p ⁢ (α , β ; z) {{}_{p}R_{q}(\alpha,\beta;z)} function. Further we give generating relations for incomplete exponential R q p ⁢ (α , β ; z) {{}_{p}R_{q}(\alpha,\beta;z)} function. Some applications related to ground water pumping (hydrology) modelling and probability theory are also discussed. [ABSTRACT FROM AUTHOR]

We derive the infinite sum of the Lerch function in terms of the incomplete gamma function and the Lerch function. Special cases are evaluated in terms of fundamentals constants. All the results in this work are new. [ABSTRACT FROM AUTHOR]

With the arrival of the first truly global energy crisis, how to precisely forecast the hydroelectric generation becomes a hot spot for allowing governments to obtain more valuable information. A novel forecasting model, Seasonal Fractional Incomplete Gamma Nonlinear Grey Bernoulli Model (SFIGNGBM(1, 1)), is proposed in this paper to precisely forecast the hydroelectric generation in some countries. First, the seasonal raw data are classified into four seasonal groups based on their significant seasonal fluctuations. Second, a novel SFIGNGBM(1, 1) model is established by combining the Bernoulli equation, the fractional-order accumulation operator, and the incomplete gamma function to further optimize partial parameters in the forecasting model and improve the forecasting performance. Third, the Whale Optimized Algorithm (WOA) is employed to optimize the Bernoulli power exponent η , the fractional order parameter r , and the incomplete coefficient h for minimizing the MAPE values and enhancing the fitting precision. Finally, our results present that our proposed model outperforms a set of baseline forecasting models with the smallest three error measure values in all fitting results, and its MAPE values converge before 10 iterations. This indicates that our proposed model has a favorable forecasting performance with fast-convergence for hydroelectric generation in the elected countries. • A novel model is proposed to precisely forecast the hydroelectric generation. • The proposed model uses the seasonal data grouping to enhance the forecasting precision. • The proposed model uses three approaches to further optimize partial parameters. • A meta-heuristic optimization algorithm is used to further enhance the forecasting precision. [ABSTRACT FROM AUTHOR]

An important task in the analysis of graphs is separating nodes into densely connected groups with little interaction between each other. Prominent methods here include flow based graph cutting procedures as well as statistical network modeling approaches. However, adequately accounting for this, the so-called community structure, in complex networks remains a major challenge. We present a novel generic Bayesian probabilistic model for graph cutting in which we derive an analytical solution to the marginalization of nuisance parameters under constraints enforcing community structure. As a part of the solution a large scale approximation for integrals involving multiple incomplete gamma functions is derived. Our multiple cluster solution presents a generic tool for Bayesian inference on Poisson weighted graphs across different domains. Applied on three real world social networks as well as three image segmentation problems our approach shows on par or better performance to existing spectral graph cutting and community detection methods, while learning the underlying parameter space. The developed procedure provides a principled statistical framework for graph cutting and the Bayesian Cut source code provided enables easy adoption of the procedure as an alternative to existing graph cutting methods. [ABSTRACT FROM AUTHOR]

This current paper is a table of definite integrals involving hyperbolic and logarithmic functions expressed in terms of the incomplete gamma function and fundamental constants. All the results in this work are new. [ABSTRACT FROM AUTHOR]

The incomplete gamma function Γ(a, u) is defined by Γ(a, u) = ∫u∞ta−1e−tdt, where u > 0. Using the incomplete gamma function, we define a new Poisson like regular matrix P(μ) = (pnkμ) given by pnkμ = n! Γ(n+1, μ) e−μμkk! (0 ≤ k ≤ n), 0 (k > n), where μ > 0 is fixed. We introduce the sequence space ℓp(P(μ)) for 1 ≤ p ≤ ∞ and some topological properties, inclusion relations and generalized duals of the newly defined space are discussed. Also we characterize certain matrix classes and compact operators related to the space ℓp(P(μ)). We obtain Gurarii's modulus of convexity and investigate some geometric properties of the new space. Finally, spectrum of the operator P(μ) on sequence space c0 has been investigated. [ABSTRACT FROM AUTHOR]

In this paper, we extend the result of Paris [R.B. Paris, The Stokes phenomenon associated with the Hurwitz zeta function ζ(s, a), Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461(2053):297–304, 2005] on the exponentially improved expansion of the Hurwitz zeta function ζ(s, z), the expansion of which can be reduced to the large-z Poincaré asymptotics of ζ(s, z). Furthermore, we deduce some new series and integral representations of the Hurwitz zeta function ζ(s, z). [ABSTRACT FROM AUTHOR]

In this paper, we concentrate on the statistical properties of Gamma-X family of distributions. A special case of this family is the Gamma-Weibull distribution. Therefore, the statistical properties of Gamma-Weibull distribution as a sub-model of Gamma-X family are discussed such as moments, variance, skewness, kurtosis and Rényi entropy. Also, the parameters of the Gamma-Weibull distribution are estimated by the method of maximum likelihood. Some sub-models of the Gamma-X are investigated, including the cumulative distribution, probability density, survival and hazard functions. The Monte Carlo simulation study is conducted to assess the performances of these estimators. Finally, the adequacy of Gamma-Weibull distribution in data modeling is verified by the two clinical real data sets. Mathematics Subject Classification : 62E99; 62E15 [ABSTRACT FROM AUTHOR]

Recently, Srivastava, Saxena and Parmar [H. M. Srivastava, R. K. Saxena and R. K. Parmar, Some families of the incomplete H-functions and the incomplete H ¯ {\overline{H}} -functions and associated integral transforms and operators of fractional calculus with applications, Russ. J. Math. Phys. 25 2018, 1, 116–138] suggested incomplete H-functions (IHF) that paved the way to a natural extension and decomposition of H-function and other connected functions as well as to some important closed-form portrayals of definite and improper integrals of different kinds of special functions of physical sciences. In this article, our key aim is to present some new integral transform (Jacobi transform, Gegenbauer transform, Legendre transform and 𝖯 δ {\mathsf{P}_{\delta}} -transform) of this family of incomplete H-functions. Further, we give several interesting new and known results which are special cases our key results. [ABSTRACT FROM AUTHOR]

In this article we study a mathematical model of the heat transfer in semi infinite material with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component. In particular, the temperature distribution in liquid and solid phases of such kind of body can be modeled by Stefan problem for the generalized heat equation. The method of solution is based on similarity principle, which enables us to reduce generalized heat equation to nonlinear ordinary differential equation. Moreover, we determine temperature solution for two phases and free boundaries which describe the position of boiling and melting interfaces. Existence and uniqueness of the similarity type solution is provided by using the fixed point Banach theorem. [ABSTRACT FROM AUTHOR]

We prove the nonintegrability of the susceptible–exposed–infected–removed (SEIR) epidemic model in the Bogoyavlenskij sense. This property of the SEIR model is different from the more fundamental susceptible–infected–removed (SIR) model, which is Bogoyavlenskij-integrable. Our basic tool for the proof is an extension of the Morales–Ramis theory due to Ayoul and Zung. Moreover, we introduce three new state variables and extend the system to a six-dimensional system to treat transcendental first integrals and commutative vector fields. We also use the fact that the incomplete gamma function Γ (α , x) is not elementary for α ⁄ ∈ N , of which a proof is included. • We prove the Bogoyavlenskij-nonintegrability of the SEIR epidemic model. • This result contrasts with the SIR model, which is Bogoyavlenskij-integrable. • Our basic tool for the proof is the generalized Morales–Ramis theory. [ABSTRACT FROM AUTHOR]

A specific type of convex functions is discussed. By examining this, we investigate new Hermite–Hadamard type integral inequalities for the Riemann–Liouville fractional operators involving the generalized incomplete gamma functions. Finally, we expose some examples of special functions to support the usefulness and effectiveness of our results. [ABSTRACT FROM AUTHOR]

Fractional relaxation equations, as well as relaxation functions timechanged by independent stochastic processes have been widely studied (see, for example, [21], [33] and [11]). We start here by proving that the upperincomplete Gamma function satisfies the tempered-relaxation equation (of index ρ ∈ (0, 1)); thanks to this explicit form of the solution, we can then derive its spectral distribution, which extends the stable law. Accordingly, we define a new class of selfsimilar processes (by means of the n-times Laplace transform of its density) which is indexed by the parameter ρ: in the special case where ρ = 1, it reduces to the stable subordinator. Therefore the parameter ρ can be seen as a measure of the local deviation from the temporal dependence structure displayed in the standard stable case. MSC 2010: Primary 26A33; Secondary 34A08, 33B20, 60G52, 60G18 [ABSTRACT FROM AUTHOR]

In the paper we present two incomplete Gaussian hypergeometric formulas in summation form by specific known formulas. We also developed each of these formulas and how they use to derive double series identities in general forms. [ABSTRACT FROM AUTHOR]

Recently, Forde et al. [The Rough Bergomi model as H → 0 - skew flattening/blow up and non-Gaussian rough volatility; preprint] found an explicit expression for the third moment of the log-price in the rough Bergomi model, in terms of a double integral, whose integrand involves a hypergeometric function. One of the parameters of this financial market model, the Hurst parameter H, is observed to be small in practice. We analyse the third moment asymptotically as H tends to zero, using as our main tools hypergeometric transformation formulas and uniform asymptotic expansions for the incomplete gamma function. [ABSTRACT FROM AUTHOR]

This paper is motivated by an open problem of inequalities of confluent hypergeometric functions. Our goal is to derive some inequalities for error, incomplete gamma and Whittaker's functions. [ABSTRACT FROM AUTHOR]

We present a computational procedure to evaluate the integral ∫yxsp-1e-μsds for 0 ≤ x < y ≤ +∞,μ = ±1, p> 0, which generalizes the lower (x=0) and upper (y=+∞) incomplete gamma functions. To allow for large values of x, y, and p while avoiding under/overflow issues in the standard double precision floating point arithmetic, we use an explicit normalization that is much more efficient than the classical ratio with the complete gamma function. The generalized incomplete gamma function is estimated with continued fractions, with integrations by parts, or, when x ≈ y, with the Romberg numerical integration algorithm. We show that the accuracy reached by our algorithm improves a recent state-of-the-art method by two orders of magnitude, and it is essentially optimal considering the limitations imposed by floating point arithmetic. Moreover, the admissible parameter range of our algorithm (0 ≤ p,x,y ≤ 1015) is much larger than competing algorithms, and its robustness is assessed through massive usage in an image processing application. [ABSTRACT FROM AUTHOR]

This paper is motivated by an open problem of inequalities of confluent hypergeometric functions. Our goal is to derive some inequalities for error, incomplete gamma and Whittaker's functions. [ABSTRACT FROM AUTHOR]

In this article, we established an equation of internal blood pressure involving incomplete H̄-functions. Next, we give some special cases by specializing the parameters of incomplete H̄-functions (for example, H̄-functions, generalized Wright hypergeometric function, generalized Riemann zeta function and generalized Wright Bessel functions) and also mentioned some known results. [ABSTRACT FROM AUTHOR]

There is no standard procedure for calculating the generalized temperature integral, instead myriads of different approximations to it are applied in the processing of thermogravimetric analysis data. This work presents first an integration procedure based on the Simpson rule that generates reference values of the generalized temperature integral. It also reviews the available representations of the temperature integral in power series, and presents the conversion of its generalized form into the form of special functions. From the comparison with the reference values from integration it was concluded that for argument values of practical interest the generalized temperature integral is best computed as the incomplete gamma function. [ABSTRACT FROM AUTHOR]

A new analytical formula for the Arrhenius integral has been proposed by using incomplete gamma functions, which is simple, accurate and reliable. The proposed formula has compared with several published the Arrhenius integral approaches, and is in agreement with the other approaches. Compared with the other approximate formulas proposed in the literature, our proposed method gives more accurate values in the precision of the activation energy as a function of x and temperature. [ABSTRACT FROM AUTHOR]

We introduce an algorithm for the evaluation of the Incomplete Gamma Function, P(m, x), for all m, x > 0. For small m, a classical recursive scheme is used to evaluate P(m, x), whereas for large m a newly derived asymptotic expansion is used. The number of operations required for evaluation is O(1) for all x and m. Nearly full double and extended precision accuracies are achieved in their respective environments. The performance of the scheme is illustrated via several numerical examples. [ABSTRACT FROM AUTHOR]

Let b ∈ ℝ, let c > 0, let x > 0, and let Gb,c(x) = e-x / xb Pc(x) with Pc(x) = ∞Σ k=0 xk / Γ(c + k) . We prove that Gb,c is completely monotonic on (0,∞) if and only if b ≥ 0 and b+c ≥ 1. Moreover, we present various functional inequalities for Pc. Among others, we show that if c ∈ (0, 1), then, for x, y > 0 we have e < Pc(1/x)x Pc(1/y)y / Pc(1/(x + y))x+y . If c > 1, then the reverse inequality holds for x, y > 0. In both cases, the constant bound e is best possible. [ABSTRACT FROM AUTHOR]

The SIR (Susceptible–Infected–Removed) is one of the simplest models for epidemic outbreaks. The present paper derives a novel, simple, analytical asymptotic solution for the I-variable, which is valid on the entire real line. Connections with the Gompertz and Gumbel distributions are also demonstrated. The approach is applied to the ongoing coronavirus disease 2019 (COVID-19) pandemic in four European countries — Belgium, Italy, Sweden, and Bulgaria. The reported raw incidence data from the outbreaks in 2020–2021 have been fitted using constrained least squares. It is demonstrated that the asymptotic solution can be used successfully for parametric estimation either in stand-alone mode or as a preliminary step in the parametric estimation using numerical inversion of the exact parametric solution. [ABSTRACT FROM AUTHOR]

Mazucheli et al. [On the one parameter unit-Lindley distribution and its associated regression model for proportion data. J. Appl. Stat. 46 (2019), pp. 700–714] introduced the unit Lindley distribution. The expressions given for the moments and incomplete moments are either not correct or not in closed form. We derive closed form expressions moments and incomplete moments. The expressions are elementary except for the exponential integral. [ABSTRACT FROM AUTHOR]

The association of birth type (single vs. double) and pregnancy type (single vs. double) with lactation performance and lactation curve was investigated in Holstein cows in Iran. Data of 243298 lactations on 138021 cows in 261 herds collected during January 2000 to December 2012 were used. The average twinning rate was 3.64% and ranged from 1.44% (primiparous) to 5.41% (multiparous). Factors associated with twining were calving season, calving year, herd, and parity. The rate of twinning increased from 2.49% in 2000 to 3.42% in 2012. The probability of twinning was higher in multiparous cows than that in primiparous [Odds ratio (OR) (95% confidence interval (CI))= 3.64 (3.44-3.84) for multiparous vs. primiparous cows]. The probability of twin pregnancy was higher in cows previously delivering twins than that in those delivering singles [OR (95% CI)= 3.17 (3.02-3.34) for cows delivering twins vs. cows delivering singles]. The lactation curve for cows delivering singles in previous gestation and pregnant with twins in subsequent gestation tended to be lower and flatter. The incidence of twin pregnancy was increased as the cow reached its peak yield earlier, and produced more milk at peak lactation. Primiparous cows delivering singletons in their previous gestation and having twins in their subsequent gestation produced more 100-d, 200-d and 305-d milk than those pregnant with singletons. Multiparous cows that delivered singleton in previous gestation and were pregnant with twin in the subsequent gestation, produced more milk during the first 100-d and 200-d of lactation than those pregnant with singles. [ABSTRACT FROM AUTHOR]

We derive single integral representations for the exact distribution of the sum of independent generalized Pareto random variables. The integrands involve the incomplete and complementary incomplete gamma functions. Applications to insurance and catastrophe bonds are described. [ABSTRACT FROM AUTHOR]

The objectives were to estimate (co)variance components and genetic parameters for lactation curve traits in Holstein dairy cows in Iran. The used data were records on Holstein cows collected during January 2003 to December 2012 by the Animal Breeding Center of Iran (Karaj, Iran). In order to describe the lactation curve, an incomplete gamma function was fitted to 691200 test-day records on 96263 lactations, 377696 test-day records on 52168 lactations, and 182143 test-day records on 24951 lactations for the first three parities, respectively. Lactation curve traits that were analyzed included a scaling factor associated with initial yield, the inclining and declining slopes of the curve, peak time, peak yield, 305-d milk yield and a lactation persistency measure derived from the incomplete gamma function. The estimated heritability of lactation curve traits for the first, second and third lactation ranged from 0.012 to 0.29, 0.017 to 0.15, and 0.02 to 0.13, respectively. Genetic correlations among lactation curve traits for the first, second and third lactation ranged from -0.68 to 0.99, -0.84 to 0.98 and -0.90 to 0.94, respectively. The estimated repeatability of lactation curve traits ranged from 0.07 to 0.40. The moderate to large positive genetic correlations of 305-d milk yield with initial yield, peak yield and lactation persistency suggest that one of these traits could be used as a selection criterion to improve all four traits. However, the peak yield and 305-d milk yield were more heritable than lactation were persistency and initial yield. [ABSTRACT FROM AUTHOR]

In this paper, we derive some formulas for higher order derivatives ofr-Lambert functions. Moreover, an integration formula involving powers ofr-Lambert function is obtained. [ABSTRACT FROM PUBLISHER]

In this paper, some recursive relations of the derivatives of the gamma function Γ(α) and incomplete gamma function Γ(a, z) are obtained for complex number α ≠ 0,-1,-2,.... Moreover, the recurrence relation of the derivative of the k-gamma function Γk(x) are also given. Thus, the partial derivative ∂p+q/∂vp∂μq B(v; μ) of the Beta function B(v; μ) can also be represented for p, q = 0, 1, 2, · · ·. Based on these results, the partial derivatives of the extended, generalized complete and incomplete gamma functions are obtained. Furthermore, the partial derivative of confluent hypergeometric function is also considered. [ABSTRACT FROM AUTHOR]

This paper is concerned with some recursive relations of the derivatives of the Gamma function Γ(a) and incomplete Gamma function Γ(a,z) for the complex value of. In particular, dnΓ/dan(-m)(n,m = 0,1,2...) can be expressed as linear forms in djΓ/daj(1)(j = 0,1,....n + 1) while ∂nΓ/∂an(-m,n) can be represented as the combination of ∂jΓ/∂aj(1,z)(j = 0,1,...,n+1) and the elementary functions. With the aid of these results, we can establish the closed forms of some special integrals associated with Γ(a) and Γ(a,z), which can be expressed by the Riemann zeta functions and some special constants. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the integral representation problem of the generalized Bessel linear functional B[v], well-known by the Pearson equation that it satisfies: ... By means of some integral estimation and technical results including important inequalities of the incomplete gamma function and the exponential integral, we obtain an integral representation of B[v], for every real number v = -n, n ≥ 0. The connection formula between B[v] and the classical Bessel linear functional B(α) allows us to obtain an integral representation of B(α); for all real number α = -(n=2), n ≥0. [ABSTRACT FROM AUTHOR]

This paper considers the distributions of spacings between successive order statistics corresponding to a random sample from a two-parameter gamma distribution Γ(α,β ) . We prove that when the shape parameter of the underlying distribution is a positive integer, these spacings can be expressed as finite gamma mixtures. We present exact formulas for computing the distributions of the spacings. Then we present a Mathematica program to implement the results. [ABSTRACT FROM AUTHOR]