The article represents the elementary and general introduction of some characterizations of the extended gamma and beta Functions and their important properties with various representations. This paper provides reviews of some of the new proposals to extend the form of basic functions and some closed-form representation of more integral functions is described. Some of the relative behaviors of the extended function, the special cases resulting from them when fixing the parameters, the decomposition equation, the integrative representation of the proposed general formula, the correlations related to the proposed formula, the frequency relationships, and the differentiation equation for these basic functions were investigated. We also investigated the asymptotic behavior of some special cases, known formulas, the basic decomposition equation, integral representations, convolutions, recurrence relations, and differentiation formula for these target functions by studying. Applications of these functions have been presented in the evaluation of some reversible Laplace transforms to the complex of definite integrals and the infinite series of related basic functions.
Researchers abstractly consider natural phenomena when modeled according to a classification that in-cludes three cases according to their specific total traceability, the first case as a deterministic value, the second as random, and the third as a chaotic phenomenon whose values are cited in the cases described according to the target models. Stochastic models include the largest aspect of the targeted phenomena. Therefore, the probability and statistics has been the wide field in which researchers go to study random issues, where the random variable is described as a function that expresses the random quantity, a function that expresses the value of the results of a random phenomenon at the resulting values of the function (Blitzstein et al., 2014).
A random variable is expressed in mathematics as a measurable function in a certain period of the pro-bability space adopted in the period (0,1), and generally takes the form of real numerical quantities, where the probability values of a random variable may represent the possible outcomes of a random experiment, or the possible outcomes of a previous experiment. It is in fact, unconfirmed (Ferreira et al., 2000). Probability is a result of many reasons, in-cluding that measurements are inaccurate, uncertain, and because of random theoretical results or are produced through incomplete random results tainted in two parts different. The random variable is attri-buted to a probability distribution that determines the probability of the values that the value of the random variable takes. It can be a discrete random variable on a countable set, connected to a partial period of real numbers, or a combination of both types, which are very rare.
The conditions of the well-known probability density function have been met, and there is still a difference between two random variables with the same pro-bability distribution from the point of view that they are randomly related or independent, and this is clear from the results of choosing the random values of the random variable according to the probability distribution function investigated (Johnson et al., 1994). Since the nineteenth century, many scholars have been developing research on probability dis-tributions that reflect statistical phenomena with a high level of stochastic inference, and allow real phenomena to be traced within appropriate propor-tions to the real world. Pearson, (1895) was able to present a model for generating continuous distri-butions emanating from differential equations, which took its name, and the distribution of Pearson models, then the models developed by the researcher Johnson, (1949), followed by Burr models, and other wonderful models that achieved stimulating results, in addition to models That radically changed the view of scientists about probability distributions and their applications (Bachioua, 2013; 2014). Tukey, (1960) was able, through alignment models, to develop methods based on quantitative functions and find a probability distribution that allows the adaptation of continuous and discrete random dis-tributions (Bachioua & Shaker, 2006). With the new developed results achieved by these models and the expansion in the field of application, many articles have appeared interested in trading the probability distributions and searching for the relationship between each other, where I heard the results obt-ained from making approximate maps of the links between most of the probability distributions and the special case involving each other (AL Jarrah et al., 2014). Interest has expanded in recent years to deve-loping and extending innovative methods for gene-rating statistical distributions, finding some distri-butions that include most possible models for other distributions that do not conflict with the law of central tendency to achieve general goals and flexi-bility to reduce margins of errors when genera-ting pseudorandom numbers. Lee et al. (2013) noted that most of the methods developed after the 1980s are methods of standardization for several reasons, including that these new methods are based on the idea of combining two existing distributions or adding additional parameters to the existing dis-tribution to create a new family distribution to rep-resent a combination of random models (Lee et al., 2013).
The probability density function (pdf) of continuous distributions is seen as a function of the random variable that fulfills the following two condi-tions:
Through the two conditions mentioned in equation (1) has been many functions that meet these two conditions, and based on the modeling of statistical distributions, many researchers were able to propose functions that took their name. In order to expand any pdf, the following formula was adopted:
Whereas, and the new pdf of the random variable . Many researchers have been able to propose a formula for the pdf as a generalized formula as follows:
The formula (3) is defined by Ferreira and Steel, (2006) in order to represent skewed distributions based on symmetrical distribution (Harold et al., 1972). Several important distribution families have been generated and derived, including the T-X family. (Alzatreh et al., 2013; Rudin &Walter, 1987) have studied some of their details. From generalized versions of equations (2) and (3), a more number of continuous distribution families can be created. In this work, we seek to address both extended gene-ralized gamma distribution and extended Beta dis-tribution, and demonstrate some properties and app-lications (Awadala et al., 2020).
History of Gamma Function and its Changes
In order to know most of the new formulas that have been derived from the generalization of the general form of the gamma function and the incomplete functions of the gamma function, it is noted that it is very important to give an integrated review of the history of the emergence and development of in-tegrative functions and how to trade the formulas emanating from the generalized formula, where the gamma function attracted the attention of some of the most prominent scientists Mathematics of all time. Initially the first function was first introduced by Leonhard Euler while pursuing the goal of generalizing the factorization of computations to incorrect values. In a letter dated 08/01/1730 to Christian Gold Bach, Sandifer, (2007) Suggest the first form of the gamma function, as follows:
The equation (4) named Euler function; this function belongs to category of the special functions and appears in various area as Zeta Riemann function, Hyper geometric series, and number theory (Abra-mowitz Milton & Stegun Irene, 1972).Until the middle of the last (twentieth) century, mathe-maticians relied on hand-made approximation tables in which values were calculated in the case of the gamma function, especially the tables computed by Gauss in 1813 and the ones computed by Legendre in 1825, which were used respectively In calculating the values of changing variables related to the significance of the gamma function values. If let in equation (1) for we get the fun-ction equation in the new form:
The equation (5) named gamma function and the relation is the important functional equation, for integer values the equation becomes .
This led Legendre in 1811 to decompose the gamma function into the incomplete gamma functions, and which represent (16): The upper incomplete gamma function is defined as:
Whereas the symbol for the definition of the lower incomplete gamma function is indicated as:
Because of the importance in simplifying the symbols, the regular form of the gamma function holds, and is defined as;
Choudhury and Zubair were able to extend the scope of these functions by introducing new formula, where when you put a new regularization variable in the form equation (5). We note the following relation equation by (16):
The formula for the gamma function can be shown in two basic parts by the incomplete upper form of the gamma function and the incomplete lower part of the incomplete gamma function, according to the formula:
Due to the huge success of the gamma function, many authors have defined and discussed different type of gamma function in recent years. Recently Kobayashi, (1991) considered a generalized gamma function in the generalized form by –
Kobayashi considered the field of flat-wave diffract-ion by tape using the Wiener-Hopf technique. Bachioua, (8) has demonstrated a new generalization of the gamma function proposed by Kobayashi earlier by introducing the parameter m the first time and representing the new gamma function through the following formula:
Bachioua later defines additional parameters, and studies the convergence of the extended generalized gamma function in terms of six parameters; it used modified forms of the generalized gamma function of the Kobayashi function, which is a special case when Bachioua fixes some parameters of the exten-ded function (8):
The properties of convergence for the integrative function were studied for the six parameters for , , , , positives and real numbers (8). Equ-ation (7) is a special case of equation (8), which is itself a special case of equation (9) and can be checked by fixing parameters at special values. The expanded gamma function can be reformulated to more broadly reflect most of the new modifications and add withdrawals to the exponential variables. The proposed formula of seven parameters is given the following formula:
In order to respond to the most important proposed formulas for expanding the formulas of the gene-ralized gamma function by expansion, the researcher seeks to continue to propose a new formula, which is the content of the later part of this article, as it is determined according to the parameters domain that makes the defined integral in terms of eight con-vergent parameters. The probability density function diagrams (pdf) for X indicate different values for the parameters of the gamma function for the most important changes in the shape of the graphics, which allows identifying the areas of possible app-lications, which are shown in the Fig 1.
The importance of the proposal is to extend the applications of the extended beta and gamma fun-ction descriptions in extending the usage forms for each of the transformation functions for both the gamma function and the beta function, and by adding weights that result from the additional para-meters, and new forms that allow the creation of general and complex integrals that allow the deve-lopment of specific tasks for the teachers. These ex-tensions add important applications in the use of probability distributions, tracking functions with high efficiency and reliability, and during the study of the range of guesses and predictions for extended problems that express mixed phenomena, which is an important part of the proposed model. The pro-posal for the extension of the gamma and beta functions, and the incomplete functions for each of them, also provides for some requirements resulting from the extension of the previous results for each of the gamma and gamma incomplete functions in a natural and simple way that allows achieving the target results, which are often difficult to produce in a simple form. Of course, some results similar to the previous results may have any extension of the extended functions indicated in the formula. What we asked for and got is that the extension results should be no less elegant or more complex than the original function of the canonical gamma, beta fun-ctions, since the numerical values of and can easily be obtained by most mathematical software of most mathematical programs can easily be obtained by alignment and simulation.
The proposed models allow providing great oppor-tunities for modeling operations of random pheno-mena, in addition to providing experiments based on pseudo-random numbers generated using simulation and skew methods, and this gives a wider scope for the applications of these functions in proposing com-prehensive probability distributions, in addition to physical applications related to generalized functions.
The author declares that this article content has no conflict of interest.
Academic Editor
Dr. Wiyanti Fransisca Simanullang Assistant Professor Department of Chemical Engineering Universitas Katolik Widya Mandala Surabaya East Java, Indonesia.
Associate Professor, Department of Basic Sciences, Prep. Year, P.O. Box 2440, University of Hail, Hail, Saudi Arabia.
Lahcene B. (2021). Some characterizations of the extended beta and gamma functions: properties and applications, Int. J. Mat. Math. Sci., 3(5), 101-112. https://doi.org/10.34104/ijmms.021.01010112