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Definite integrals containing gamma functions and other special functions are obtained. [ABSTRACT FROM AUTHOR]

Relying on the Hurwitz formula, we find closed-form formulas for the series over sine and cosine functions through the Hurwitz zeta functions, and using them and another summation formula for trigonometric series, we obtain a finite sum for some series over the Riemann zeta functions. We apply these results to the series over Bessel functions, expressing them first as series over the Riemann zeta functions. [ABSTRACT FROM AUTHOR]

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue computations with the help of some well-known results of Eisenstein series given by Ramanujan, Berndt, et al. As our main results, several series involving hyperbolic functions are evaluated and expressed in terms of z = F 1 2 (1 / 2 , 1 / 2 ; 1 ; x) and z ′ = d z / d x . When a certain parameter in these series is equal to π , the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented. [ABSTRACT FROM AUTHOR]

We introduce new extension of the extended Pochhammer symbol and gamma function by using the extended Mittag-Leffler function. We also present extension of the generalized hypergeometric function as well as some of their special cases by using this extended Pochhammer symbol. Further, we define the extension of the τ-Gauss hypergeometric function. Integral and derivative formulas involving the Mellin transform and fractional calculus techniques associated with this extended τ-Gauss hypergeometric function are also given. Also, new extended τ-Gauss hypergeometric function also provides a few more interesting and well-known results. This enriches the theory of special functions. The obtained results are believed to be newly presented. [ABSTRACT FROM AUTHOR]

Sample surveys are conducted to estimate parameters of subpopulation such as mean, variance and totals, etc. The utility of the supporting information lies in the calibration technique, we use to estimate the subpopulation parameter. In this article, we propose calibration-based estimators for a total of the subpopulations. The defined estimators are based on gamma and exponential models. Since both models have right-skewed or positively skewed forms, the defined estimators are appropriate for right-skewed or positively skewed supportive variables. Two situations are considered in which the total of the supporting variable hold for all subpopulations: (i) the total of subpopulations of the supporting variable is known in advance and (ii) the total subpopulations of the supportive variable are not known. Due to difficulty arising in the calibration weight, we have carried out a simulation study in terms of absolute relative bias and simulated relative standard error using Sweden Municipalities 1984 (Sarndal, Swensson, and Wretman, 1992, Appendix B). A simulation study shows that the estimator based on the gamma function is more efficient than ratio, generalized regression, and exponential estimators for all subpopulations in the considered situations. [ABSTRACT FROM AUTHOR]

The concept of special functions plays an important role in mathematical analysis and physics as well. In this paper, we study some different types of the special Gamma function defined on the two-fold fuzzy complex field, where we combine the classical Gamma function with the two-fold fuzzy algebra defined on complex numbers. On the other hand, many elementary properties of this new special function will be determined in terms of theorems and proofs. [ABSTRACT FROM AUTHOR]

In this study, we proposed a normalized time-fractional diffusion equation and conducted a numerical investigation of the dynamics of the proposed equation. We discretized the governing equation by using a finite difference method. The proposed normalized time-fractional diffusion equation features a different time scale compared to the conventional time-fractional diffusion equation. This distinct time scale provides an intuitive understanding of the fractional time derivative, which represents a weighted average of the temporal history of the time derivative. Furthermore, the sum of the weight function is one for all values of the fractional parameter and time. The primary advantage of the proposed model over conventional time-fractional equations is the unity property of the sum of the weight function, which allows us to investigate the effects of the fractional order on the evolutionary dynamics of time-fractional equations. To highlight the differences in performance between the conventional and normalized time-fractional diffusion equations, we have conducted several numerical experiments. [ABSTRACT FROM AUTHOR]

This research investigates specific classes of fractional integro-differential equations using a straightforward fractional calculus technique. The employed methodology yields a variety of compelling outcomes, including a generalized version of the well-established classical Frobenius method. The approach presented in this study primarily relies on fundamental theorems concerning the specific solutions of fractional integro-differential equations, utilizing the Mohand transform and binomial series extension coefficients. Additionally, advanced techniques for solving fractional integro-differential equations effectively are showcased. [ABSTRACT FROM AUTHOR]

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Several sharp approximations of the generalized-binomial-coefficient function having real arguments are presented on the basis of Stirling's approximation formula for Γ function. [ABSTRACT FROM AUTHOR]

In this study, we proposed a normalized time-fractional diffusion equation and conducted a numerical investigation of the dynamics of the proposed equation. We discretized the governing equation by using a finite difference method. The proposed normalized time-fractional diffusion equation features a different time scale compared to the conventional time-fractional diffusion equation. This distinct time scale provides an intuitive understanding of the fractional time derivative, which represents a weighted average of the temporal history of the time derivative. Furthermore, the sum of the weight function is one for all values of the fractional parameter and time. The primary advantage of the proposed model over conventional time-fractional equations is the unity property of the sum of the weight function, which allows us to investigate the effects of the fractional order on the evolutionary dynamics of time-fractional equations. To highlight the differences in performance between the conventional and normalized time-fractional diffusion equations, we have conducted several numerical experiments.

Zero factorial, defined to be one, is often counterintuitive to students but nonetheless an interesting concept to convey in a classroom environment. The challenge is to delineate the concept in a simple and effective way through the practice of justification, a familiar concept in mathematics and science education. In this regard two algebraic and one statistical justification using the squeeze theorem are presented. To assess the effectiveness of the justifications, a student survey was conducted at a comprehensive university incorporating the analysis of the pre- and post-presentation statements. They clearly present that the justifications are useful in giving credence to zero factorial equals one. Overall, the result from the online survey supports that the students preferred Justification 1. The justifications provide instructors alternative ways to initiate exploration of students’ intuitive set up of comprehending unobvious facts like zero factorial equals one. For a range of learners with their varied abilities to perform various mental activities most closely associated with learning and problem-solving, the justifications as simple alternative methods offer the potential to raise the current level of cognitive skills to inspire differentiated paths of learning. These are evident from survey results noting the role of statistical thinking and techniques.

We offer intriguing new insights into parameterized finite binomial sums, revealing elegant identities such as ∑ k = 0 , k ≠ n m + n (− 1) k n − k m + n k = (− 1) n m + n n (H m − H n) , where n , m are non-negative integers and H n is the harmonic number. These formulas beautifully capture the intricate relationship between harmonic numbers and binomial coefficients, providing a fresh and captivating perspective on combinatorial sums. [ABSTRACT FROM AUTHOR]

We investigate the relations between the rings E , G and D of values taken at algebraic points by arithmetic Gevrey series of order either −1 (E -functions), 0 (analytic continuations of G -functions) or 1 (renormalization of divergent series solutions at ∞ of E -operators) respectively. We prove in particular that any element of G can be written as multivariate polynomial with algebraic coefficients in elements of E and D , and is the limit at infinity of some E -function along some direction. This prompts to defining and studying the notion of mixed functions, which generalizes simultaneously E -functions and arithmetic Gevrey series of order 1. Using natural conjectures for arithmetic Gevrey series of order 1 and mixed functions (which are analogues of a theorem of André and Beukers for E -functions) and the conjecture D ∩ E = Q ‾ (but not necessarily all these conjectures at the same time), we deduce a number of interesting Diophantine results such as an analogue for mixed functions of Beukers' linear independence theorem for values of E -functions, the transcendence of the values of the Gamma function and its derivatives at all non-integral algebraic numbers, the transcendence of Gompertz constant as well as the fact that Euler's constant is not in E. [ABSTRACT FROM AUTHOR]

Equations are obtained for the quantum distribution functions over discrete states in systems of non-interacting fermions and bosons with an arbitrary, including small, number of particles. The case of systems with two levels is considered in detail. The temperature dependences of entropy, heat capacities and pressure in two-level Fermi and Bose systems are calculated for various multiplicities of degeneracy of levels. [ABSTRACT FROM AUTHOR]

The Kullback–Leibler divergence is a measure of the divergence between two probability distributions, often used in statistics and information theory. However, exact expressions for it are not known for multivariate or matrix-variate distributions apart from a few cases. In this paper, exact expressions for the Kullback–Leibler divergence are derived for over twenty multivariate and matrix-variate distributions. The expressions involve various special functions. [ABSTRACT FROM AUTHOR]

The zeta function is like a milestone in analytic number theory. Considering Euler's definition of the zeta function in this article, we will find some recursion at even integral points. We will also discuss the Dirichlet eta function, even Bernoulli numbers, and the gamma–zeta relation. Toward the end, we will define something known as the odd zeta function and find a recurrence for the same. [ABSTRACT FROM AUTHOR]

The beta (type 1) and inverted beta distributions are of vital importance in science, engineering and economics. In this article, we introduce a matrix variate generalization of the inverted beta distribution. By using linear and quadratic transformations, we obtain several matrix variate distributions including matrix variate generalized beta type 1 distribution. [ABSTRACT FROM AUTHOR]

The study introduces a new generalised family of distributions for cancer data modelling using a generalisation of the gamma function and a Topp-Leone-G distribution called the Gamma-Odd Topp-Leone-G (GOTL-G). Cancer data is normally characterised by complex heterogeneous properties like skewness, kurtosis, and presence of extreme values which makes it difficult to model using classical distributions. We derived multiple statistical properties including the linear representation, Re«yi entropy, quantile functions, distribution of order statistics, and maximum likelihood estimates which normally guarantees a positive effect on the generalisability of cancer data. Interestingly, we observed that these derived statistical properties make it possible for the generalisation of different models which are useful in the analysis, control, insurance, and survival of cancer patients. Our results show that this new family of distributions can be applied to a variety of data sets such as bladder and breast cancer data which exhibited high level of skewness and kurtosis as well as symmetric attributes. Therefore, we can conclude that the GOTL-G family of distributions can be extremely useful in capturing distinct complex heterogeneous properties normally exhibited by cancer patients. We recommend that this new family of distributions can be useful in modelling complex real-life applications including cancer data. [ABSTRACT FROM AUTHOR]

In this paper, we mainly prove the following conjectures of Z.-W. Sun (J. Number Theory 133 (2013), 2914–2928): let $p>2$ be a prime. If $p=x^2+3y^2$ with $x,y\in \mathbb {Z}$ and $x\equiv 1\ ({\rm {mod}}\ 3)$ , then \[ x\equiv\frac14\sum_{k=0}^{p-1}(3k+4)\frac{f_k}{2^k}\equiv\frac12\sum_{k=0}^{p-1}(3k+2)\frac{f_k}{({-}4)^k}\ ({\rm{mod}}\ p^2), \] and if $p\equiv 1\pmod 3$ , then \[ \sum_{k=0}^{p-1}\frac{f_k}{2^k}\equiv\sum_{k=0}^{p-1}\frac{f_k}{({-}4)^k}\ ({\rm{mod}}\ p^3), \] where $f_n=\sum _{k=0}^n\binom {n}k^3$ stands for the $n$ th Franel number. [ABSTRACT FROM AUTHOR]

The goal of the current paper is to present an integral type Favard-Szasz operators including Apostol-Genocchi poynomials. With the help of the moments, we investigate the order of convergence in terms of the first and the second order modulus of continuity and Peetres K-functional. We also examine the convergence in the weighted spaces of functions by means of weighted Korovkin type theorem. [ABSTRACT FROM AUTHOR]

Two summation theorems concerning the F 1 2 (1 / 2) -series due to Gauss and Bailey will be examined by employing the "coefficient extraction method". Forty infinite series concerning harmonic numbers and binomial/multinomial coefficients will be evaluated in closed form, including eight conjectured ones made by Z.-W. Sun. The presented comprehensive coverage for the harmonic series of convergence rate " 1 / 2 " may serve as a reference source for readers. [ABSTRACT FROM AUTHOR]

The Wallis power function is defined on \left (-\min \left \{ p,q\right \},\infty \right) by \begin{equation*} W_{p,q}\left (x\right) =\left (\dfrac {\Gamma \left (x+p\right) }{\Gamma \left (x+q\right) }\right) ^{1/\left (p-q\right) }\text { if }p\neq q\text { and }W_{p,p}\left (x\right) =e^{\psi \left (x+p\right) }. \end{equation*} Let p_{i},q_{i}\in \mathbb {R} with 0<\delta _{i}=p_{i}-q_{i}\leq 1, \theta _{i}=\left (1-\delta _{i}\right) /2 for i=1,2 and p_{1}+q_{1}=p_{2}+q_{2}=2\sigma +1. We prove that, if q_{1}>q_{2} then for any integer m\in \mathbb {N}, the function \begin{equation*} x\mapsto \left (-1\right) ^{m}\left [ \ln \frac {W_{p_{1},q_{1}}\left (x\right) }{W_{p_{2},q_{2}}\left (x\right) }-\sum _{k=1}^{m}\frac {a_{2k}\left (\theta _{1},\theta _{2}\right) }{\left (2k+1\right) \left (2k\right) \left (x+\sigma \right) ^{2k}}\right ] \end{equation*} is completely monotonic on \left (-\sigma,\infty \right), where \begin{equation*} a_{2k}\left (\theta _{1},\theta _{2}\right) =\frac {B_{2k+1}\left (\theta _{2}\right) }{\theta _{2}-1/2}-\frac {B_{2k+1}\left (\theta _{1}\right) }{\theta _{1}-1/2}. \end{equation*} This extends and generalizes some known results. [ABSTRACT FROM AUTHOR]

The Selberg integral has a twin ('the Dotsenko–Fateev integral') of the following form. We replace real variables x k in the integrand ∏ | x k | σ - 1 | 1 - x k | τ - 1 ∏ | x k - x l | 2 θ of the Selberg integral by complex variables z k , integration over a cube we replace by an integration over the whole complex space C n . According to Dotsenko, Fateev, and Aomoto, such integral is a product of Gamma functions. We define and evaluate a family of beta integrals over spaces C m × C m + 1 × ⋯ × C n , which for m = n gives the complex twin of the Selberg integral (with three additional integer parameters). [ABSTRACT FROM AUTHOR]

In this paper, we solve the question completely raised by Feng Qi and Dongkyu Lim in paper "Monotonicity properties for a ratio of finite many gamma functions" published in Advances in Difference Equations and get some properties about ratios of finitely many gamma functions such as complete monotonicity, logarithmically complete monotonicity, the Bernstein function property, null point and extreme value. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to introduce a new extension of the generalized Hurwitz-Lerch Zeta functions of two variables [abbreviated as H-L-Z-F-2V] and three variables [abbreviated as H-L-ZF-3V] involving the extended Beta function. We then systematically investigate its several interesting properties and related formulas as (for example) various integral representations, which provide certain new and known extensions of earlier corresponding results, a derivative formula, and Mellin-transform. We also consider some important special cases [ABSTRACT FROM AUTHOR]

In this paper, we define generalized Fourier and inverse Fourier transforms containing the h-exponential function in their kernels and give the fundamental properties of these transforms. We also compute the transforms of both the classical and the generalized Riemann-Liouville and Caputo fractional operators. In addition, we compute the transforms of some elementary and generalized special functions as well. Finally, as applications, we obtain the solutions of two differential equations with ordinary and fractional derivatives using the transforms we have defined. [ABSTRACT FROM AUTHOR]

A collection of functions organized according to their indexing based on non-negative integers is grouped by the common factor of fixed integer N. This grouping results in a summation of N series, each consisting of functions partitioned according to this modulo N rule. Notably, when N is equal to two, the functions in the series are divided into two subseries: one containing even-indexed functions and the other containing odd-indexed functions. This partitioning technique is widely utilized in the mathematical literature and finds applications in various contexts, such as in the theory of hypergeometric series. In this paper, we employ this partitioning technique to establish four distinct families of summation formulas for F 3 4 (1) hypergeometric series. Subsequently, we leverage these summation formulas to introduce eight categories of integral formulas. These integrals feature compositions of Beta function-type integrands and F 2 3 (x) hypergeometric functions. Additionally, we highlight that our primary summation formulas can be used to derive some well-known summation results. [ABSTRACT FROM AUTHOR]

In this paper, two new approximation formulas with monotonic remainders for the gamma function have been presented. Also, we present some numerical comparisons between our new approximation formulas and some known ones, which demonstrate the superiority of our results. [ABSTRACT FROM AUTHOR]

In the paper, the authors establish several identities and relations involving q-analogues of the Pochhammer k-symbol. Moreover, the authors generalize several identities and relations for q-analogues of the Catalan numbers and the Catalan–Qi numbers. [ABSTRACT FROM AUTHOR]

The generalized $M$-series is a hybrid function of generalized Mittag-Leffler function and generalized hypergeometric function. The principal aim of this paper is to investigate certain properties resembling those of the Mittag-Leffler and Hypergeometric functions including various differential and integral formulas associated with generalized $M$-series. Certain corollaries involving the generalized hypergeometric function are also discussed. Further, in view of Hadamard product of two analytic functions, we have represented our main findings in Hadamard product of two known functions.

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue computations with the help of some well-known results of Eisenstein series given by Ramanujan, Berndt, et al. As our main results, several series involving hyperbolic functions are evaluated and expressed in terms of z=F12(1/2,1/2;1;x) and z′=dz/dx. When a certain parameter in these series is equal to π, the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented.

Relying on the Hurwitz formula, we find closed-form formulas for the series over sine and cosine functions through the Hurwitz zeta functions, and using them and another summation formula for trigonometric series, we obtain a finite sum for some series over the Riemann zeta functions. We apply these results to the series over Bessel functions, expressing them first as series over the Riemann zeta functions.

In this paper, a new approximation formula for the gamma function and some of its symmetric inequalities are established. We prove the superiority of our results over Yang and Tian's approximation formula for the gamma function of order v − 9 . [ABSTRACT FROM AUTHOR]

The article focuses on exploring relations involving special functions like the gamma, beta, and zeta functions. Topics including their connections to integrals, infinite sums, inequalities and specific value calculations. It delves into the historical context and applications of special functions in mathematics, physics and engineering; and highlighting their diverse nature and importance in various mathematical disciplines.

In the field of histopathology, it has been observed that most images are of poor quality as a result of inadequate luminance and contrast. The first step in identifying and analyzing them is to enhance their quality while maintaining their inherent characteristics. This paper presents a novel method for enhancing histopathology images' important features, brightness preservation, and contrast by utilizing the adaptive intensity transformation gamma function in the wavelet domain. The RGB histopathological color image is first converted to the HSV color model. The V component is then stretched in order to compensate for any color distortion. In the proposed scheme, the stretched V component is split into four sub‐bands using discrete wavelet transform (DWT). Next, it determines the optimal value of the gamma correction function in the DWT lower sub‐band. A correction factor is applied using the singular value decomposition (SVD) technique to determine the optimal gamma function value using the nature‐inspired particle swarm optimization method (NIPSO). The corrected LL sub‐band is then applied to the inverse DWT (IDWT) together with the other unprocessed sub‐bands. In addition, the contrast of each feature is enhanced by employing an adaptive histogram equalization model that is contrast‐limited. The proposed method was evaluated through experiments and validations on a variety of histopathology images retrieved from multiple databases. Several quantitative studies have demonstrated that the proposed method outperforms existing enhancement techniques in terms of entropy, edge preservation index, contrast ratio, and Universal Image Quality Index (UIQI). This proposed technique increases contrast, while preserving the inherent characteristics of the original image, resulting in better quality histopathology images that can be used in disease diagnosis and inspection. [ABSTRACT FROM AUTHOR]

Tsionas [Ann Oper Res 323:229–245, 2023] considered parameter estimation under linex and double-linex loss functions. He was not able to provide details for the double-linex loss function because of not being able to derive a closed form. In this note, we show that a closed form can be derived. Further, we give details of estimation using the double-linex loss function by the method of maximum likelihood and Bayesian method. [ABSTRACT FROM AUTHOR]

This paper investigates an improved fixed-time stability theory together with a state feedback controller for a class of nonlinear stochastic systems. First, a delicate transformation is performed, and next, a Gamma function is utilized to directly derive the value of the integral function, which ultimately yields a fixed-time stabilization theorem with a higher precision upper bound for the settling time. Unlike the existing estimation process of amplifying twice, we only performed one amplification, which weakens the effect of amplification. Then, a state feedback controller is constructed for stochastic systems by the method of adding a power integrator. Utilizing the proposed stochastic fixed-time stability theory, simulations show that the intended controller ensures that the trivial solution of the suggested system is fixed-time stable in probability. The results of the simulation demonstrate that the suggested control scheme is meaningful. [ABSTRACT FROM AUTHOR]

Let a,b,c,d ∈ R with a+b = c+d = 2r +1. Then In Γ(x+a) Γ (x+b)/Γ(x+c) Γ (x+d) ~ ∞/Σ/k=1 B2k (θ1) - B2k (θ2)/k (2k-1) (x+r)2k-1 as x→∞, where (δ1,δ2) = (|a−b|,|c−d|) = (1−2θ1,1−2θ2). When 0 ≤ δ2 < δ1 ≤ 1, the function x→ (-1)m [1n Γ(x+a) Γ (x+b)/Γ(x+c) Γ (x+d) - m/Σ/k=1 B2k (θ1) - B2k (θ2)/k (2k-1) (x+r)2k-1 ] for m ∈ N is completely monotonic on (−r,∞). This yields some known and new results. [ABSTRACT FROM AUTHOR]

A novel enhancement method to improve resolution and contrast has been proposed to address the issues of blurring and distortion commonly encountered in traditional patterns. Initially, a discrete wavelet transform, a stationary wavelet transform, and an interpolation algorithm are used to obtain high-resolution images of traditional patterns. Subsequently, improved singular value matrix coefficients and reconstructed gamma function are used to enhance the image contrast to obtain high-resolution and contrast-enhanced patterns. Experimental results demonstrate the efficacy of this method, as evidenced by improved evaluation indexes, such as mean square error, peak signal-to-noise ratio, and structural similarity, in comparison to other existing methods. The proposed method effectively improves the quality of traditional patterns and offers significant contributions to research on the restoration and protection of traditional patterns. [ABSTRACT FROM AUTHOR]

The extension of the exponential function is the Mittag-Leffler function. In physical implications, the MittagLeffler function is becoming more and more crucial, as a result a number of researchers are investigating different generalizations and extensions associated with the Mittag-Leffler function. In this present article, we define the bicomplex r-parameter Mittag-Leffler function which is an extension of bicomplex one parameter Mittag-Leffler function. In addition, various properties of this function like integral representation, differential relation, duplication formula, bicomplex C-R equation, analyticity, region of convergence, order and type are established. Additionally, the bicomplex r-parameter MittagLeffler function’s Laplace transform and Caputo fractional derivative are produced. [ABSTRACT FROM AUTHOR]

In the present article, our main goal is to study the new extended beta, Gauss and confluent hypergeometric functions defined by using the two Fox-Wright functions as their kernel. Some applications of the new extended beta function to distribution theory are also discussed. [ABSTRACT FROM AUTHOR]

Using the Fourier–Gegenbauer series, we prove several identities that generalize known results. In particular, it is proved, that ∑ n = 0 ∞ 1 4 n ( 2 n n ) z − 2 n ( z − 1 / 2 n ) ( z n ) 3 = tan ⁡ (π z) π for all complex numbers z such that ℜ (z) > − 1 2 and z ∉ 1 2 + Z . In addition, for all nonnegative integers m, we obtain the Ramanujan-type series 2 π = ∑ n = m ∞ (− 1) n − m (1 + 4 n) 64 n ( 2 n − 2 m n − m ) ( 2 n n ) ( 2 m + 2 n m + n ) . 2 4 m + 5 π 2 = ∑ n = 0 ∞ (1 + 2 m + 4 n) (2 m + 1) (2 m + 3) (m + n + 1) 2 (2 n − 1) 2 256 n ( 2 n n ) 2 ( 2 m + 2 n m + n ) 2. Which are known in the case m=0. [ABSTRACT FROM AUTHOR]

In this work, by constructing a new kernel function in general form which includes both the homogeneous and the non-homogeneous cases, a half-discrete Hilbert-type inequality involving the newly constructed kernel function is established. Additionally, the equivalent Hardy-type inequalities are considered, and all the constant factors in the newly obtained inequalities are proved to be the best possible. Furthermore, by specializing the kernel function and introducing some special functions such as Beta function and Gamma function, some existing results and new examples are presented at the end of the paper. [ABSTRACT FROM AUTHOR]

In the present study it has been endeavored to estimate the fatigue crack propagation in V-notch Charpy specimens of 2024 T351 Al-alloy. For this purpose, a new application of fatigue crack growth (FCG) is developed based on the “Gamma function.” Experimental fatigue tests are conducted for stress ratios from 0.1 to 0.5 under constant amplitude loading. The empiric model depends principally on physical parameters and materials’ properties in non-dimensional form. Deviation percentage, prediction ratio, and band error are used for validation of the performance of the fatigue life. The results determined from Gamma application are in good agreement with experimental FCG rates and those obtained from using Paris law.

In this paper, a summation formula for a general family of a finite secant sum has been extended by making use of a particularly convenient integration contour method. The main theorem derived from this approach is the finite sum involving the Hurwitz-Lerch zeta function. This theorem for particular values is used to derive the finite product of the fifth roots of the quotient product of the gamma function along with finite sums and functional equations involving trigonometric functions.

The Kullback–Leibler divergence is a measure of the divergence between two probability distributions, often used in statistics and information theory. However, exact expressions for it are not known for multivariate or matrix-variate distributions apart from a few cases. In this paper, exact expressions for the Kullback–Leibler divergence are derived for over twenty multivariate and matrix-variate distributions. The expressions involve various special functions.

We offer intriguing new insights into parameterized finite binomial sums, revealing elegant identities such as ∑k=0,k≠nm+n(−1)kn−km+nk=(−1)nm+nn(Hm−Hn), where n,m are non-negative integers and Hn is the harmonic number. These formulas beautifully capture the intricate relationship between harmonic numbers and binomial coefficients, providing a fresh and captivating perspective on combinatorial sums.