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

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]

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 study, new prediction equations are proposed to determine the variables of gamma envelope-shape Function. Envelope-shape functions are employed for the simulation of ground motion using a stochastic method in order to form the white noise. A variety of Envelope-shape functions have been already proposed, among which gamma equations are more used in computational programs for the simulation of ground motion, e.g. SMSIM or EXSIM. In addition to the detection of influential variables of gamma envelope-shape function, their relationship with important seismic criteria such as moment magnitude, closest distance to rupture plane, site conditions and mechanism of faulting are determined and suitable models are presented in this study. Limited research has been already undertaken to specify the variables of envelope-shape equations and their relationships with the main seismic criteria. Hence the results of this study can almost demonstrate this relationship with seismic criteria. The new equations are developed using Iran strong ground motion database, which includes 2228 records of 749 earthquakes that covering up to 2018. The proposed equations can be used to determine exponential window function for earthquakes with moment magnitude M w from 4.5 to 7.6 and rupture distance up to 200 km. The proposed model is compared with some existing models and the results suggest that the equations can lead to better and more reasonable results in the ground motion simulation process, particularly for the estimation of response spectrum of velocity and displacement. [ABSTRACT FROM AUTHOR]

We give some necessary and some sufficient conditions for the complete monotonicity on the negative half-line of a Mittag–Leffler function of Le Roy type. It is conjectured that the underlying positive random variable, when it exists, must be logarithmically infinitely divisible. [ABSTRACT FROM AUTHOR]

In the paper, the author presents bounds for completely monotonic degree of a remainder for an asymptotic expansion of the trigamma function. This result partially confirms one in a series of conjectures on completely monotonic degrees of remainders of asymptotic expansions for the logarithm of the gamma function and for polygamma functions. [ABSTRACT FROM AUTHOR]

We investigate the relationship between the sign of the discrete fractional sequential difference ( Δ 1 + a − μ ν Δ a μ f) (t) and the monotonicity of the function t ↦ f (t) in the case where the orders of the fractional difference are mixed – i.e. μ ∈ (0 , 1) but ν ∈ (1 , 2) with μ + ν ∈ (1 , 2). We consider the special case in which the bound ( Δ 1 + a − μ ν Δ a μ f) (t) ≥ − ε f (a) , for some ε > 0 and where f (a) ≥ 0 , is satisfied. Even though the fractional difference is negative, we demonstrate that there still exists a relationship between this quantity and the monotonicity of f. We augment our analytical results with numerical simulations. [ABSTRACT FROM AUTHOR]

The coefficients of regression are usually estimated for minimization problems with asymmetric loss functions. In this paper, we rather correct predictions so that the prediction error follows a generalized Gaussian distribution. In our method, we not only minimize the expected value of the asymmetric loss, but also lower the variance of the loss. Predictions usually have errors. Therefore, it is necessary to use predictions in consideration of these errors. Our approach takes into account prediction errors. Furthermore, even if we do not understand the prediction method, which is a possible circumstance in, e.g. deep learning, we can use our method if we know the prediction error distribution and asymmetric loss function. Our method can be applied to procurement of electricity from electricity markets. [ABSTRACT FROM AUTHOR]

We investigate relationships between the sign of the discrete fractional sequential difference ( Δ 1 + a − μ ν Δ a μ f) (t) and the convexity of the function t ↦ f (t). In particular, we consider the case in which the bound ( Δ 1 + a − μ ν Δ a μ f) (t) ≥ ε f (a) , for some ε > 0 and where f (a) < 0 , is satisfied. Thus, we allow for the case in which the sequential difference may be negative, and we show that even though the fractional difference can be negative, the convexity of the function f can be implied by the above inequality nonetheless. This demonstrates a significant dissimilarity between the fractional and non-fractional cases. We use a combination of both hard analysis and numerical simulation. [ABSTRACT FROM AUTHOR]

We establish some supercongruences related to a supercongruence of Van Hamme, such as ∑ k = 0 (p + 1) / 2 (− 1) k (4 k − 1) (− 1 2) k 3 k ! 3 ≡ p (− 1) (p + 1) / 2 + p 3 (2 − E p − 3) (mod p 4) , ∑ k = 0 (p + 1) / 2 (4 k − 1) 5 (− 1 2) k 4 k ! 4 ≡ 16 p (mod p 4) , where p is an odd prime and E p − 3 is the (p − 3) th Euler number. Our proof uses some congruences of Z.-W. Sun, the Wilf–Zeilberger method, Whipple's 7 F 6 transformation, and the Mathematica package Sigma developed by Schneider. We also put forward two related conjectures. [ABSTRACT FROM AUTHOR]

Differintegral methods, namely those techniques using differential and integral operators on the same footing, currently exploited in calculus, provide a fairly unexhausted source of tools to be applied to a wide class of problems involving the theory of special functions and not only. The use of integral transforms of Borel type and the associated formalism will be shown to be an effective means, allowing a link between umbral and operational methods. We merge these two points of view to get a new and efficient method to obtain integrals of special functions and the summation of the associated generating functions as well. [ABSTRACT FROM AUTHOR]

Mathematical inequalities and other results involving such widely- and extensively-studied special functions of mathematical physics and applied mathematics as (for example) the Bessel, Struve and Lommel functions as well as the associated hypergeometric functions are potentially useful in many seemingly diverse areas of applications, especially in situations in which these functions are involved in solutions of mathematical, physical and engineering problems which can be modeled by ordinary and partial differential equations. With this objective in view, our present investigation is motivated by some open problems involving inequalities for a number of particular forms of the hypergeometric function 1F2(a; b, c; z). Here, in this paper, we apply a novel approach to such problems and obtain presumably new two-sided inequalities for the Struve function, the associated Struve function and the modified Struve function by first investigating inequalities for the general hypergeometric function 1F2(a; b, c; z). We also briefly discuss the analogous new inequalities for the Lommel function under some conditions and constraints. Finally, as special cases of our main results, we deduce several inequalities for the modified Lommel function and the normalized Lommel function. [ABSTRACT FROM AUTHOR]

van Hamme conjectured that, for any odd primep,This result was proved by Mortenson using atransformation, and was reproved by Zudilin via the Wilf–Zeilberger method. In this paper, we propose a conjectural generalization of van Hamme's supercongruence and prove some special cases, such aswherepis a prime with. We also state some related conjectures. [ABSTRACT FROM PUBLISHER]

The zeros of the digamma function are known to be simple and real, but up to now few identities involving them have appeared in the literature. By establishing a Weierstrass infinite product for a particular regularization of the digamma function, we are able to find interesting formulas for the sums of thenth powers of the reciprocals of its zeros, for. We make a parallel study of the zeros of the logarithmic derivative of the BarnesG-function. We also compare asymptotic estimates of the zeros of the digamma function and those of its BarnesG-function analogue. [ABSTRACT FROM PUBLISHER]

In nano-medicine, attempts of using the carbon-nanotubes (CNTs) as drug-carriers are undertaken especially in the treatment of cancer. These (CNTs) are first injected into the blood which then reach the tumor cite under the actions of the waves propagated by the walls of the arteries with an external force such as a magnetic field or laser beams. The flow near the boundary of the artery may be treated as a boundary layer flow only for simplification as we consider here. In applied science, the flow and heat transfer of CNTs are usually described by systems of nonlinear differential equations. Due to nonlinearities, the exact solutions of such systems cannot be obtained in most cases. In this paper, an effective analytical procedure is proposed to deduce the exact solution of a system of nonlinear differential equations describing the effect of a convective heat condition on the flow and the heat transfer of carbon-nanotube suspended nanofluids with suction/ injection in the presence of a magnetic field. The heat transfer equation is solved via applying Laplace transform and the solution is expressed in terms of the generalized incomplete gamma function. Also, it is proved that the present exact solutions for the flow and the heat transfer reduce to those in literature in the absence of the suction/injection and the convective parameters. The results declare that the temperature profiles are very sensitive regarding the value assigned to the convective parameter. Moreover, the effects of other physical parameters on the studied phenomena are displayed through graphs. Besides, possible applications of the current results have been also discussed. [ABSTRACT FROM AUTHOR]

We provide new representations for the finite parts at the poles and the derivative at zero of the Barnes zeta function in any dimension in the general case. These representations are in the forms of series and limits. We also give an integral representation for the finite parts at the poles. Similar results are derived for an associated function, which we term homogeneous Barnes zeta function. Our expressions immediately yield analogous representations for the logarithm of the Barnes gamma function, including the particular case also known as multiple gamma function. [ABSTRACT FROM AUTHOR]

In 1997, Wilf posed the following elegant infinite product formula as a problem:which contains some of the most important mathematical constants such asπ,e, and the Euler–Mascheroni constantγ. In 2009, Mortini posed the following problem to determineIn this paper, we present the connection between generalized Wilf's and Mortini's problems. We also consider the connection between another infinite product formula and limit relation with several parameters. [ABSTRACT FROM PUBLISHER]

In this article, we generalize the univariate Macdonald distribution to the matrix case and give its derivation using matrix variate gamma distribution. We study several properties such as cumulative distribution function, marginal distribution of submatrix, triangular factorization, moment generating function, and expected values of several functions of the Macdonald matrix. Some of these results are expressed in terms of special functions of matrix arguments and zonal polynomials. [ABSTRACT FROM PUBLISHER]

Recently, [Veestraeten D. On the inverse transform of Laplace transforms that contain (products of) the parabolic cylinder function. Integr Transf Spec F 2015;26:859–871] derived inverse Laplace transforms for Laplace transforms that contain products of two parabolic cylinder functions by exploiting the link between the parabolic cylinder function and the transition density and distribution functions of the Ornstein–Uhlenbeck process. This paper first uses these results to derive new integral representations for (products of two) parabolic cylinder functions. Second, as the Brownian motion process with drift is a limiting case of the Ornstein–Uhlenbeck process also limits can be calculated for the product of gamma functions and (products of) parabolic cylinder functions. The central results in both cases contain, in stylized form,andsuch that the recurrence relation of the parabolic cylinder function straightforwardly allows to obtain integral representations and limits also for countless other combinations in the orders such asand. [ABSTRACT FROM AUTHOR]

The Laplace transforms of the transition probability density and distribution functions for the Ornstein–Uhlenbeck process contain the product of two parabolic cylinder functions, namelyand, respectively. The inverse transforms of these products have as yet not been documented. However, the transition density and distribution functions can be obtained by alternatively applying Doob's transform to the Kolmogorov equation and casting the problem in terms of Brownian motion. Linking the resulting transition density and distribution functions to their Laplace transforms then specifies the inverse transforms to the aforementioned products of parabolic cylinder functions. These two results, the recurrence relation of the parabolic cylinder function and the properties of the Laplace transform then enable the calculation of inverse transforms also for countless other combinations in the orders of the parabolic cylinder functions such as,and. [ABSTRACT FROM AUTHOR]

The paper is about calculating in closed form the following multiple fractional part integralwhere,are integers anddenotes the fractional part ofx. We prove that the integralcan be expresses in terms of a difference of two rational numbers and a series involving Riemann zeta function values and some binomial coefficients. [ABSTRACT FROM AUTHOR]

Brychkov [Brychkov YuA. Evaluation of some classes of definite and indefinite integrals. Integral Transforms Spec Funct. 2002;13:163–167] evaluated some interesting classes of definite and indefinite integrals involving various special functions and the logarithmic function. Motivated essentially by Brychkov's work [Brychkov YuA. Evaluation of some classes of definite and indefinite integrals. Integral Transforms Spec Funct. 2002;13:163–167], we aim at presenting several classes of definite integrals involving hypergeometric functions and the logarithmic functions, which are expressed explicitly in terms of Psi and generalized zeta functions. Some interesting special cases of our main formulas are given. Relevant connections of the results presented here with those earlier ones are also pointed out. [ABSTRACT FROM AUTHOR]

In this short note, we point out an interesting connection between series with zeta values and certain infinite products. Using this connection we give a closed-form evaluation of a special class of series with zeta values in the coefficients. [ABSTRACT FROM AUTHOR]

In this note, we develop new, simple and very accurate asymptotic approximations for non-integer order derivatives of monomial functions by using the more accurate asymptotic approximations for large factorials that have recently appeared in the literature. [ABSTRACT FROM AUTHOR]

This note provides asymptotic formulas for approximating the sequence factorial of members of a finite arithmetic progression by using Stirling, Burnside and other more accurate asymptotic formulas for large factorials that have appeared in the literature. [ABSTRACT FROM PUBLISHER]

In this note, we provide basic asymptotic formulas for approximating large g-gonal sequence factorials by using Stirling and Burnside asymptotic approximation formulas for large factorials. More accurate asymptotic approximation formulas for large g-gonal sequence factorials resulting from some recent, more accurate asymptotic formulas for large factorials that have appeared in the literature are presented. [ABSTRACT FROM AUTHOR]

In a recent paper the following inequality has been deduced Γ(x+1)<(x2+2)/(x+2),x∈(0, 1). We shall improve this result determining the smallestcfor which the inequality Γ(x+1)<(x2+c)/(x+c),x∈(0, 1) holds. We shall compare this inequality with some known bounds of the Γ function. [ABSTRACT FROM PUBLISHER]

Two interesting infinite product formulas were presented by Choi, Lee and Srivastava [A generalization of Wilf's formula. Kodai Math J. 2003;26:44–48], who generalized an infinite product formula posed by Wilf as a problem in American Mathematical Monthly, 1997. Chen and Choi established in 2013 two infinite product formulas, expressed in terms of the gamma function, whose special cases contain the above generalized infinite product formulas. In this paper, we generalize Chen and Choi's results. [ABSTRACT FROM PUBLISHER]

The multiple gamma function Γn(z), defined by a recurrence-functional equation as a generalization of the Euler gamma function, is used in many applications of pure and applied mathematics, and theoretical physics. The theory of the multiple gamma function has been related to certain spectral functions in mathematical physics, to the study of functional determinants of Laplacians of then-sphere, to HeckeL-functions, to the Selberg zeta function, and to the random matrix theory. There is a wide class of definite integrals and infinite sums appearing in statistical physics (the Potts model) and the lattice theory, which can be computed by means of the Γn(z) function. This paper presents new integral representations for the multiple gamma function and other mathematical functions and constants. [ABSTRACT FROM AUTHOR]

Asymptotic expansions are derived for the reciprocal of the gamma function. We show that the coefficients of the expansion are the same, up to a sign change, as the asymptotic expansions for the gamma function obtained by exponentiating the expansions of its logarithm due to Stirling and de Moivre. Expressions for the coefficients are given in terms of Bell polynomials. [ABSTRACT FROM PUBLISHER]

Computing the matrix elements of the linear operator, which transforms the spherical basis ofSO(3, 1)-representation space into the hyperbolic basis, we present an integral formula involving the product of two Legendre functions of the first kind expressed in terms of4F3-hypergeometric function. Also, using the general Mehler–Fock transform, we obtain another integral formula for the Legendre function of the first kind. A relevant connection of one of the results presented here with a known integral formula is also pointed out. [ABSTRACT FROM PUBLISHER]

For the first time, we derive closed-form expressions for the moments of the type I generalized logistic distribution, the type II generalized logistic distribution, the type III generalized logistic distribution and the type IV generalized logistic distribution. As by products, we give several hitherto unknown expressions for derivatives of general order of products of two gamma functions. [ABSTRACT FROM PUBLISHER]

A new view on classical asymptotic expansions of the logarithm of gamma function is given. Then general formulae for the asymptotic expansions of the logarithm of gamma function and the Wallis power function through polygamma functions are derived and analysed. [ABSTRACT FROM PUBLISHER]

In the paper the authors show that neither the function ... nor its negative is completely monotonic on (0,∞), where ai >0 and bi ≥0 are real numbers, i ≥2, Γ(x) is the classical Euler's gamma function, and ... are polygamma functions. Moreover, some other results are given in the form of remarks. [ABSTRACT FROM AUTHOR]

Euler's gamma function Γ(x) is logarithmically convex on (0, ∞). Additivity of logarithmic convexity implies that the functionx→∑fkΓ(x+k) is also log-convex (assuming convergence) if the coefficients are non-negative. In this paper, we investigate the series ∑fkΓ(x+k)−1, where each term is clearly log-concave. Log-concavity is not preserved by addition, so that non-negativity of the coefficients is now insufficient to draw any conclusions about the sum. We demonstrate that the sum is log-concave ifand is discrete Wright log-concave if. We conjecture that the latter condition is in fact sufficient for the log-concavity of the sum. We exemplify our general theorems by deriving known and new inequalities for the modified Bessel, Kummer and generalized hypergeometric functions and their parameter derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we establish a continued product approximation for the gamma function:where the powers γican be determined by a recurrence. It can be seen that one of Mortici's [A class of integral approximations for the factorial function. Comput. Math. Appl. 2010;59(6):2053–2058] results is a special case of our approximation and can be viewed as the zeroth truncation. Some numerical computations are also made to show the performance of this formula. [ABSTRACT FROM PUBLISHER]

Several representations for the logarithm of the Gamma function exist in the literature. There are four important expansions which bear the name of Binet. Hermite generalized Binet's first formula to the logarithm of the Gamma function with shifted argument. The generalization of Binet's second formula is apparently not known; however, it follows easily from another result of Hermite. The aim of this paper is to give possible generalizations of the third and fourth Binet formulas. [ABSTRACT FROM PUBLISHER]

An inequality concerning the Γ function is improved, and the best possible version is determined. The technique of developments in function series is used. [ABSTRACT FROM AUTHOR]

The gamma function, which has the property to interpolate the factorial whenever the argument is an integer, is a special case (the caseg = 2) of the general term of the sequence factorial ofg-gonal numbers. In relation to this special case, a formula for calculating the general term of the sequence factorial of anyg-gonal number was obtained after considering some specific cases. Important properties on functional equations ofg-gonal sequence factorials are derived. [ABSTRACT FROM PUBLISHER]

Two interesting infinite product formulas were presented by Choiet al. [A generalization of Wilf's formula, Kodai Math. J. 26 (2003), pp. 44–48], who generalized an infinite product formula posed by Wilf as a problem inAmerican Mathematical Monthly[H.S. Wilf,Problem 10588, problems and solutions, Amer. Math. Monthly 104 (1997), p. 456]. Here, we aim mainly at establishing two infinite product formulas, expressed in terms of the Gamma function, whose special cases contain the above generalized infinite product formulas. Our formulas presented here are themselves interesting and their another advantage is the production, easily, of various known or (presumably) new formulas as their special cases. [ABSTRACT FROM AUTHOR]

In the present study, we prove the monotonicity of two functions involving the gamma function Γ(x) and relating to the n-dimensional volume of the unit ball 𝔹 n in ℝ n . [ABSTRACT FROM AUTHOR]

By means of the familiar incomplete gamma functions γ(s, x) and Γ(s, x), we introduce the incomplete Pochhammer symbols that lead us to a natural generalization and decomposition of a class of hypergeometric and other related functions which are potentially useful in closed-form representations of definite and semi-infinite integrals of various special functions. Applications of these functions in (for example) communication theory, probability theory and groundwater pumping modelling are shown. [ABSTRACT FROM AUTHOR]

In this article, a necessary and sufficient condition and a necessary condition are established for a class of functions involving the gamma function to be logarithmically completely monotonic on . As applications of the necessary and sufficient condition, several two-sided bounding inequalities for the psi and polygamma functions and the ratio of two gamma functions are derived. [ABSTRACT FROM AUTHOR]

We propose a new sharp family of approximations which improve the results of Sandor and Debnath [On certain inequalities involving the constant e and their applications, J. Math. Anal. Appl. 249 (2000), pp. 569–582], Batir [Sharp inequalities for factorial n, Proyecciones 27 (2008), pp. 97–102] and other recent results. Then, we prove the complete monotonicity of a function involving the gamma function to establish new sharp upper and lower bounds for the gamma and the digamma functions. [ABSTRACT FROM PUBLISHER]

Asymptotic expansions of the function [Γ(x+t)/Γ(x+s)]1/(t−s) involving exponential function are given and analysed. An efficient algorithm for calculating coefficients of these expansions is obtained. An application to the asymptotic expansion of the central binomial coefficient is given. [ABSTRACT FROM AUTHOR]

Historically, the normal variance model has been used to describe stock return distributions. This model is based on taking the conditional stock return distribution to be normal with its variance itself being a random variable. The form of the actual stock return distribution will depend on the distribution for the variance. In practice, the distributions chosen for the variance appear to be very limited. In this note, we derive a comprehensive collection of formulas for the actual stock return distribution, covering some sixteen flexible families. The corresponding estimation procedures are derived by the method of moments and the method of maximum likelihood. We feel that this work could serve as a useful reference and lead to improved modelling with respect to stock market returns. [ABSTRACT FROM PUBLISHER]

In this paper, we shall locate Plana's summation formula à la Koshlyakov as a form of the modular relation for the Riemann zeta-function and an analytic continuation technique furnished by Lemma 1. Then by Plana's summation formula, we shall prove an important integral representation for the Hurwitz–Lerch zeta-function through confluent hypergeometric functions, which cover a wide range of integral representations attributed to famous mathematicians. We shall also locate the genesis of Mikolás’ formula and prove that the functional equations for the Hurwitz zeta-function and the Riemann zeta-function are equivalent. [ABSTRACT FROM PUBLISHER]

Motivated essentially by their potential for applications in a wide range of mathematical and physical problems, the log-sine integrals have been evaluated, in the existing literature on the subject, in many different ways. The main object of this paper is to present explicit evaluations of some families of log-sine integrals by making use of a familiar algorithm, and also those of log-cosine and other related integrals by using some known integral formulas expressed in terms of the Gamma functions. [ABSTRACT FROM AUTHOR]

We present some properties associated with the monotonicity and the complete monotonicity of the psi (or digamma) function and establish the higher-order estimate for the familiar Euler–Mascheroni constant γ. Relevant connections of the results presented here with those derived in earlier works are also pointed out. [ABSTRACT FROM AUTHOR]