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In the present paper we establish a unified treatment of Hilbert–Pachpatte-type inequalities for a class of non-homogeneous kernels. Our results are derived in both discrete and integral versions. A particular emphasis is devoted to constants and weight functions appearing on the right-hand sides of the established inequalities. As an application, we obtain inequalities with constants and weight functions expressed in terms of generalized harmonic numbers, the incomplete Beta and Gamma function, and the logarithmic integral function.

The aim of this paper is to construct asymptotic series associated to some expressions involving the volume of the n-dimensional unit ball. New refinements and sharpenings of some old and recent inequalities on the volume of the n-dimensional unit ball are presented.

We define ?(x) by the equality ? ( x + 1 ) = π ( x e ) x ( 8 x 3 + 4 x 2 + x + 1 30 ) 1 / 6 e ? ( x ) . We call ?(x) the remainder of Ramanujan's formula. In this paper, we present some properties for ?(x), including monotonicity properties, inequalities and asymptotic expansions. Furthermore, we present some full asymptotic expansions for the gamma function related to the Nemes, Ramanujan and Burnside formulas.

In this paper, we present new asymptotic expansions of the gamma function related to the Nemes, Gosper and Burnside formulas.

In the article, we present several asymptotic formulas for the gamma function in terms of the bivariate means. As applications, some sharp upper and lower bounds for the gamma function and factorial n are given.

In this paper, we present the greatest value a ? (1/15, ∞) and the least value b ? (4/15, ∞) such that the double inequality L ( x , a ) 0, where L ( x , a ) = 1 90 a 2 + 2 log ( x 2 + x + 3 a + 1 3 ) + 45 a 2 90 a 2 + 2 log ( x 2 + x + 15 a - 1 45 a ) and ?(x) is the classical psi function.

We present some inequalities and asymptotic expansions associated with the Ramanujan and Nemes formulas for the gamma function.

We prove that some functions associated with the products ? k = 1 n 2 k - 1 2 k , ? k = 1 n 3 k - 2 3 k - 1 , ? k = 1 n 3 k - 2 3 k and ? k = 1 n 3 k - 1 3 k are completely monotonic. By using the results obtained, we derive some inequalities for the previous products. Our results improve some known inequalities.

Nonexistence of periodic solution is proved for periodic right-hand side.Existence of S-asymptotically periodic solutions is discussed.Existence and uniqueness results are obtained for S-asymptotically periodic solution.Nontrivial estimations of a ratio of gamma functions are used. In this paper, it is shown that a fractional difference equation with a periodic right-hand side can not possess a periodic solution but it can have an S-asymptotically periodic solution. Sufficient conditions are proved for the existence of a unique S-asymptotically periodic solution. An example is also given illustrating the obtained results.

In this paper, we construct some new approximations of the gamma function based on Nemes' formula, Ramanujan's formula and Burnside's formula. Using these approximations, some inequalities are established. Finally, for demonstrating the superiority of our new approximation over Mortici's formula and other classical ones, some numerical computations are also given.

In this paper, we present some completely monotonic functions and asymptotic expansions related to the gamma function. Based on the obtained expansions, we provide new bounds for Γ ( x + 1 ) / Γ x + 1 2 and Γ x + 1 2 .

The main purpose of the present article is to introduce the classes of generalized fractional order difference sequence spaces l ∞ ( Γ , Δ α , p ) , c 0 ( Γ , Δ α , p ) and c ( Γ , Δ α , p ) by defining the fractional difference operator Δ α x k = ∑ i = 0 ∞ ( - 1 ) i Γ ( α + 1 ) i ! Γ ( α - i + 1 ) x k + i , where α is a positive proper fraction and k ∈ N = { 1 , 2 , 3 … . } . Results concerning the linearity and various topological properties of these spaces are established and also the alpha-, beta-, gamma- and N -duals of these spaces are obtained. The matrix transformations from these classes into Maddox spaces are also characterized. Throughout the article we use the notation Γ ( n ) as the Gamma function of n , defined by an improper integral Γ ( n ) = ∫ 0 ∞ e - t t n - 1 dt , where n ∉ { 0 , - 1 , - 2 , … } and Γ ( n + 1 ) = n Γ ( n ) .

The aim of this paper is to present new approximations for the gamma function in terms of x x - 1 . The monotonicity and convexity of some involved functions are studied. As applications, sharp bounds are presented. It is also solved an open problem posed by Mortici (2013).

In this paper, we establish some new refinements and similar extensions of Hermite-Hadamard inequality for fractional integrals and present several applications in the Beta function.

In this paper a multidimensional Hilbert-type integral inequality with the best possible constant factor associated to the gamma function is proved. An equivalent form and some reverses are obtained. We also consider the operator expressions and some particular results related to the non-homogeneous kernels and a homogeneous kernel of degree zero.

The constants of Landau and Lebesgue are defined, for all integers n ? 0 , in order, by G n = ? k = 0 n 1 16 k 2 k k 2 and L n = 1 2 π ? - π π sin n + 1 2 t sin 1 2 t d t , which play important roles in the theories of complex analysis and Fourier series, respectively. Diverse inequalities and approximations for these constants have been investigated and developed by many authors. Here, in this paper, we establish new asymptotic expansions for the constants G n and L n / 2 of Landau and Lebesgue, respectively, in terms of the digamma and polygamma functions. Based on our expansion for the Landau constants G n , we present new bounds for the Landau constants G n in terms of the digamma and polygamma functions. We also establish inequalities for the Lebesgue constants L n / 2 , which are applied to derive an asymptotic expansion for L n / 2 in terms of 1 / ( n + 1 ) . Furthermore, by giving numerical calculations to be compared, among several developed asymptotic expansions for the constants G n and L n / 2 , it is shown that our expansions presented here would be best ones.

Let the sequence r n be defined by n ! = 2 π n n / e n e r n . We establish new estimates for Stirling's formula remainder r n . This improves some known results. Let ? ( x ) be defined by the relation: ? ( x + 1 ) = 2 π ( x / e ) x e ? ( x ) / ( 12 x ) . We prove that ? ? ( x ) is completely monotonic on ( 0 , ∞ ) . This implies the result given by Mortici, who proved that - x - 1 ? ? ( x ) is completely monotonic on ( 0 , ∞ ) .

In this paper, we obtain the complete monotonicities regarding both the functions σ x and λ x defined by Γ ( x + 1 ) = 2 π x x e x e σ x 12 x = 2 π x x e x ( 1 + λ x ) . As direct consequences, σ x is strictly increasing and strictly concave on ( 0 , ∞ ) and λ x is strictly decreasing and strictly convex on ( 0 , ∞ ) . Furthermore, we show that σ x consists the properties lim x → 0 + σ x = 0 and lim x → ∞ σ x = 1 as well as λ x holds the properties lim x → 0 + λ x = ∞ and lim x → ∞ λ x = 0 .

We develop Windschitl's approximation formula for the gamma function to two asymptotic expansions and provide a formula for determining the coefficients of each asymptotic expansion.

The constants of Landau and Lebesgue are defined, for all integers n ⩾ 0 , in order, by G n = ∑ k = 0 n 1 16 k 2 k k 2 and L n = 1 2 π ∫ - π π sin ( n + 1 2 ) t sin ( 1 2 t ) d t , which play important roles in the theories of complex analysis and Fourier series, respectively. In this paper, we establish new bounds and asymptotic expansions for the constants of Landau and Lebesgue.

Motivated by the work of Chen and Qi (2005) [3] we study the products ∏k=1n3k-23k and ∏k=1n3k-13k. We prove that some functions associated to the previous products are completely monotonic and we establish some sharp inequalities.

Integral means are important class of bivariate means. In this paper we prove the very general algorithm for calculation of coefficients in asymptotic expansion of integral mean. It is based on explicit solving the equation of the form $B(A(x))=C(x)$, where $B$ and $C$ have known asymptotic expansions. The results are illustrated by calculation of some important integral means connected with gamma and digamma functions.

In this article, we give a sufficient condition and a necessary and sufficient condition for a function class associated with the gamma function to be logarithmically completely monotonic. Some applications are also presented.

There is an abundant literature on inequalities for the (Euler's) Gamma function @C and its various related functions. Yet, only very recently, several authors began to study inequalities for the (Barnes') double Gamma function @C"2. Here, in this paper, we aim at presenting several two-sided inequalities for the multiple Gamma functions @C"n(n=2,3,4,5). In our investigation of these two-sided inequalities for the multiple Gamma functions @C"n(n=2,3,4,5), we employ and extend a method based upon Taylor's formula and express [email protected]"n(1+x) as series involving the Zeta functions. We also give a more convenient explicit form of the multiple Gamma functions @C"n([email protected]?N),N being the set of positive integers. The main two-sided inequalities for the multiple Gamma functions @C"n(n=2,3,4,5) (which we have presented in this paper) are presumably new and their derivations provide a fruitful insight into the corresponding problem for the multiple Gamma functions @C"n when [email protected]?6.

A very simple approach using series manipulations and the Weierstrass Representation Theorem yield surprisingly nice and non-trivial series identities involving special functions. In our paper we present a large number of examples of these identities. At the end, several number theoretical sums are deduced.

In the paper the authors introduce the concept of completely monotonic degrees of nonnegative functions on ( 0 , ∞ ) and proves that a function involving the difference between the tri-gamma function ψ ′ ( x ) and a proper fraction of x is completely monotonic of degree 1 on ( 0 , ∞ ) .

In this paper, we present new completely monotonic functions and Bernstein functions relating to the gamma and psi functions. As consequences we establish new upper and lower bounds for the gamma function in terms of ψ -function. Also, we present some estimates for Stirling’s formula remainder.

In this work, by the use of Guseinov’s one-center expansion formulas and Lowdin- α radial function, the series expansion relations in molecular coordinate system are established for the two-center nuclear attraction integrals of noninteger n ∗ Slater type orbitals in terms of basic two-center nuclear attraction integrals over integer n Slater functions. The Lowdin α -radial function convoluted with the Guseinov’s one-center expansion formulas is one of the most important ingredients for accurate and efficient implementation of electronic structure calculation methods regardless of Hartree–Fock–Roothaan (HFR) method. The proposed algorithm shows better performance in arbitrary quantum numbers, screening constants and location of orbitals leading to significantly reduced run times.

Several inequalities involving gamma function are obtained. They are established using elementary properties of logarithmically convex functions.

There is an abundant literature on inequalities for the Gamma function Γ and its various related functions as well as their approximations. Only very recently, several authors began to investigate various inequalities for the double Gamma function Γ 2 and its approximation. Here, in this sequel to some of these recent works, we aim at presenting an integral representation of the triple Gamma function Γ 3 , which is then used to derive an asymptotic formula for Γ 3 . As a by-product of the results presented here, integral representations and asymptotic formulas for the Gamma function Γ and the double Gamma function Γ 2 are also given. The methods and techniques used in this paper can easily be extended to derive the corresponding integral representations and asymptotic formulas for the multiple Gamma functions Γ n ( n ≧ 4).

The Barnes’ G -function G ( x ) = 1/ Γ 2 , satisfies the functional equation log G ( x + 1) − log G ( x ) = log Γ ( x ). We complement W. Krull’s work in Bemerkungen zur Differenzengleichung g ( x + 1) − g ( x ) = φ ( x ), Math. Nachrichten 1 (1948), 365–376 with additional results that yield a different characterization of the function G , new expansions and sharp bounds for G on x > 0 in terms of Gamma and Digamma functions, a new expansion for the Gamma function and summation formulae with Polygamma functions.

We present a Fourier transform representation of the generalized gamma functions, which leads to a distributional representation for them as a series of Dirac-delta functions. Applications of these representations are shown in evaluation of the integrals of products of the generalized gamma function with other functions. The results for Euler’s gamma function are deduced as special cases. The relation of the generalized gamma function with the Macdonald function is exploited to deduce new identities for it.

The aim of this paper is to propose some improvements of the Stirling formula for approximation of the factorial function.

In this paper we discuss analytical and numerical properties of the function V�,µ(α, β, z) = R 1 0 e zt (t + α) � (t + β) µ dt, with α, β, ℜz > 0, which can be viewed as a generalization of the complementary error function, and in fact also as a generalization of the Kummer U −function. The function V�,µ(α, β, z) is used for certain values of the parameters as an approximant in a singular perturbation problem. We consider the relation with other special functions and give asymptotic expansions as well as recurrence relations. Several methods for its numerical evaluation and examples are given. 2000 Mathematics Subject Classification: 33B20, 33C15, 41A60, 65D20, 65Q05.

The aim of this paper is to introduce an approximations family of the factorial function that contains Stirling's formula, Burnside's formula and Gosper's formula. The parameters which provide the best approximations are indicated. Finally, numerical computations are made to show the superiority of our formulas over other known formulas.

In many seemingly diverse physical contexts (including, for example, certain radiation field problems, studies of crystallographic minimal surfaces, the theory of scattering of acoustic or electromagnetic waves by means of an elliptic disk, studies of elliptical crack problems in fracture mechanics, and so on), a remarkably large number of general families of elliptic-type integrals, and indeed also many definite integrals of such families with respect to their modulus (or complementary modulus), are known to arise naturally. Motivated essentially by these and many other potential avenues of their applications, we present here a systematic account of the theory of a certain family of incomplete elliptic integrals in a unified and generalized manner. By means of the familiar Riemann-Liouville fractional differintegral operators, we obtain several explicit hypergeometric representations and apply these representations with a view to deriving various associated definite integrals, not only with respect to the modulus (or complementary modulus), but also with respect to the amplitude of the incomplete elliptic integrals involved therein.

In the article, sufficient and necessary conditions that a class of functions involving ratio of Euler’s gamma functions and originating from Wendel’s and Gautschi’s inequalities are logarithmically completely monotonic are presented. From this, Wendel’s, Gautschi’s, Kershaw’s, Laforgia’s, Bustoz-Ismail’s, Merkle’s and Elezovic-Giordano-Pecaric’s inequalities are refined, extended and sharpened, and a double inequality on the divided differences of the psi and polygamma functions is deduced straightforwardly.

Some new inequalities for the q-gamma, the q-beta and the q-analogue of the Psi functions are established via some q-integral inequalities.

In this paper, we first investigate several further interesting properties of the multiple Hurwitz–Lerch Zeta function Φ n ( z , s , a ) which was introduced recently by Choi et al. [J. Choi, D.S. Jang, H.M. Srivastava, A generalization of the Hurwitz–Lerch Zeta function, Integral Transform. Spec. Funct., 19 (2008)]. We then introduce and investigate some q -extensions of the multiple Hurwitz–Lerch Zeta function Φ n ( z , s , a ), the Apostol–Bernoulli polynomials B k ( n ) ( x ; λ ) of order n , and the Apostol–Euler polynomials E k ( n ) ( x ; λ ) of order n . Relevant connections of the results presented here with those obtained in earlier works are also indicated precisely.

In this article, the authors present a necessary condition, a sufficient condition and a necessary and sufficient condition for a class of functions associated with the gamma function to be logarithmically completely monotonic. As a consequence of these results, supplements to the recent investigation by Chen and Qi [C.-P. Chen, F. Qi, Logarithmically completely monotonic functions relating to the gamma function, J. Math. Anal. Appl. 321 (2006) 405–411] are provided and a new Keckic–Vasic type inequality is concluded.

Series expansions of Kapteyn type in terms of Gamma function and a generalization of Bessel function of first kind were considered by Exton. Here we show that the convergence conditions given by him can be relaxed and the achieved results are optimal.

The theory of multiple Gamma functions was studied in about 1900 and has been revived in the study of determinants of Laplacians in the middle or so of 1980s. There are some classes of mathematical constants involved naturally in the Gamma and multiple Gamma functions. Here we introduce those known mathematical constants associated with the Gamma and multiple Gamma functions and will show how they are involved, if possible.

The history of problems of evaluation of series associated with the Riemann Zeta function can be traced back to Christian Goldbach (1690–1764) and Leonhard Euler (1707–1783). Many different techniques to evaluate various series involving the Zeta and related functions have since then been developed. The authors show how elegantly certain families of series involving the Hurwitz–Lerch Zeta function can be evaluated by starting with a single known identity for the Hurwitz–Lerch Zeta function. Some of the special cases of these series identities are also shown to lead to certain known families of summation formulas involving the Hurwitz Zeta function.

In reliability studies, the sense of variation of the hazard rate is of major concern since it indicates system wear out or burn in or in some cases a non-monotonic hazard rate. In this paper, we consider a generalized gamma distribution and study, in detail, the monotonicity of its failure rate and the mean residual life function. This involves the investigation of the monotonic structure of a transform function and then using this structure, we study the monotonic structure of the failure rate and the mean residual life function. Some of the known results follow as special cases of our results.

A general volume formula in terms of beta-gamma functions for a wide range of symmetric solids is established and a program code for computational purposes is generated.

The infinite product @?"n"="0^~(1-aq^n), plays a part in the theory of basic functions, similar to that of the Gamma function in the theory of ordinary hypergeometric functions, so it is pertinent to enquire how such products can be evaluated numerically. In this paper we will discuss the numerical evaluation of the infinite products using the method of gridded data interpolation.

The theory of multiple Gamma functions, which was recently revived in the study of the determinants of the Laplacians, was applied in several earlier works in order to evaluate some families of series involving the Riemann Zeta function as well as to compute the determinants of the Laplacians. Here, in the present paper, the authors address the converse problem and apply various (known or new) formulas for series associated with the Zeta and related functions with a view to developing the corresponding theory of multiple Gamma functions and then using these series to compute the determinants of the Laplacians on the n-dimensional unit sphere S^n (n=5,6,7) explicitly.

The main object of this paper is to present an easily accessible integral representation of the partial sum L"u(x,a)=@?"0"=

In this work, the solution of the integral equation of the mixed type in the domain @W={(x,y,z)@[email protected]:-~

The authors give several new (and potentially useful) relationships between the multiple Gamma functions and other mathematical functions and constants. As by-products of some of these relationships, a classical definite integral due to Euler and other definite integrals are also considered together with closed-form evaluations of some series involving the Riemann and Hurwitz Zeta functions.