In our recent work we proposed a generalization of the beta integral method for derivation of the hypergeometric identities which can by analogy be termed "the G function integral method". In this paper we apply this technique to the cubic and the degenerate Miller-Paris transformations to get several new transformation and summation formulas for the generalized hypergeometric functions at a fixed argument. We further present an alternative approach for reducing the right hand sides resulting from our method to a single hypergeometric function which does not require the use of summation formulas.
In this paper, a general integral identity for a twice differentiable functions is derived. By using of this identity, the author establishes some new Hermite-Hadamard type and Simpson type inequalities for differentiable (α,m)-convex functions via Riemann Liouville fractional integral.
In this paper, a general integral identity for twice differentiable functions is derived. By using of this identity, the author establish some new estimates on Hermite-Hadamard type and Simpson type inequalities for s-convex via Riemann Liouville fractional integral.
In this paper, we obtain new estimates on generalization of Hermite-Hadamard, Simpson and Ostrowski type inequalities for functions whose second derivatives is $φ$-convex via fractional integrals.
Using the Dirichlet integrals, which are employed in the theory of Fourier series, this paper develops a useful method for the summation of series and the evaluation of integrals.
In this paper a new integral for the remainder of $π(x)$ is obtained. It is proved that there is an infinite set of the formulae containing miscellaneous parts of this integral.
In this paper, the author obtains new estimates on generalization of Hadamard, Ostrowski and Simpson type inequalities for Lipschitzian functions via Hadamard fractional integrals. Some applications to special means of positive reals numbers are also given.
In this paper, the authors gives a new identity for Hadamard fractional integrals. By using of this identity, the authors obtains new estimates on generalization of Hadamard, Ostrowski and Simpson type inequalities for (alpha?;m)-GA-convex function via Hadamard fractional integral.
Based on known definite integrals of Bessel functions of the first kind, we obtain exact solutions to unknown definite integrals using the method of integral transforms from Hankel's transform.
This paper considers various integrals where the integrand includes the log gamma function (or its derivative, the digamma function) multiplied by a trigonometric or hyperbolic function. Some apparently new integrals and series are evaluated.
In this paper, the author introduces the concept of the quasi-geometrically convex and defines a new identity for fractional integrals. By using of this identity, author obtains new estimates on generalization of Hadamard, Ostrowski and Simpson type inequalities for GA-s-convex, quasi-geometrically convex and (s,m)-GA-convex functions via Riemann Liouville fractional integral.
We obtain definite integrals for products of associated Legendre functions with Bessel functions, associated Legendre functions, and Chebyshev polynomials of the first kind using orthogonality and integral transforms.
In this paper a double integral containing two Gaussian hypergeometric functions is discussed. The integral is not found in the literature and a direct computation is not (yet) possible. Therefore, a complete different integral is computed by two methods. The first method gives a combination of hypergeometric functions and the requested original integral, while with the second method this different integral can be computed directly. Equalization of the results from both methods gives the requested integral.
Based on the total integrability we first define an integral of a real valued function f as an interval function associated to its antiderivative F. By introducing the concept of the residue of a function into the real analysis, the relationship between the so defined integral and the generalized Riemann integral is established.
A new method is presented for obtaining indefinite integrals of common special functions. The approach is based on a Lagrangian formulation of the general homogeneous linear ordinary differential equation of second order. A general integral is derived which involves an arbitrary function, and therefore yields an infinite number of indefinite integrals for any special function which obeys such a differential equation. Techniques are presented to obtain the more interesting integrals generated by such an approach, and many integrals, both previously known and completely new are derived using the…
This paper considers some integrals where the integrand comprises the log gamma function or the digamma function multiplied by exponential or trigonometric functions.
We derive some Euler type double integral representations for hypergeometric functions in two variables. In the first part of this paper we deal with Horn's $H_2$ function, in the second part with Olsson's $F_P$ function. Our double integral representing the $F_P$ function is compared with the formula for the same integral representing an $H_2$ function by M. Yoshida (Hiroshima Math. J. 10 (1980), 329-335 and M. Kita (Japan. J. Math. 18 (1992), 25-74). As specified by Kita, their integral is defined by a homological approach. We present a classical double integral version of Kita's integral, w…
The aim of the present paper is to obtain some new fractional integral inequalities for convex functions. Saigo fractional integral operator is used to establish the results.
A number of new definite integrals involving Bessel functions are presented. These have been derived by finding new integral representations for the product of two Bessel functions of different order and argument in terms of the generalized hypergeometric function with subsequent reduction to special cases. Connection is made with Weber's second exponential integral and Laplace transforms of products of three Bessel functions.
We have recently established some integral inequalities for convex functions via the Hermite-Hadamard's inequalities. In continuation here, we also establish some interesting new integral inequalities for convex functions via the Hermite--Hadamard's inequalities and Jensen's integral inequality. Useful applications involving special means are also included.
This paper deals with generalized elliptic integrals and generalized modular functions. Several new inequalities are given for these and related functions.
We consider a certain definite integral involving the product of two classical hypergeometric functions having complicated arguments. We show in this paper the surprising fact that this integral does not depend on the parameters of the hypergeometric functions.
Dawson's integral and related functions in mathematical physics that include the complex error function (Faddeeva's integral), Fried-Conte (plasma dispersion) function, (Jackson) function, Fresnel function and Gordeyev's integral are analytically evaluated in terms of the confluent hypergeometric function.And hence, the asymptotic expansions of these functions on the complex plane $\mathbb{C}$ are derived using the asymptotic expansion of the confluent hypergeometric function.
The theory of self-reciprocal functions is applied to the study Mordell type integrals. We find two particular eigenfunctions of the double cosine Fourier transform and then use them to evaluate certain one- and two-dimensional Mordell type integrals in closed form. A reduction formula is given for a certain family of double Mordell integrals in terms of one-dimensional integrals.
The classical table of integrals by I. S. Gradshteyn and I. M. Ryzhik contains many definite integrals where the integrand is the product of a rational function times the logarithm of another rational function. We begin the systematic evaluation of these integrals.
By starting with Durand's double integral representation for a product of two Jacobi functions of the second kind, we derive an integral representation for a product of two Jacobi functions of the second kind in kernel form. We also derive a Bateman-type sum for a product of two Jacobi functions of the second kind. From this integral representation we derive integral representations for the Jacobi function of the first kind in both the hyperbolic and trigonometric contexts. From the integral representations for Jacobi functions, we also derive integral representations for products of limiting …
In this paper, firstly we have established Hermite--Hadamard-Fejér inequality for fractional integrals. Secondly, an integral identity and some Hermite-Hadamard-Fejer type integral inequalities for the fractional integrals have been obtained. The some results presented here would provide extensions of those given in earlier works.
By using Cauchy integral formula in the theory of complex functions, the authors establish some integral representations for the principal branches of several complex functions involving the logarithmic function, find some properties, such as being operator monotone function, being complete Bernstein function, and being Stieltjes function, for these functions, and verify a conjecture on complete monotonicity of a function involving the logarithmic function.
In this paper, a new identity for fractional integrals is established. Then by making use of the established identity, some new Ostrowski type inequalities for harmonically s-convex functions via Riemann--Liouville fractional integral are established.
We have shown that in some region where the Euler integral of the first kind diverges, the Euler formula defines a generalized function. The connected of this generalized function with the Dirac delta function is found.
Integral Mittag-Leffler, Whittaker and Wright functions with integrands similar to those which already exist in mathematical literature are introduced for the first time. For particular values of parameters, they can be presented in closed-form. In most reported cases, these new integral functions are expressed as generalized hypergeometric functions but also in terms of elementary and special functions. The behavior of some of the new integral functions is presented in graphical form. By using the MATHEMATICA program to obtain infinite sums that define the Mittag-Leffler, Whittaker, and Wrigh…
In this paper, we extend some estimates of the right hand side of a Hermite-Hadamard type inequality for prequasiinvex functions via fractional integrals.
A new generalization called $\mathtt{k}$-Struve function and its properties given by Nisar and saiful very recently. In this paper, we establish the pathway fractional integral representation of $\mathtt{k}$-Struve function. Many special cases also established to obtain the pathway integral representation of classical Struve function.
In this paper, the author established Hermite-Hadamard's inequalities for harmonically convex functions via fractional integrals and obtained some Hermite-Hadamard type inequalities of these classes of functions.
A new parametric integral is obtained as a consequence of the Riemann hypothesis. An asymptotic multiplicability is the main property of this integral.
This paper establishes a real integral representation of the reciprocal $Γ$ function in terms of a regularized hypersingular integral. The equivalence with the usual complex representation is demonstrated. A regularized complex representation along the usual Hankel path is derived.
In this paper, we establish several inequalities for different convex mappings that are connected with the Riemann-Liouville fractional integrals. Our results have some relationships with certain integral inequalities in the literature.
In this paper, firstly we have established Hermite-Hadamard's inequalities for s-convex functions in the second sense and m-convex functions via fractional integrals. Secondly, a Hadamard type integral inequality for the fractional integrals are obtained and these result have some relationships with KBOP.
We obtain a necessary and sufficient condition for embeddings of integral Lipschitz classes Lip(α; p) into classes ΛBV of functions of bounded Λ-variation.
We present a systematic derivation of some definite integrals in the classical table of Gradshteyn and Ryzhik that can be reduced to the gamma function.
In this paper we study the integral of type \[_{δ,a}Γ_{ρ,b}(x) =Γ(δ,a;ρ,b)(x)=\int_{0}^{\infty}t^{x-1}e^{-\frac{t^δ}{a}-\frac{t^{-ρ}}{b}}dt.\] Different authors called this integral by different names like ultra gamma function, generalized gamma function, Kratzel integral, inverse Gaussian integral, reaction-rate probability integral, Bessel integral etc. We prove several identities and recurrence relation of above said integral, we called this integral as Four Parameter Gamma Function. Also we evaluate relation between Four Parameter Gamma Function, p-k Gamma Function and Classical Gamma Func…
Based on a Problem and its solution published on the pages of SIAM Review, we give an interesting integral representation for the Lambert $W$ function in this short note. In particular, our result yields a new integral representation for the $Ω=W(1)$ constant as well.
New identity for fractional integrals have been defined. By using of this identity, we obtained new estimates on generalization of Hadamard, Ostrowski and Simpson type inequalities for s-convex, quasi-convex, m-convex functions via Riemann Liouville fractional integral.
In this paper, we establish several inequalities for s-convex mappings that are connected with the Riemann-Liouville fractional integrals. Our results have some relationships with certain integral inequalities in the literature.
In this paper we introduce the iterations of the Jacob's ladder and the new type of integral containing certain product of the factors $|ζ|^2$ corresponding to the components of some disconnected set of the critical line. Next, we obtain an asymptotic formula for this integral, its factorization and, for example, the essential generalization of two Selberg's formulae (1946).
Several estimates for singular integrals, maximal functions and the spherical summation operator are given in the spaces $L^p_{\text{rad}}L^2_{\text{ang}}(\mathbb{R}^n)$, $n\geq 2$.
This paper aims to investigate properties associated with fractional integral operators involving the three-parameters Mittag-Leffler function in the kernels with respect to another function. We prove that the Cauchy problem and the Volterra integral equation are equivalent. We find a closed-form to the solution of the Cauchy problem using successive approximations method and $ψ$-Caputo fractional derivative.
We prove the endpoint weak type estimate for square functions of Marcinkiewicz type with fractional integrals associated with non-isotropic dilations. This generalizes a result of C. Fefferman on functions of Marcinkiewicz type by considering fractional integrals of mixed homogeneity in place of the Riesz potentials of Euclidean structure.
In this paper, first we have established Hermite- Hadamard's inequalities for preinvex functions via fractional integrals. Second we extend some estimates of the right side of a Hermite- Hadamard type inequality for preinvex functions via fractional integrals.
The table of Gradshteyn and Rhyzik contains some trigonometric integrals that can be expressed in terms of the beta function. We describe the evaluation of some of them.
Many integrals in the classical table by Gradshteyn and Ryzhik can be evaluated in terms of the digamma function (= the logarithmic derivative of the gamma function). Some of them are presented here.
An integral representation is provided for the parabolic cylinder function product $D_μ(x)D_μ(-y)$ where $Re\,μ<0$ and $x>y$ are unrelated. A few simple consequences are given in the form of hyperbolic integrals and a sum rule.
This paper deals with a Euler type integral operator involving k-Mittag-Leffler function defined by Gupta and Parihar [8]. Furthermore, some special cases are also taken into consideration.
In this paper, Hermite-Hadamard type inequality for Sugeno integrals based on log-convex functions is studied. Some examples are given to illustrate the results.
