Abstract: In this paper, two Holder mean inequalities for the complete elliptic integrals are established.

Abstract: A closed expression for Q ν(a, b) with integer ν in terms of a confluent Appell function, differentiation formulas with respect to a and b, generating functions and other relations are given.

Abstract: In this paper, making use of the generalized Srivastava–Attiya operator , we introduce a new subclass of spiral-like functions and discuss some subordination results for functions in this generalized function class Furthermore, we mention some known and new results, which follow as special cases of our results

Abstract: From the Kummer 2 F 1-summation theorem, the Dixon–Kummer 4 F 3-summation theorem and the Dougall–Dixon 5 F 4-summation theorem, we establish, by means of the Bell polynomials, three general formulas related to the generalized harmonic numbers and the Riemann zeta function. Based on these three general formulas, we further find series of harmonic number identities. Some of these identities involve both finite summation and infinite series, so that we can determinate the explicit expressions of numerous infinite series. In particular, we show that several interesting analogues of the Euler sums can be evaluated.

Abstract: Inspired by the recent work of Opps, Saad and Srivastava, who gave the recursion formulas of Appell’s function F 2 by the contiguous relation of the Gauss hypergeometric series 2 F 1, we establish the recursion formulas for Appell functions F 1, F 2, F 3 and F 4 by the contiguous relations of hypergeometric series.

Abstract: We establish sharp Wilker and Huygens-type inequalities for inverse trigonometric and inverse hyperbolic functions.

Abstract: We establish Wilker- and Huygens-type inequalities for inverse trigonometric and inverse hyperbolic functions. We also provide a laconic proof to Oppenheim’s problem associated with inequalities involving the sine and cosine functions.

Abstract: 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: The fractional wavelet transform is defined and its properties are studied. The continuity of the fractional wavelet transform on Gelfand–Shilov spaces of type S is shown.

Abstract: This paper is intended to investigate a fractional telegraph equation of the form with positive real parameters a, b and c. Here , and are operators of the Riemann–Liouville fractional derivative, where 0<α≤1 and 0<β≤1. A symbolic operational form of the solutions in terms of the Mittag–Leffler functions is exhibited. Using the Banach fixed point theorem, the existence and uniqueness of solutions are studied for this kind of fractional differential equations.

Abstract: 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: In studying the behaviour of series, defined by means of the multi-index Mittag-Leffler functions, on the boundary of their domains of convergence in the complex plane, we prove Cauchy–Hadamard-, Abel-, Tauber- and Littlewood-type theorems. Asymptotic formulae for the cases of ‘large’ values of indices that are used in the proofs of the convergence theorems are also proved.

Abstract: We present a new characterization of Dunkl-classical symmetric orthogonal polynomials. Moreover, we establish an example of a non-symmetric sequence of Dunkl-classical orthogonal polynomials, which proved that the set of this last kind of polynomials is not empty.

Abstract: It is shown that the S-transform with the Gaussian window is a continuous linear map of the space of type S defined on ℝ into the space of type [Stilde] defined on ℝ×(ℝ∖{0}). The S-transforms of certain ultradifferentiable functions are also investigated.

Abstract: The Lambert W function has a number of integral expressions, including integrals of Bernstein, Thorin, Poisson, Stieltjes, Pick and Burniston–Siewert types. We give explicit integral expressions for W for each of these types. We also give integrals for a number of functions containing W.

Abstract: In this paper, we study the Dunkl–Gabor transform on ℝ d . We also give the practical real inversion formulas for this transform using the theory of reproducing kernels.

Abstract: In some recent investigations involving differential operators for generalized Lagrange polynomials, Chan et al. [The Lagrange polynomials in several variables, Integral Transforms Spec. Funct. 12 (2001), pp. 139–148] encountered and proved a certain summation identity for the Lagrange polynomials in several variables, which are popularly known as the Chan–Chyan–Srivastava polynomials. In the present paper, we investigate several families of bilateral generating functions for the Chan–Chyan–Srivastava polynomials and the (Srivastava–Daoust) generalized Lauricella functions.

Abstract: The main objective of this paper is to derive some inequality properties, convolution properties, subordination and superordination properties, and sandwich-type results involving the generalized Srivastava–Attiya operator.

Abstract: The purpose of this paper is to define a particular fractional analog of the Laplace operator in a rectangular domain in the plane by exploiting the Riemann–Liouville fractional derivatives. Such a definition allows the introduction of fractional boundary value problems which correspond to the classical Dirichlet, Neumann and mixed boundary value problems for the Laplace operator. By exploiting a suitable Integration by Parts Formula and the positiveness of the corresponding energy integral, we verify some uniqueness results for the solutions of the boundary value problems and show the existence of particular solutions.

Abstract: An orthogonal polynomial sequence with respect to a regular form (linear functional) u is said to be semi-classical if there exist a monic polynomial Φ and a polynomial Ψ, with deg Ψ≥1, such that (Φ u)′+Ψ u=0. Recently, all semi-classical monic orthogonal polynomial sequences of class one satisfying a three-term recurrence relation with β n =(−1) n β0, n≥0, β0∈ℂ∖{0} have been determined (see [B. Bouras and A. Alaya, A large family of semi-classical polynomials of class one, Integral Transforms Spec. Funct. 18 (2007), pp. 913–931]). In this paper, the sequences of the above family such that their corresponding Stieltjes function S(u)(z)=−∑ n≥0⟨ u, x n ⟩/z n+1 satisfies a quadratic relation of the form BS 2(u)+CS(u)+D=0, where B, C, D are polynomials, are described.

Abstract: Linear homogeneous differential equations with finite number of left-hand sided Liouville fractional derivatives, being analogues of Euler-type ordinary differential equations, are investigated. Using the direct and inverse Mellin transforms, the residue theory and properties of fractional derivatives and Euler Γ and psi functions, general solutions of the considered equations are established. The corresponding results are deduced for Euler-type ordinary differential equations.

Abstract: In this paper, we show that the circular prolate spheroidal wave functions (CPSWFs) are the most concentrate energy function on (0, T) among Hankel band-limited functions, here T is a positive real number. Hence, they best approximate each function in the set of essentially time- and Hankel band-limited signals than any other subspace of L 2(0,+∞). More precisely, using the theory of the CPSWFs, we show that the space spanned by the N first CPSWFs best approximate the set of essentially time- and Hankel band-limited signals than any other subspace of L 2(0,+∞) of the same dimension N.

Abstract: A closed expression, recurrency relation, differentiation formulas and other new relations for the generalized Bernoulli and Euler polynomials are given. Formulas of summation containing the generalized Bernoulli and Euler polynomials and various special functions, are derived.

Abstract: In this paper, we obtain a new real-variable inverse formula for the Laplace transform and investigate its convergence rate.

Abstract: In this paper, we define a sequential analytic Feynman integral of functionals on Wiener space. We then give the existence of the sequential analytic Feynman integral of functionals in L 2(C 0[0, T]). The expansion of functionals in L 2(C 0[0, T]) in terms of Fourier–Hermite functionals plays a key role in the study.

Abstract: Inequalities connecting Jacobian elliptic functions sn and sc with their inverses are established. Applications to circular and hyperbolic functions are included.

Abstract: Product formulas for Jacobian elliptic functions s n, s c, s d and their reciprocals are established. Applications to Legendre’s incomplete elliptic integral of the first kind and to the arc lemniscate sine function are given. Lower and upper bounds for elliptic functions and elliptic integrals mentioned above are also derived.

Abstract: Let C[0, t] denote the function space of all real-valued continuous paths on [0, t]. Define and by X n (x)=(x(t 0), x(t 1), …, x(t n )) and X n+1(x)=(x(t 0), x(t 1), …, x(t n ), x(t n+1)), where 0=t 0

Abstract: This paper deals with a new singular differential-difference operator Y t, A on the real line including, as a particular case, the Dunkl, Dunkl–Heckman and Dunkl–Cherednik operators. We establish some results of Harmonic analysis related to this operator, such that a product formula for the related eigenfunction G λ is given as integral with an explicit kernel. This product formula is an important tool to define the generalized translation operator which is used to set up a convolution structure. Next, we establish an inversion formula and prove a generalized Plancherel theorem for this operator. As a direct application, we give a maximum principle of the operators and we solve the heat equation associated with the generalized Dunkl operator on the real line.