The convolution product is associated with exponential kernel transforms such as the Fourier and Laplace transforms. It's feasibility is due the important property of the simple addition operation for the arguments of the exponential kernels product. This is behind the simple form of the convolution product. No procedure that is remotely close to this exists for the transforms with non-exponential

We establish a functional identity for Mahler measures of the two-parametric family P a , c ( x , y ) = a ( x + 1 / x ) + y + 1 / y + c . Our result extends an identity proven in a paper of Lalín, Zudilin and Samart. As a by-product, we obtain evaluations of m ( P a , c ) for some algebraic values of a and c in terms of special values of L-functions and logarithms. We also give a sufficient condition

The linear canonical transform has been found various applications in several areas, such as signal processing and optics. Several uncertainty principles were established for this transform in the last years. However, most results only focused on the uncertainty relations in two different time-frequency domains. In this paper, we study the uncertainty principle for measurable sets associated with several

We consider a problem for the zero-order Bessel equation with complex-valued physical and spectral parameters in the boundary condition. The spectral parameter is squared in the boundary condition. We study the basis property for the system of eigenfunctions.

A fractional power interpretation of the Laguerre derivative ( D x D ) α , D ≡ d / d x is discussed. The corresponding fractional integrals are introduced. Mapping and semigroup properties, integral representations and Mellin transform analysis are presented. A relationship with the Riemann–Liouville fractional integrals is demonstrated. Finally, a second kind integral equation of the Volterra type

Some new relations for the Horn function H 1 a , b , c ; d ; w , z and confluent Horn function H 1 ( c ) a , b ; d ; w , z are obtained including differentiation and integration formulas, series and values for specific choice of parameters and variables. Some generating functions for various special functions are given in terms of these Horn functions.

ABSTRACT A method has been presented recently for deriving integrals of special functions using two kinds of integrating factor for the homogeneous second-order linear differential equations which many special functions obey. The classical orthogonal polynomials are well-suited for this method, and results are given here for the Gegenbauer, Hermite and Laguerre polynomials. All the integrals presented

ABSTRACT An integrating factor f ~ x is presented involving the terms in y ′ ′ x and q x y x of the general homogenous second-order linear ordinary differential equation. The new integrating factors obey second-order differential equations, rather than being given by quadrature. The new factors provides old and new integrals for special functions which obey such differential equations. The functions

We study integral transforms mapping a function on the Euclidean plane to the family of its integration on plane curves, that is, a function of plane curves. The plane curves we consider in the present paper are given by the graphs of functions with a fixed axis of the independent variable and are imposed some symmetry with respect to the axes. These transforms contain the parabolic Radon transform

The aim of this paper is to prove some new uncertainty principles for the windowed Hankel transform. They include uncertainty principle for orthonormal sequence, local uncertainty principle, logarithmic uncertainty principle and Heisenberg-type uncertainty principle. As a side result, we obtain the Shapiro's dispersion theorem for the windowed Hankel transform.

We evaluate Euler-like sums involving harmonic numbers using expansions for powers of logarithmic, arctan and a r c t a n h functions.

A polynomial sequence (sn(x))n∈N is symmetric, or self-dual, if sn(m)=sm(n) for all m,n=0,1,2,…. In this paper, we give the characterization of the exponential symmetric Sheffer sequences with weight sequences, and show that all the symmetric Sheffer sequences with weight (λn), (⟨λ⟩n) or ((λ)n) are in fact constant multiples of the Charlier polynomial sequence, the Meixner polynomial sequence, or the

We construct formulas for analytic continuation of the Lauricella functions FA(N), FB(N) and FD(N) into the exterior of the domains, where they are initially defined by N-multiple hypergeometric series. To construct the analytic continuation, we use a new method, which allows to obtain continuation formulas for arbitrary hypergeometric series belonging to Horn's class, which, in particular, contains

In this paper, we study properly the fractional integrals Δw−μ/2, associated with the Weinstein operator, for all μ>0. In the order to give sense to the higher order Riesz Weinstein transforms, we prove that, for a function f∈D∗(Rd) and n=(n1,…,nd)∈Nd∖{(0,…,0)}, Δw−|n|/2 is |n| times differentiable for x outside the support of f and |n|−d times differentiable (if |n|>d) inside the support of f.

Some new relations for the Horn functions G2(a,a′,b,b′;w,z), G3(a,a′;w,z) and confluent Horn function Γ1(a,b,b′;w,z) are obtained including differentiation and integration formulas, series and values for specific choice of parameters and variables. Some generating functions for various special functions are given in terms of these Horn functions.

In this paper, we establish a general q-series identity based on Z.-G. Liu's work, from which a handful of basic hypergeometric transformations are derived. Further, their applications to Rogers–Ramanujan type identities will be discussed.

The (generalized) Mellin transforms of Gegenbauer polynomials have polynomial factors pnλ(s), whose zeros all lie on the ‘critical line’ ℜs=1/2 (called critical polynomials). The transforms are identified in terms of combinatorial sums related to H. W. Gould's S:4/3, S:4/2, and S:3/1 binomial coefficient forms. Their ‘critical polynomial’ factors are then identified in terms of 3F2(1) hypergeometric

We derive a convergent expansion of the generalized hypergeometric function p−1Fp in terms of the Bessel functions 0F1 that holds uniformly with respect to the argument in any horizontal strip of the complex plane. We further obtain a convergent expansion of the generalized hypergeometric function pFp in terms of the confluent hypergeometric functions 1F1 that holds uniformly in any right half-plane

Description of convergence domains for multiple power series is a quite difficult problem. In 1889 J.Horn showed that the case of hypergeomteric series is more favourable. He found a parameterization formula for surfaces of conjugative radii of such series. But until recently almost nothing was known about the description of convergence domains in terms of functional inequalities ρj(|a1|,…,|am|)<0

Some new relations for the Horn function G1a,b,b′;w,z and confluent Horn function Γ2b,b′;w,z are obtained including differentiation and integration formulas, series and values for specific choice of parameters and variables. Some generating functions for various special functions are given in terms of these Horn functions.

The problem of analytic continuation is considered for the general hypergeometric Horn series with an arbitrary number of variables. An approach is proposed that allows one to find formulas for the continuation of such series into the exterior of the set of their convergence in the form of linear combinations of other hypergeometric series. These new hypergeometric series belong also to the Horn class

We prove a Heisenberg type uncertainty principle for the Gabor spherical mean transform, and we study its generalization. Next, we extend local uncertainty principle for sets of finite measure to the latter transform.

In this paper, we present a mixed type integral-sum representation of the cylinder functions Cμ(z), which holds for unrestricted complex values of the order μ and for any complex value of the variable z. Particular cases of these representations and some applications, which include the discussion of limiting forms and representations of related functions, are also discussed.

In this paper, we present the basic k-Hankel wavelet theory. Next, we study the boundedness and compactness of localization operators associated with k-Hankel wavelet transforms on L k p ( R ) , 1 ≤ p ≤ ∞ . Finally, we give some typical examples of localization operators.

We prove that the function Ga(x)=a−log⁡(1−x)K(x)(a∈R) is strictly concave on (0,1) if and only if a≥8/5. This solves a problem posed by Yang and Tian and complements their result that 1/Ga (a≥0) is strictly concave on (0,1) if and only if a=4/3. Moreover, we apply our concavity theorem to present several functional inequalities involving K. Among others, we prove that if a≥8/5, then 2aπ+1

In this paper, we introduce the definitions of the Fourier transform and spherical Fourier transform on quaternionic Heisenberg group, then we present some of their properties, in particular, we will determine the spherical functions and the heat kernel of quaternionic Heisenberg group. Finally, we prove a qualitative uncertainty principle ‘Donoho–Stark's uncertainty principle’.

In this paper, using the Macdonald's identities for the products of modified Bessel functions of first and second kinds, we derive new integral representations for the products of Airy functions and their derivatives. Manipulating the integrands of Macdonald's identities with various integral representations lead us to get new representations for the products of Airy functions and their derivatives

An orthogonal polynomial sequence with respect to a regular form (linear functional) w is said to be semiclassical if there exists a monic polynomial φ and a polynomial ψ with degψ≥1, such that (φw)′+ψw=0. Recently, all semiclassical monic orthogonal polynomial sequences {Wn}n≥0, of class one obtained from the cubic decompositions (CD) satisfying the relation W3n(x)=Pn(x3+qx+r) have been determined

In this paper, we present some new elements of harmonic analysis related to the right-sided multivariate continuous quaternion wavelet transform. The main objective of this article is to introduce the concept of the right-sided multivariate continuous quaternion wavelet transform and investigate its different properties using the machinery of multivariate quaternion Fourier transform. Last, we have

By making use of divided differences, new proofs are presented for Dougall's summation theorem for well-poised 7 F 6 -series and Whipple's transformation between well-poised 7 F 6 -series and balanced 4 F 3 -series. Several new summation formulae of higher order are established, including three identities of Fox–Wright function and an extension of Karlsson–Minton formula.

Discrete analogues of the classical Mehler–Fock transforms are introduced and investigated. It involves series with the associated Legendre function P i n − 1 / 2 μ ( x ) , x>1, R e μ < 1 / 2 , n ∈ N , i is the imaginary unit, and the so-called incomplete Legendre integrals. Several expansions of arbitrary functions and sequences in terms of these series and integrals are established.

An integration formula for generating indefinite integrals which was presented in Conway JT [A Lagrangian method for deriving new indefinite integrals of special functions. Integral Transforms Spec Funct. 2015;26(10):812–824], which involves an arbitrary function h x , is generalized to include two arbitrary functions s x and h x . In the new formula the Lagrangian factor f x of the original formula

We define the maximal and fractional maximal operators associated with the Kontorovich–Lebedev transform. We study the boundedness of these operators in the space L μ p ( R + , d x ) . Using these results we establish the Hardy–Littlewood–Sobobev theorem for the Riesz potential operator associated with Kontorovich–Lebedev transform.

This paper investigates some system of integral inequalities of one independent variable on time scales. The conclusion can be obtained by using Hadamard-type fractional differential equations and Greene's method which bring together and expand some integral inequalities on time scales. The established inequalities give explicit bounds on unknown functions which can be utilized as a key in examining

In this paper some new classes of two-variable orthogonal functions by using Fourier transforms of two-variable orthogonal polynomials are introduced. Orthogonality relations are obtained by using the Parseval identity. Recurrence relations for new families of orthogonal functions are also presented.

We introduce poly-Cauchy numbers with level 2. Poly-Cauchy numbers may be interpreted as a kind of generalizations of the classical Cauchy numbers by using the inverse relation of exponentials and logarithms. On the contrary, poly-Bernoulli numbers can be from the inverse relation of logarithms and exponentials. In this similar stream, poly-Cauchy numbers with level 2 may be yielded from the inverse

Let σ j n ( a ) be the sum of ( − j ) th powers of the zeros of the generalized Bessel polynomial θ n ( z ; a ) . In this note, we generalize the well-known Burchnall's result σ 1 n ( 2 ) = − 1 , σ 2 j − 1 n ( 2 ) = 0 ( j = 2 , … , n ) . In particular, we find the simple explicit formulae for σ j n ( k − n ) ( j = 1 , … , n + k − 2 ) with k=2,3,4 and establish the new identity for certain Bessel polynomials

The aim of this article is to formulate some new uncertainty principles for the continuous shearlet transforms in arbitrary space dimensions. Firstly, we derive an analogue of Pitt's inequality for the continuous shearlet transforms, then we formulate Beckner's uncertainty principle via two approaches: one based on a sharp estimate from Pitt's inequality and the other from the classical Beckner inequality

The paper discusses some properties of the modulus | W k , m ( z ) | of the Whittaker function W k , m ( z ) . In particular, completely monotone functions expressed in terms of | W k , m ( z ) | are found. The results follow from an integral representation for products of Whittaker functions due to Erdélyi (1938). An application to the Bessel function K ν ( z ) is given.

In this paper we treat certain elliptic and hyper-elliptic integrals in a unified way. We introduce a new basis of these integrals coming from certain basis φ n ( x ) of polynomials and show that the transition matrix between this basis and the traditional monomial basis is certain upper triangular band matrix. This allows us to obtain explicit formulas for the considered integrals. Our approach, specified

(2020). Correction. Integral Transforms and Special Functions: Vol. 31, No. 5, pp. (i)-(i).

In 1990, van Eijndhoven and Meyers provide a special orthonormal basis for the Bargmann Hilbert space consisting of holomorphic Hermite functions. Then it was be natural to look for its orthogonal complement in the underlying L2-Hilbert space. In this paper, we describe the orthogonal complement of this Hilbert space. More precisely, a polyanalytic orthonormal basis is given and the explicit expressions

Let u be a non-zero linear functional acting on the space of polynomials. Let Dq,ω be a Hahn operator acting on the dual space of polynomials. Suppose that there exist polynomials φ and ψ, with deg⁡φ≤2 and deg⁡ψ≤1, so that the functional equation Dq,ω(φu)=ψu holds, where the involved operations are defined in a distributional sense. In this note we state necessary and sufficient conditions, involving

In this article, we discuss the existence and uniqueness of solutions to a class of integral boundary value problem of fractional differential equations. Different fractional order derivations are included in the nonlinear term and boundary conditions. The unique solution is derived by a new fixed point theorem of increasing Ψ − ( h , r ) -concave operators defined on ordered sets. Further, we construct

In this paper, we consider the Bessel Fourier transform on the Weyl chamber which generalizes Hankel transform and coincides with the restriction of the Dunkl transform associated to the reflection group Bq=Sq⋉Z2q. We deal with partial Bessel integrals and we give an interesting estimation of the modified partial Bessel integrals on L2(ω~μ)∩L∞(ω~μ).

ABSTRACT The Laplace transforms of the transition probability density and distribution functions of the Feller process contain products of a Kummer and a Tricomi confluent hypergeometric function. The intricacies caused by the singularity at 0 of the Feller process imply that ultimately seven new inverse Laplace transforms can be derived of which four contain the Marcum Q function. The results of this

We obtain a multidimensional Tauberian theorem for Laplace transforms of Gelfand-Shilov ultradistributions. The result is derived from a Laplace transform characterization of bounded sets in spaces of ultradistributions with supports in a convex acute cone of Rn, also established here.

In this article, we first give some basic properties of generalized Hermite polynomials associated with parabolic cylinder functions. We next use Weisnerś group theoretic method and operational rules method to establish new generating functions for these generalized Hermite polynomials. The operational methods we use allow us to obtain unilateral, bilinear and bilateral generating functions by using

Schlömilch's series is named after the German mathematician Oscar Xavier Schlömilch, who derived it in 1857 as a Fourier series type expansion in terms of the Bessel function of the first kind. However, except for Bessel functions, here we consider an expansion in terms of Struve functions or Bessel and Struve integrals as well. The method for obtaining a sum of Schlömilch's series in terms of the

We provide an explicit formula for a Sonine-Gegenbauer integral, which seems to be unknown in the literature so far. For another type of these integrals, we show a dependence relation over the rational functions, including the explicit calculation of the coefficient functions.