In this paper, we consider the third-order differential equation of the quadratic products of Hermite functions and present the integral representations of products of Hermite functions (Lebedev identities) in terms of the confluent hypergeometric functions. Manipulating the integrands of these identities with various integral representations lead us to get new representations for the products of Hermite

We derive relations for a certain class of terminating 4F3(4) hypergeometric series with three free parameters. The invariance group composed of these relations is shown to be isomorphic to the symmetric group S3. We further study relations for terminating 3F2(4) series that fall under two families. By using a series reversal, we examine the corresponding terminating 4F3(1/4) and 3F2(1/4) series relations

Discrete analogues of the classical Fourier-Jacobi transform are introduced and investigated. It involves series and integrals with respect to parameters of the Gauss hypergeometric function 2F1(a+in/2,a−in/2;c;−x2),x>0,n∈N,a,c>0,i is the imaginary unit. The corresponding inversion formulas for suitable functions and sequences in terms of these series and integrals are established.

We investigate the interplay between the Caputo fractional operators D±1/2 and Fourier–Legendre expansions of hypergeometric functions, resulting in transformation formulas for q+1Fq(z) series with quarter-integer parameters and in explicit evaluations of many series, including: ∑n≥0((12)nn!)2(−1)n2n+1,∑n≥0((12)nn!)214n+3,∑n≥0((12)nn!)31n+1,∑n≥0((12)nn!)31(n+1)2,∑n≥0((12)nn!)2H2n2n(n+1),∑n≥0((12)nn

Discrete analogues of the index transforms with squares of Bessel functions of the first and second kind Jν(z),Yν(z) are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.

ABSTRACT The purpose of this work is to prove analogues of the classical Titchmarsh theorem and Younis' theorem associated with the Fourier–Jacobi transform of a set of functions satisfying a generalized Lipschitz condition of a certain order in suitable weighted spaces Lp([0,+∞)), 1

We consider a function that is obtained by the first kind Hankel function of order zero by removing its logarithmic singularity. This function, apart from a multiple of the zero-order Bessel function as an additive term, can be considered as the ‘entire part’ of the Hankel function. We constitute the Fourier transform of this function by considering a fifth-order ordinary differential equation which

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 In this work, we will study a Bessel potential spaces Wαs,p(R+n), ( s∈R, 1≤p<+∞, α=(α1,…,αn)∈Rn, α1>−12,…,αn>−12) associated with the Poly-axially operator Δα and their properties. We define the Sobolev-type spaces Eαm,p(R+n) (m∈N, 0∈N, 1≤p<+∞) and we prove an analogue of Calderón's theorem for Fourier–Bessel transform. We define the nonlinear Bessel potentials associated with a measure ν

A class of integral transforms which satisfies certain assumptions is considered. Several examples, including the Fourier, Laplace and Mellin transforms, are in this class as well as transforms which are yet to be named. The assumptions allow for certain properties to exist for these integral transforms, such as the shifting and convolution properties. Further conditions are presented which allow for

The coupled fractional Fourier transform is a two-dimensional fractional Fourier transform Fα,β that depends on two angles α,β that are coupled in such a way that the transform parameters are γ=(α+β)/2 and η=(α−β)/2. It generalizes the two-dimensional Fourier transform and it serves as useful tool in some applications in optics and signal processing. In this article we derive new properties of the

Thermal equilibrium of a two-dimensional model of mobile point-like charges with pair logarithmic interactions, immersed in a fixed neutralising background, is exactly solvable at a special coupling constant. Assuming gauge invariance of the statistical mechanics and the requirement of local charge neutrality in the bulk regime of the system, the exact solution implies explicit expressions for inverse

While it is known that one can consider the Cauchy problem for evolution equations with Caputo derivatives, the situation for the initial value problems for the Riemann–Liouville derivatives is less understood. In this paper, we propose new type initial, inner, and inner-boundary value problems for fractional differential equations with the Riemann–Liouville derivatives. The results on the existence

It is shown that an appropriate combination of methods, relevant to operational calculus and to special functions, can be a very useful tool to establish and treat a new modified 2D-Laguerre polynomials. We explore the formal properties of the operational identities to derive a number of properties of the new modified 2D-Laguerre polynomials and discuss the operational links with various known polynomials

In this paper, we consider the ∗w-product on function space Ca,b[0,T] introduced in Chang et al. [A new approach method to obtain the L1 generalized analytic Fourier–Feynman transform. Integr Transf Spec Funct. 2018;29:745–760]. We then obtain various integration formulas involving the ∗w-product and the first variation. Finally, we have relationships between the ∗w-product and the generalized integral

The exponential integral function Ei(x) is given as an indefinite integral of an elementary expression. This allows a second-order linear differential equation for the function to be constructed, which is of conventional form. A limitless number of differential equations can be derived from the original by elementary transformations, and many integrals are given by applying the method of fragments

ABSTRACT New asymptotic expansions are derived of the Kummer functions M(a,b,z) and U(a,b+1,z) for large positive values of a and b, with z fixed. For both functions we consider b/a≤1 and b/a≥1, with special attention for the case a∼b. We use a uniform method to handle all cases of these parameters.

ABSTRACT For fixed reals c>0, a>0 and α > − 1 2 , the circular prolate spheroidal wave functions (CPSWFs) or 2d-Slepian functions are the eigenfunctions of the finite Hankel transform operator, denoted by H c α , which is the integral operator defined on L 2 ( 0 , 1 ) with kernel H c α ( x , y ) = c x y J α ( c x y ) . Also, they are the eigenfunctions of the positive, self-adjoint compact integral

ABSTRACT Sufficient conditions are obtained for which localized fractional derivatives of the Riemann–Liouville and Marshaud type can be represented as localized fractional derivatives of Djrbashian–Caputo type. The compositional properties of localized fractional derivatives and ordinary derivatives are proved. The localized derivatives for the exponential, trigonometric, power and Weierstrass's functions

Some new relations for the Horn function H 4 ( a , b ; c , c ′ ; w , z ) and confluent Horn function H 7 ( c ) ( a ; c , c ′ ; w , z ) are obtained including differentiation and integration formulas, series and for special values of parameters and variables. Some generating functions for various special functions are given in terms of these Horn functions.

Discrete analogs of the Lebedev–Skalskaya transforms are introduced and investigated. It involves series and integrals with respect to the kernels R e   K α + i n ( x ) , I m   K α + i n ( x ) , x > 0 , n ∈ N , | α | < 1 ,   i is the imaginary unit and K ν ( z ) is the modified Bessel function. The corresponding inversion formulas for suitable functions and sequences in terms of these series and integrals

ABSTRACT In this paper, generalized Bessel potential is studied. A generalization is achieved by considering the singular Laplace–Bessel operator instead of the Laplace operator in the construction of the Bessel potential. First, we give some auxiliary statements as weighted plane wave relation and generalized Bochner formula. Next, we consider the Bessel kernel and prove some of its properties. After

ABSTRACT For a given sequence of real numbers a = ( a m ) m ≥ 1 , we define the function U μ , ν a ( x ) = 2 μ Γ ( μ + 1 ) x μ ∑ m ≥ 1 a m j m , ν μ J μ ( j m , ν x ) , x ∈ ( 0 , + ∞ ) , where μ , ν > − 1 , J μ denotes the Bessel function of order μ, and ( j m , ν ) m ≥   1 are the positive zeros of  J ν . In this paper, we study the function U μ , ν a and some outstanding instances of it on the whole

ABSTRACT In this paper, the idea of wave packet transform has been generalized. We have obtained reconstruction formula, characterization range, orthogonality, some important estimates and convolution of linear canonical wave packet transform. We have also discussed uncertainty principles and some properties on periodic functions for this transform.

In this paper, we characterize the discrete Hahn-classical d-orthogonal polynomial sets (and their iterated) by a difference distributional matrix equation satisfied by the associated d-dimensional functional vector, where the coefficient matrices are explicitly given. As a result we describe this set of polynomials via a structure relation. We illustrate the preceding results with some examples.

ABSTRACT We introduce in this paper the concept of Dunkl-coherent pair of forms (linear functionals) in the symmetric case. We prove that if { u , v } is a Dunkl-symmetrically coherent pair of symmetric forms, then at least one of them has to be a symmetric Dunkl-classical form, i.e. Generalized Hermite or Generalized Gegenbauer polynomials.

Several types of conical Radon transforms have been studied since the introduction of the Compton camera. Several factors of a cone of integration can be considered as variables, for example, a vertex, a central axis, and an opening angle. In this paper, we study the conical Radon transform with a fixed central axis and opening angle. Furthermore, we consider the attenuation effect in the conical Radon

ABSTRACT In this paper, using the Lipschitz–Hankel identities we obtain some new integral representations for the toroidal functions in terms of the elementary, Bessel, parabolic cylinder and hypergeometric functions. Manipulating the integrands of Lipschitz–Hankel identities with several integral representations lead us to present the results. In this sense, we also derive some integral representations

Let a function f belong to the Lebesgue class , , and let be the Fourier transform of f. The classical theorem of E. Titchmarsh states that if the function f belongs to the Lipschitz class , , then belongs to the Lebesgue classes for . In this paper we prove an analogue of this result for the the Hankel transform of functions from Nikol'ski type function classes on the half-line .

ABSTRACT A method was 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 Jacobi polynomials. Results have been previously presented for Gegenbauer, Hermite

We introduce the directional short-time Fourier transform for which we prove a new inversion formula. We also prove for this transform several uncertainty principles as Heisenberg inequalities, Faris–Price uncertainty principles and local uncertainty principles.

ABSTRACT In the article ‘Convolution and product theorems for the special affine Fourier transform’ [In: Nashed MZ, Li X, editors. Frontiers in orthogonal polynomials and q-series. World Scientific; 2018. p. 119–137], a convolution structure is presented in the realm of the special affine Fourier transform. In continuation of the study, we introduce a novel integral transform coined the special affine

ABSTRACT Some new relations for the Horn function H 3 a , b ; c ; w , z , confluent Horn function H 6 ( c ) a ; c ; w , z and Humbert function Φ 3 b ; c ; w , z are obtained including differentiation and integration formulas, series and values for specific choice of parameters and variables. Some generating fuctions for various special functions are given in terms of these Horn functions.

ABSTRACT Given complex parameters x, ν, α, β and γ ∉ − N , consider the infinite lower triangular matrix A ( x , ν ; α , β , γ ) with elements A n , k ( x , ν ; α , β , γ ) = ( − 1 ) k n + α k + α ⋅ F ( k − n , − ( β + n ) ν ; − ( γ + n ) ; x ) for 1 ⩽ k ⩽ n , depending on the Hypergeometric polynomials F ( − n , ⋅ ; ⋅ ; x ) , n ∈ N ∗ . After stating a general criterion for the inversion of infinite

Time–frequency analysis theory has relatively recent developments in pure and applied mathematics. Motivated by Wong's approach, we will study, in this paper, the localization operators associated with the generalized Wigner transform. Results on the L p -boundedness and L p -compactness of these localization operators are given. The proofs are based on some estimates on the generalized hypergeometric

In this paper, a two-parameter extension of the operational calculus of Mikusiński's type for the Caputo fractional derivative is presented. The first parameter is connected with the rings of functions that are used as a basis for construction of the convolution quotients fields. The convolutions by themselves are characterized by another parameter. The obtained two-parameter operational calculi are

ABSTRACT A complete characterization of a measure μ governing the boundedness of fractional integral operators defined on a quasi-metric measure space ( X , d , μ ) (non-homogeneous space) from one grand Lebesgue spaces L μ p ) , θ 1 ( X ) into another one L μ q ) , θ 2 ( X ) is established. As a corollary, we have a generalization of the Sobolev inequality for potentials with measure. An appropriate

ABSTRACT The aim of this paper is to develop a new harmonic analysis related to a Bessel type operator Δνm on the real line: We define the canonical Fourier Bessel transform Fνm and study some of its important properties. We prove a Riemann–Lebesgue lemma, inversion formula and operational formulas for this transformation. We derive Plancherel theorem and Babenko inequality for Fνm. In the present

ABSTRACT In this work, we will study a Bessel potential spaces Wαs,p(R+n), ( s∈R,1≤p<+∞,α=(α1,…,αn)∈Rn,α1>−12,…,αn>−12) associated with the Poly-axially operator Δα and their properties. We define the Sobolev type spaces Eαm,p(R+n)(m∈N,0∈N,1≤p<+∞) and we prove an analogue of Calderón's theorem for Fourier-Bessel transform. We define the nonlinear Bessel potentials associated with a measure ν and the

ABSTRACT Some new relations for the Horn function H2(a,b,c,c′;d;w,z) and confluent Horn function H2(c)(a,b,c;d;w,z) are obtained including differentiation and integration formulas, series and reduction formulas. Some generating functions for various special functions are given in terms of these Horn functions.

We prove that, for p∈(1,∞) and β∈R, the function x↦β−log⁡1−xpKp(xp) is strictly concave on (0,1) if and only if β≥λ(p):=2p(p2−2p+2)(p−1)(2p2−3p+3), where Kp represents the generalized complete p-elliptic integrals of the first kind defined by Kp(r):=∫0πp/2dθ(1−rpsinpp⁡θ)1−1/p, where πp:=2pB(1/p,1−1/p), π2=π, and sinp is the generalized sine function, with sin2=sin. This extends the recently obtained

The work is devoted to obtaining explicit formulas for analytic continuation of quite general hypergeometric series depending on three variables and belonging to the Horn class. We have derived such continuation formulas with respect to one variable for an arbitrary series belonging to the class under consideration. In addition, on the base of several examples we have demonstrated the use of the obtained

This paper reports computational manipulation to express xmpn(x), pn(x)qm(x) and ∏j=1spnj(x) in terms of generalized Laguerre polynomials, where pn(x) and qm(x) are polynomials of degrees n, m, respectively. The presented analytical formulae are discussed when pn(x) and qm(x) are generalized Laguerre, Hermite and Jacobi polynomials. Also, the corresponding formulae for the cases of Legender polynomials

ABSTRACT In this article, we present some boundedness results of fractional Hankel transform on Gelfand–Shilov spaces of type S. Certain inequalities are obtained for pseudo-differential operators involving the fractional Hankel transform on Gelfand–Shilov spaces of type S. This article made further discussion on the continuity properties of fractional Hankel transform and pseudo-differential operator

ABSTRACT Quantitative Shapiro's dispersion uncertainty principle and umbrella theorem are proved for the multivariate continuous shearlet transform SHψ introduced earlier in Dahlke et al. [The continuous shearlet transform in arbitrary space dimensions. Preprint Nr. 2008-7, Philipps-Universität Marburg; 2008; The continuous shearlet transform in arbitrary space dimensions. J Fourier Anal Appl. 2010;16:340–364]

ABSTRACT Hankel- K{Mp} spaces, as introduced by the second-named author, play the same role in the theory of the Hankel transformation as the Gelfand-Shilov K{Mp} spaces in the theory of the Fourier transformation. For μ>−1/2, under suitable restrictions on the weights {Mp}p=0∞, the topology of a Hankel- K{Mp} space E can be generated by norms of Lq-type (1≤q≤∞) involving the Bessel operator Sμ. In

In this paper, we establish the Cowling–Price's, Hardy's and Morgan's uncertainty principles for the Opdam–Cherednik transform on modulation spaces associated with this transform. The proofs of the theorems are based on the properties of the heat kernel associated with the Jacobi–Cherednik operator and the versions of the Phragmén–Lindlöf type result for the modulation spaces.

ABSTRACT The classical 2-orthogonal polynomials share the so-called Hahn property, this means that they are 2-orthogonal polynomials whose the sequences of their derivatives are also 2-orthogonal polynomials. Based only on this property, a new class of classical 2-orthogonal polynomials is obtained as particular solution of the non-linear system governing the coefficients involved in the recurrence

ABSTRACT Our aim in this paper is to prove an analogue of the classical Titchmarsh theorem on the image under the discrete Fourier–Laplace transform of a set of functions satisfying a generalized Lipschitz condition in the space L 2 on the sphere.

ABSTRACT The main aim of this article is to establish a summation formula for the second type Neumann series which members contain a product of two modified Bessel functions of the first kind of not necessarily equal orders and arguments.

ABSTRACT In this paper, some basic results and notations related to Lebedev–Skalskaya transforms (LS-transforms) are introduced and then estimates of translation and convolution operators associated to LS-transform are obtained. Continuity of LS-transforms on Lebesgue space as well as on function spaces Sβ,k and Gα,k are discussed. Pseudo-differential operators in terms of LS-transforms are defined

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

ABSTRACT We establish a functional identity for Mahler measures of the two-parametric family Pa,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(Pa,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 for validity of a certain

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 (DxD)α,D≡d/dx 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, involving

Some new relations for the Horn function H1a,b,c;d;w,z and confluent Horn function H1(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

ABSTRACT 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