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 Δ ν m on the real line: We define the canonical Fourier Bessel transform F ν m 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 F ν m . In the present

ABSTRACT In this work, we will study a Bessel potential spaces W α s , p ( R + n ) , ( s ∈ R , 1 ≤ p < + ∞ , α = ( α 1 , … , α n ) ∈ R n , α 1 > − 1 2 , … , α n > − 1 2 ) associated with the Poly-axially operator Δ α and their properties. We define the Sobolev type spaces E α m , p ( R + n ) ( m ∈ N , 0 ∈ N , 1 ≤ p < + ∞ ) and we prove an analogue of Calderón's theorem for Fourier-Bessel transform

Some new relations for the Horn function H 2 ( a , b , c , c ′ ; d ; w , z ) and confluent Horn function H 2 ( 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 − x p K p ( x p ) is strictly concave on ( 0 , 1 ) if and only if β ≥ λ ( p ) := 2 p ( p 2 − 2 p + 2 ) ( p − 1 ) ( 2 p 2 − 3 p + 3 ) , where K p represents the generalized complete p-elliptic integrals of the first kind defined by K p ( r ) := ∫ 0 π p / 2 d θ ( 1 − r p sin p p ⁡ θ ) 1 − 1 / p , where π p := 2 p B ( 1 / p , 1

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 x m p n ( x ) , p n ( x ) q m ( x ) and ∏ j = 1 s p n j ( x ) in terms of generalized Laguerre polynomials, where p n ( x ) and q m ( x ) are polynomials of degrees n, m, respectively. The presented analytical formulae are discussed when p n ( x ) and q m ( x ) are generalized Laguerre, Hermite and Jacobi polynomials. Also, the corresponding

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 on some

Quantitative Shapiro's dispersion uncertainty principle and umbrella theorem are proved for the multivariate continuous shearlet transform S H ψ 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]

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

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.

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 relation

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 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.

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 S β , k and G α , k are discussed. Pseudo-differential operators in terms of LS-transforms are defined and

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

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

ABSTRACT 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.

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

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

ABSTRACT A polynomial sequence ( s n ( x ) ) n ∈ N is symmetric, or self-dual, if s n ( m ) = s m ( 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 ( λ n ) , ( ⟨ λ ⟩ n ) or ( ( λ ) n ) are in fact constant multiples of the Charlier polynomial

ABSTRACT We construct formulas for analytic continuation of the Lauricella functions F A ( N ) , F B ( N ) and F D ( 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,

ABSTRACT In this paper, we study properly the fractional integrals Δ w − 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 ∗ ( R d ) and n = ( n 1 , … , n d ) ∈ N d ∖ { ( 0 , … , 0 ) } , Δ w − | n | / 2 is | n | times differentiable for x outside the support of f and | n | −

ABSTRACT Some new relations for the Horn functions G 2 ( a , a ′ , b , b ′ ; w , z ) , G 3 ( a , a ′ ; w , z ) and confluent Horn function Γ 1 ( 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.

ABSTRACT 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.

ABSTRACT The (generalized) Mellin transforms of Gegenbauer polynomials have polynomial factors p n λ ( s ) , whose zeros all lie on the ‘critical line’ ℜ s = 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

ABSTRACT We derive a convergent expansion of the generalized hypergeometric function p − 1 F p in terms of the Bessel functions 0 F 1 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 p F p in terms of the confluent hypergeometric functions 1 F 1 that holds uniformly in

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 ρ j ( | a 1 | , …

ABSTRACT Some new relations for the Horn function G 1 a , b , b ′ ; w , z and confluent Horn function Γ 2 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.

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 C μ ( 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 G a ( 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 / G a ( 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

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 deg ψ ≥ 1 , such that ( φ w ) ′ + ψ w = 0 . Recently, all semiclassical monic orthogonal polynomial sequences { W n } n ≥ 0 , of class one obtained from the cubic decompositions (CD) satisfying the relation W 3 n ( x ) = P n