Showing papers in "Integral Transforms and Special Functions in 2018"
TL;DR: The fractional Fourier transform (FrFT) as mentioned in this paper is a generalization of the Fourier Transform (FT) and has become the focus of many research papers in recent years because of its applications in electrical engineering.
Abstract: The fractional Fourier transform (FrFT), which is a generalization of the Fourier transform, has become the focus of many research papers in recent years because of its applications in electrical e...
30 citations
TL;DR: The q-analogues of Ramanujan-type evaluations and related supercongruences were discussed in this paper. But the q-nalogues were not discussed in this paper.
Abstract: The hypergeometric formulae designed by Ramanujan more than a century ago for efficient approximation of π, Archimedes' constant, remain an attractive object of arithmetic study. In this note we discuss some q-analogues of Ramanujan-type evaluations and of related supercongruences.
29 citations
TL;DR: In this paper, sharp lower and upper bounds for the branch W−1 of the Lambert W function are provided by means of functional analysis methods and monotonicity properties, and they are shown to be tight.
Abstract: In this paper, sharp lower and upper bounds for the branch W−1 of Lambert W function are provided by means of functional analysis methods and monotonicity properties.
20 citations
TL;DR: In this paper, a short note on the asymptotic behavior of the two parameter Mittag-Leffler function is given, along with explicit estimation formulas for t.
Abstract: In this paper, we give a short note on the asymptotic behaviour of the two parameter Mittag–Leffler function. Useful results are collected for the reader and also explicit estimation formulas for t...
18 citations
TL;DR: In this article, the positivity of 1F2 generalized hypergeometric functions is investigated and necessary and sufficient conditions in terms of parameters are presented to determine the region of positivity.
Abstract: As for the positivity of 1F2 generalized hypergeometric functions, we present a list of necessary and sufficient conditions in terms of parameters and determine the region of positivity by certain ...
17 citations
TL;DR: In this paper, the Lauricella function FD(N) and a corresponding system of partial differential equations are considered for an arbitrary N, and a generalized hypergeometric function of N variables is considered.
Abstract: The Lauricella function FD(N), which is a generalized hypergeometric function of N variables, and a corresponding system of partial differential equations are considered. For an arbitrary N, we giv...
17 citations
TL;DR: In this article, an integral representation of the solution of the Cauchy problem with discontinuous and continuous initial conditions for linear fractional differential symmetric problems was obtained, where the initial conditions were defined by a linear FDS.
Abstract: The aim of the present paper is to obtain an integral representation of the solution of the Cauchy problem with discontinuous and continuous initial conditions for linear fractional differential sy...
16 citations
TL;DR: In this article, the estimates of convolution for zero-order Mehler-Fock transform with various approaches were found out, including pseudo-differential operator in terms of zero order Mehler Fock transform.
Abstract: The main aim of this paper is to find out the estimates of convolution for zero-order Mehler–Fock transform with various approaches. Pseudo-differential operator in terms of zero-order Mehler–Fock ...
16 citations
TL;DR: In this paper, the leading asymptotic behavior of the Humbert functions of two variables becomes simultaneosly large when the absolute values of the two independent variables become simultaneously large.
Abstract: The leading asymptotic behaviour of the Humbert functions $\Phi_2$, $\Phi_3$, $\Xi_2$ of two variables is found, when the absolute values of the two independent variables become simultaneosly large. New integral representations of these functions are given. These are re-expressed as inverse Laplace transformations and the asymptotics is then found from a Tauberian theorem. Some integrals of the Humbert functions are also analysed.
12 citations
TL;DR: In this article, the authors give several independent extensions of the Karlsson-Minton summation formula for generalized hypergeometric functions with integral parameters differences and explore some alternative forms of the first Miller-Paris transformation, including one expressed in terms of Meijer-Norlund G function.
Abstract: In this paper, we give several independent extensions of the Karlsson–Minton summation formula for the generalized hypergeometric function with integral parameters differences. In particular, we examine the ‘prohibited’ values for the integer top parameter in Minton's formula, extend one unit negative difference in Karlsson's formula to a finite number of integer negative differences and establish known and new summation and transformation formulas when the unit negative difference is allowed to take arbitrary values. We also present a recurrence relation reducing the case of integer negative difference to the Karlsson–Minton case of unit negative difference. Further, we explore some alternative forms of the first Miller–Paris transformation, including one expressed in terms of Meijer–Norlund G function.
10 citations
TL;DR: In this article, a generalized Fefferman-Phong type condition on a pair of weights u and v is shown to be sufficient for the boundedness of the potential type operator from Lvp(⋅) into Luq( ⋅).
Abstract: We prove that a generalized Fefferman–Phong type conditions on a pair of weights u and v is sufficient for the boundedness of the potential type operator from Lvp(⋅) into Luq(⋅) We also obtain an
TL;DR: In this paper, the Mellin transform of shifted Airy functions in terms of the Riesz fractional derivatives of the Airy function has been presented, where the derivative D: = −d2/dx2.
Abstract: In this paper, using the Riesz fractional derivative D:=−d2/dx2, we present the Mellin transforms of shifted Airy functions in terms of the Riesz fractional derivatives of Airy functions. We also s...
TL;DR: In this paper, a new method for deriving indefinite integrals involving quotients of special functions is presented, which combines an integration formula given previously with the recursion relations obeyed by the function.
Abstract: A new method is presented for deriving indefinite integrals involving quotients of special functions. The method combines an integration formula given previously with the recursion relations obeyed by the function. Some additional results are presented using an elementary method, here called reciprocation, which can also be used in combination with the new method to obtain additional quotient integrals. Sample results are given here for Bessel functions, Airy functions, associated Legendre functions and the three complete elliptic integrals. All results given have been numerically checked with Mathematica.
TL;DR: In this paper, the generalized Le Roy-type hypergeometric (α-Mittag-Leffler in other words) function Fp,q(α),α>0 on the real axis was derived.
Abstract: Definite integral expression is derived for the generalized Le Roy-type hypergeometric (α-Mittag–Leffler in other words) function Fp,q(α), α>0 on the real axis. Its important corollaries are the on...
TL;DR: In this paper, sequences of new recurrence relations are presented for Bessel functions, parabolic cylinder functions and associated Legendre functions, which correspond to values of an integer variable r and are generalizations of each conventional recurrence relation.
Abstract: Sequences of new recurrence relations are presented for Bessel functions, parabolic cylinder functions and associated Legendre functions. The sequences correspond to values of an integer variable r and are generalizations of each conventional recurrence relation, which correspond to r=1. The sequences can be extended indefinitely, though the relations become progressively more intricate as r increases. These relations all have the form of a first-order linear inhomogeneous differential equation, which can be solved by an integrating factor. This gives a very general indefinite integral for each recurrence. The method can be applied to other special functions which have conventional recurrence relations. All results have been checked numerically using Mathematica.
TL;DR: Closed expressions for derivatives of symbolic order with respect to parameters for the hypergeometric functions, Laguerre, Gegenbauer, Jacobi and some other polynomial as discussed by the authors.
Abstract: Closed expressions are obtained for derivatives of symbolic order with respect to parameters for the hypergeometric functions, Laguerre, Gegenbauer, Jacobi and some other polynomial
TL;DR: In this article, a method for obtaining indefinite integrals involving quotients of some common special functions is applied to obtain indefinite integral of some quotient of Gauss hypergeometric fun...
Abstract: A method recently applied to obtain indefinite integrals involving quotients of some common special functions is applied to obtain indefinite integrals of some quotients of Gauss hypergeometric fun...
TL;DR: In this paper, the Fock-type space of analytic functions on the plane is studied, invariant with respect to rotations of the plane by e 2iπ/r, with weight generated in a special way from the solutions.
Abstract: In this paper we study the Fock-type space of analytic functions on the plane, invariant with respect to rotations of the plane by e2iπ/r, with weight generated in a special way from the solutions ...
TL;DR: In this paper, the authors define and study the continuous wavelet transform Tψ associated with the Weinstein operator, and analyse the concentration of this transform on sets of finite measure.
Abstract: In this paper, we define and study the continuous wavelet transform Tψ associated with the Weinstein operator. Next, we analyse the concentration of this transform on sets of finite measure. We also establish an analogue of Heisenberg's inequality for wavelet transform. Last, we prove a quantitative version of Shapiro's mean dispersion theorem for the continuous wavelet transform.
TL;DR: In this article, the authors consider settings of the Dunkl Laplacian, the Dunk l harmonic oscillator, and the Ornstein-Uhlenbeck operator with the underlying group of reflections isomorphic to Zd 2.
Abstract: We consider settings of the Dunkl Laplacian, the Dunkl harmonic oscillator, and the Dunkl Ornstein-Uhlenbeck operator with the underlying group of reflections isomorphic to Zd
2. In each of these contexts, we admit all real-valued multiplicity functions, not necessarily bounded from below, and construct the corresponding ‘exotic’ transform or orthogonal system. This leads to new Dunkl operator-based frameworks, which generalize those yet well known, and in which harmonic analysis can reasonably be developed. To support the last claim, in all the cases we study the associated heat semigroup maximal operators and prove that they are bounded on Lp, p>1, and from L1to weak L2.
TL;DR: In this paper, the real Clifford Fourier transform (CFT) is used to deal with real Clifford-Fourier transform introduced by Hitzer [2]...
Abstract: Recently, many surveys are devoted to study the Clifford-Fourier transform (CFT). Dealing with the real Clifford-Fourier transform introduced by Hitzer [The Clifford Fourier transform in real Cliff...
TL;DR: In this article, a family of semiclassical orthogonal polynomial sequences of class two having the cubic decomposition W3n(x)=Pn(n)x3, n ≥ 0.
Abstract: We deal with a family of semiclassical orthogonal polynomial sequences of class two having the cubic decomposition W3n(x)=Pn(x3), n≥0. Only four monic orthogonal polynomial sequences (MOPS) appear in which their recurrence coefficients are explicitly given.
TL;DR: In this paper, the authors prove monotonicity, log-convexity, and logconcavity properties for incomplete Volterra functions, and as consequences of these results, they present some...
Abstract: In this paper we prove some monotonicity, log–convexity and log–concavity properties for the Volterra and incomplete Volterra functions. Moreover, as consequences of these results, we present some ...
TL;DR: In this paper, a class of non-terminating 3F2-series with unit argument is studied and several closed formulae are presented as applications. But they do not consider the case of 3F1-series without unit argument.
Abstract: By means of the linearization method, we establish analytical formulae for a class of non-terminating 3F2-series with unit argument. Several closed formulae are presented as applications.
TL;DR: In this article, the first associated Meixner-Pollaczek polynomials arising from nonlinear coherent states with anti-holomorphic coefficients were identified as orthogonal polynomial arising from coherent states.
Abstract: While considering nonlinear coherent states with anti-holomorphic coefficients z¯n/xn!, we identify as first-associated Meixner–Pollaczek polynomials the orthogonal polynomials arising from...
TL;DR: In this paper, a convergent expansion of 2F1(a,b;c;z) in terms of the function (1−z)−a and of rational functions of z was derived.
Abstract: We consider the hypergeometric function 2F1(a,b;c;z) for z∈C∖[1,∞). For ℜa≥0, we derive a convergent expansion of 2F1(a,b;c;z) in terms of the function (1−z)−a and of rational functions of z that i...
TL;DR: An earlier method for obtaining indefinite integrals of special functions from the second-order linear equations which define them has been reformulated in terms of Riccati equations, which are defined in this article.
Abstract: An earlier method for obtaining indefinite integrals of special function from the second-order linear equations which define them has been reformulated in terms of Riccati equations, which ...
TL;DR: In this paper, a nonterminating 7F6-series formula was conjectured by Gosper and confirmed subsequently by Gasper and Rahman, who evaluated another similar 7f6-sum by means of the modified Abel lemma.
Abstract: A strange nonterminating 7F6-series formula was conjectured by Gosper and was confirmed subsequently by Gasper and Rahman, who evaluated another similar 7F6-sum By means of the modified Abel lemma
TL;DR: In this paper, it was shown that the Weinstein multipliers of Laplace transform type are Calderon-Zygmund operators in the sense of the associated space of homogeneous type, thus their mapping properties follow from the general theory.
Abstract: In this paper, we establish that Weinstein multipliers of Laplace transform type are Calderon–Zygmund operators in the sense of the associated space of homogeneous type, thus their mapping properties follow from the general theory. As an application, we deduce the Lp-boundedness properties for the imaginary powers of the Weinstein operators Δw. Finally we analyse the negative powers of Δw and we give another result of boundedness of type (p,q) for this operator.
TL;DR: In this article, a modified generalized integral transform (MGIT) of functionals on function space is introduced, and some basic formulas with respect to the MGIT and the first variation are established.
Abstract: In this paper, we obtain very natural basic formulas for the modified generalized integral transform (MGIT) on function space. In order to do this, we first introduce an MGIT of functionals on function space. We next establish some basic formulas with respect to the MGIT and the first variation. Finally, we obtain a new version of the Cameron–Storvick theorem via the translation theorem. Some applications are demonstrated as examples which are used in classification of nanoparticles.