In this paper, we consider the normalized Bessel function of index $α> -\frac{1}{2}$, we find an integral representation of the term $x^nj_{α+n}(x)j_α(y)$. This allows us to establish a product formula for the generalized Hankel function $B^{κ,n}_λ$ on $\mathbb{R}$. $B^{κ,n}_λ$ is the kernel of the integral transform $\mathcal{F}_{κ,n}$ arising from the Dunkl theory. Indeed we show that $B^{κ,n}_λ(x)B^{κ,n}_λ(y)$ can be expressed as an integral in terms of $B^{κ,n}_λ(z)$ with explicit kernel invoking Gegenbauer polynomials for all $n\in\mathbb{N}^\ast$. The obtained result generalizes the product formulas proved by M. Rösler for Dunkl kernel when n=1 and by S. Ben Said when $n=2$. \\ As application, we define and study a translation operator and a convolution structure associated to $B^{κ,n}_λ$. They share many important properties with their analogous in the classical Fourier theory.
A number of new definite integrals involving Bessel functions are presented. These have been derived by finding new integral representations for the product of two Bessel functions of different order and argument in terms of the generalized hypergeometric function with subsequent reduction to special cases. Connection is made with Weber's second exponential integral and Laplace transforms of products of three Bessel functions.
The main object of this article is to present an interesting double integral involving generalized Bessel-Maitland function defined by Ghaysuddin et al., which is expressed in terms of generalized (Wright) hypergeometric function. We also considered some special cases as an application of the main result.
Simple inequalities are established for some integrals involving the modified Bessel functions of the first and second kind. In most cases these inequalities are tight in certain limits. As a consequence, we deduce a tight double inequality, involving the modified Bessel function of the first kind, for a generalized hypergeometric function. We also present some open problems that arise from this research.
Generalized integral formulas involving the generalized modified k-Bessel function $J_{k,ν}^{c,γ,λ}\left( z\right) $ of first kind are expressed in terms generalized $k-$Wright functions. Some interesting special cases of the main results are also discussed
We develop a new and further generalized form of the fractional kinetic equation involving generalized k-Bessel function. The manifold generality of the generalized k-Bessel function is discussed in terms of the solution of the fractional kinetic equation in the present paper. The results obtained here are quite general in nature and capable of yielding a very large number of known and (presumably) new results.
In this paper, we obtain uniform bounds for a number of expressions that involve derivatives and integrals of modified Bessel functions. These uniform bounds are motivated by the need to bound such expressions in the study of variance-gamma and product normal approximations via Stein's method.
We investigate the asymptotic expansion of integrals analogous to Ball's integral \[\int_0^\infty \left(\frac{Γ(1+ν)|J_ν(x)|}{(x/2)^ν}\right)^{\!n}dx\] for large $n$ in which the Bessel function $J_ν(x)$ is replaced by the modified Bessel functions $I_ν(x)$ and $K_ν(x)$ together with appropriate exponential factors $e^{\mp x}$, respectively. The above integral with $J_ν(x)$ replaced by a hyper-Bessel function of the type recently discussed in Aktas {\it et al.} [The Ramanujan J., 2019] and taken over a finite interval determined by the first positive zero of the function is also considered for…
We consider a generalisation of a definite integral involving the Bessel function of the first kind. It is shown that this integral can be expressed in terms of the Fox-Wright function ${}_pΨ_q(z)$ of one variable. Some consequences of this representation are explored by suitable choice of parameters. In addition, two closed-form evaluations of infinite series of the Fox-Wright function are deduced.
Simple inequalities are established for some integrals involving the modified Bessel functions of the first and second kind. In most cases, we show that we obtain the best possible constant or that our bounds are tight in certain limits. We apply these inequalities to obtain uniform bounds for several expressions involving integrals of modified Bessel functions. Such expressions occur in Stein's method for variance-gamma approximation, and the results obtained in this paper allow for technical advances in the method. We also present some open problems that arise from this research.
Spectral decomposition of dynamical equations using curl-eigenfunctions has been extensively used in fluid and plasma dynamics problems using their orthogonality and completeness properties for both linear and non-linear cases. Coefficients of such expansions are integrals over products of Bessel functions in problems involving cylindrical geometry. In this paper, certain identities involving products of two and three general solutions of Bessel's equation have been derived. Some of these identities have been useful in the study of Turner relaxation of annular magnetized plasma [S.K.H. Auluck,…
Fractional kinetic equations are investigated in order to describe the various phenomena governed by anomalous reaction in dynamical systems with chaotic motion. Many authors have provided solutions for various families of fractional kinetic equations involving special functions. Here, in this paper, we aim at presenting solutions of certain general families of fractional kinetic equations associated with the generalized modified $k$-Bessel function of the first kind. It is also pointed out that the main results presented here are general enough to be able to be specialized to yield many known…
Discrete analogs of the index transforms, involving Bessel and the modified Bessel functions are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.
Simple upper and lower bounds are obtained for the integral $\int_0^x\mathrm{e}^{-γt}t^νI_ν(t)\,\mathrm{d}t$, $x>0$, $ν>-\frac{1}{2}$, $0<γ<1$. Most of our bounds for this integral are tight as $x\rightarrow\infty$. We apply one of our inequalities to bound some expressions involving this integral. Two of these expressions appear in Stein's method for variance-gamma approximation, and our bounds will allow for a technical advancement to be made to the method.
n this paper, we present $q$-Bernoulli and $q$-Euler polynomials generated by the third Jackson $q$-Bessel function to construct new types of $q$-Lidstone expansion theorem. We prove that the entire function may be expanded in terms of $q$-Lidstone polynomials which are $q$-Bernoulli polynomials and the coefficients are the even powers of the $q$-derivative $\frac{δ_q f(z)}{δ_q z}$ at $0$ and $1$. The other forms expand the function in $q$-Lidstone polynomials based on $q$-Euler polynomials and the coefficients contain the even and odd powers of the $q$-derivative $\frac{δ_q f(z)}{δ_q z}$.
A recent paper by C.G. Kokologiannaki published in J. Math. Anal. Appl. \cite{Kolo:2012:BFI} gives some properties for ratios of modified Bessel functions and, in particular, some bounds. These bounds are said to improve the range of some inequalities in \cite{Segura:2011:BRM} or to be sharper. Unfortunately, Kokologiannaki made some mistakes in the comparison and no improvement or extension is made over the results in \cite{Segura:2011:BRM}. We explain the errors in these comments and show that the bounds given in \cite{Kolo:2012:BFI} are already contained in \cite{Segura:2011:BRM}.
New index transforms, involving the square of Bessel functions of the first kind as the kernel are considered. Mapping properties such as the boundedness and invertibility are investigated for these operators in the Lebesgue spaces. Inversion theorems are proved. As an interesting application, a solution to the initial value problem for the third order partial differential equation, involving the Laplacian, is obtained.
Discrete analogs of the index transforms, involving Bessel and Lommel functions are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.
Generalized integral formulas involving the generalized Bessel-Maitland function are considered and it expressed in terms of generalized Wright hypergeometric functions. By assuming appropriate values of the parameters in the main results, we obtained some interesting results of ordinary Bessel function.
We introduce Bernoulli-Dunkl and Apostol-Euler-Dunkl polynomials as generalizations of Bernoulli and Apostol-Euler polynomials, where the role of the derivative is now played by the Dunkl operator on the real line. We use them to sum a bunch of series involving the zeros of Bessel functions.
In this paper, we discuss the generalized integral formula involving Bessel-Struve kernel function $S_{α}\left( λz\right) $, which expressed in terms of generalized Wright functions. Many interesting special cases also obtained in this study.
We introduce a symbolic method for the evaluation of definite integrals containing combinations of various functions, including exponentials, logarithm and products of Bessel functions of different types. The method we develop is naturally suited for the evaluation of integrals associated with specific Feynman diagrams.
In this paper we consider the evolution equation $\partial_t u=Δ_μu+f$ and the corresponding Cauchy problem, where $Δ_μ$ represents the Bessel operator $\partial_x^2+(\frac{1}{4}-μ^2)x^{-2}$, for every $μ>-1$. We establish weighted and mixed weighted Sobolev type inequalities for solutions of Bessel parabolic equations. We use singular integrals techniques in a parabolic setting.
We establish good numerical estimates for a certain class of integrals involving sixfold products of Bessel functions. We use relatively elementary methods. The estimates will be used in the study of a sharp Fourier restriction inequality on the circle.
New index transforms of the Lebedev type are investigated. It involves the real part of the product of the modified Bessel functions as the kernel. The boundedness and invertibility are examined for these operators in the Lebesgue weighted spaces. Inversion theorems are proved. Important particular cases are exhibited. The results are applied to solve an initial value problem for the fourth order PDE, involving the Laplacian. Finally, it is shown that the same PDE has another fundamental solution, which is associated with the generalized Lebedev index transform, involving the square of the mod…
We consider a new identity involving integrals and sums of Bessel functions. The identity provides new ways to evaluate integrals of products of two Bessel functions. The identity is remarkably simple and powerful since the summand and integrand are of exactly the same form and the sum converges to the integral relatively fast for most cases. A proof and numerical examples of the identity are discussed.
In this paper we give some conditions for a class of functions related to Bessel functions to be positive definite or strictly positive definite . We present some properties and relationships involving logarithmically completely monotonic functions and strictly positive definite functions. In particular, we are interested with the modified Bessel functions.
New index transforms are investigated, which contain as the kernel products of the Bessel and modified Bessel functions. Mapping properties and invertibility in Lebesgue spaces are studied for these operators. Relationships with the Kontorovich-Lebedev and Fourier cosine transforms are established. Inversion theorems are proved. As an application, a solution of the initial value problem for the fourth order partial differential equation, involving the Laplacian is presented.
Two integral operator involving the Appell's functions, or Horn's function in the kernel are considered. Composition of such functions with generalized Bessel functions of the first kind are expressed in term of generalized Wright function and generalized hypergeometric series. Many special cases, including cosine and sine function are also discussed.
The generalized operators of fractional integration involving Appell's function $F_{3}(.) $ due to Marichev-Saigo-Maeda, is applied to the Bessel Struve kernel function $S_{α}\left( λz\right),λ,z\in \mathbb{C}$ to obtain the results in terms of generalized Wright functions.The pathway integral representations Bessel Struve kernel function and its relation between many other functions also derived in this study.
In this article we prove dimension free $L^p$-boundedness of Riesz transforms associated with a Bessel diferential operator. We obtain explicit estimates of the $L^p$-norms for the Bessel-Riesz transforms in terms of p, establishing a linear behaviour with respect to p. We use the Bellman function technique to prove a bilinear dimension free inequality involving Poisson semigroups defined through this Bessel operator.
We show a connection formula of the Hahn-Exton $q$-Bessel function around the origin and the infinity. We introduce the $q$-Borel transformation and the $q$-Laplace transformation following C. Zhang to obtain the connection formula. We consider the limit $p\to 1^-$ of the connection formula.
We derive formulas for the derivatives of general order for the functions $z^{-ν}h_ν(z)$ and $z^νh_ν(z)$, where $h_ν(z)$ is a Bessel, Struve or Anger--Weber function.
The derivatives with respect to order ν for the Bessel functions of argument x (real or complex) are studied. Representations are derived in terms of integrals that involve the products pairs of Bessel functions, and in turn series expansions are obtained for these integrals. From the new integral representations, asymptotic approximations involving Airy functions are constructed for the order derivatives, for ν large and uniformly valid for unbounded positive real x.
Our purpose in this present paper is to investigate generalized integration formulas containing the generalized $k$-Bessel function $W_{v,c}^{k}(z)$ to obtain the results in representation of Wright-type function. Also, we establish certain special cases of our main result.
We determine the asymptotic behaviour of the $n$th derivatives of the Bessel functions $J_ν(a)$ and $K_ν(a)$, where $a$ is a fixed positive quantity, as $n\to\infty$. These results are applied to the asymptotic evaluation of two incomplete Laplace transforms of these Bessel functions on the interval $[0,a]$ as the transform variable $x\to+\infty$. Similar evaluation of the integrals involving the Bessel functions $Y_ν(t)$ and $I_ν(t)$ is briefly mentioned.
Simple inequalities for some integrals involving the modified Bessel functions $I_ν(x)$ and $K_ν(x)$ are established. We also obtain a monotonicity result for $K_ν(x)$ and a new lower bound, that involves gamma functions, for $K_0(x)$.
We prove a single sum formula for the linearization coefficients of the Bessel polynomials. In two special cases we show that our formula reduces indeed to Berg and Vignat's formulas in their proof of the positivity results about these coefficients (Constructive Approximation, 27(2008), 15-32). As a bonus we also obtain a generalization of an integral formula of Boros and Moll (J. Comput. Appl. Math. 106 (1999), 361-368).
In this paper, a single sum formula for the linearization coefficients of the Bessel polynomials is given. In three special cases this formula reduces indeed to either Atia and Zeng's formula (Ramanujan Journal, Doi 10.1007/s11139-011-9348-4) or Berg and Vignat's formulas in their proof of the positivity results about these coefficients (Constructive Approximation, {\bf 27} (2008), 15-32). As a bonus, a formula reducing a sum of hypergeometric functions $_3F_2$ to $_2F_1$ is obtained.
In this present paper our aim is to deal with two integral transforms which involving the Gauss hypergeometric function as its kernels. We prove some compositions formulas for such a generalized fractional integrals with k Bessel function. The results are established in terms of generalized Wright type hypergeometric function and generalized hypergeometric series. Also, the authors presented some corresponding assertions for Riemann Liouville and Erdelyi Kober fractional integral transforms.
We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by Dirichlet series $$\sum_{n=1}^{\infty}\frac{a(n)}{λ_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{μ_n^{s}},$$ satisfying a familiar functional equation involving the gamma function $Γ(s)$. Two general identities are established. The first involves the modified Bessel function $K_μ(z)$, and can be thought of as a 'modular' or 'theta' relation wherein modified Bessel functions, instead of exponential functions, appear. Appearing in the second identity are $K_μ(z)$, the Bessel functions of imaginary argument…
Through the theory of Jack polynomials we give an iterative method for integral formula of Dunkl-Bessel functions of type $A_{N-1}$ and a partial product formula for it.
In this note, we shall prove a formula for the Fourier transform of spherical Bessel functions over complex numbers, viewed as the complex analogue of the classical formulae of Hardy and Weber. The formula has strong representation theoretic motivations in the Waldspurger correspondence over the complex field.
In this paper, we prove an exponential integral formula for the Fourier transform of Bessel functions over complex numbers, along with a radial exponential integral formula. The former will enable us to develop the complex spectral theory of the relative trace formula for the Shimura-Waldspurger correspondence and extend the Waldspurger formula from totally real fields to arbitrary number fields.
In a rectangular domain, a boundary-value problem is considered for a mixed-type equation with a regularized Caputo-like counterpart of hyper-Bessel differential operator and the bi-ordinal Hilfer's fractional derivative. Using the method of separation of variables, Laplace transform, a unique solvability of the considered problem has been established. Moreover, we have found the explicit solution of initial problems for a differential equation with the bi-ordinal Hilfer's derivative and regularized Caputo-like counterpart of the hyper-Bessel differential operator with the non-zero starting po…
In this paper we study the positivity of the generalized $q$-translation associated with the $q$-Bessel Hahn Exton function which is deduced by a new formulation of the Graf's addition formula related to this function.
In this paper we obtain quantitative weighted $L^p$-inequalities for some operators involving Bessel convolutions. We consider maximal operators, Littlewood-Paley functions and variational operators. We obtain $L^p(w)$-operator norms in terms of the $A_p$-characteristic of the weight $w$. In order to do this we show that the operators under consideration are dominated by a suitable family of sparse operators in the space of homogeneous type $((0,\infty),|\cdot |,x^{2λ}dx)$.
In this paper we obtain a set of five new transmutations of the mother formula. Further, we obtain the second set of ten exact metafunctional equations by crossbreeding on every two elements of the previous set. Elements of the last set represent a kind of synergetic formulae describing cooperative interactions between corresponding sets of values of basic classical functions.
This paper introduces and studies a generalization of the $\mathtt{k}$-Bessel function of order $ν$ given by \[\mathtt{W}^{\mathtt{k}}_{ν, c}(x):= \sum_{r=0}^\infty \frac{(-c)^r}{Γ_{\mathtt{k}}\left( r \mathtt{k} +ν+\mathtt{k}\right) r!} \left(\frac{x}{2}\right)^{2r+\fracν{\mathtt{k}}}. \] Representation formulae are derived for $\mathtt{W}^{\mathtt{k}}_{ν, c}.$ Further the function $\mathtt{W}^{\mathtt{k}}_{ν, c}$ is shown to be a solution of a second order differential equation. Monotonicity and log-convexity properties for the generalized $\mathtt{k}$-Bessel function $\mathtt{W}^{\mathtt{k}…
In this article, we shall study fundamental Bessel functions for $\mathrm{GL}_n(\mathbb{F})$ arising from the Voronoï summation formula for any rank $n$ and field $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$, with focus on developing their analytic and asymptotic theory. The main implements and subjects of our study of fundamental Bessel functions are their formal integral representations and Bessel differential equations. We shall prove the asymptotic formulae for fundamental Bessel functions and explicit connection formulae for the Bessel differential equations.
