Bernard J. Laurenzi
Special values of the Lommel functions allow the calculation of Fresnel like integrals. These closed form expressions along with their asymptotic values are reported.
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In this paper, we obtain inequalities for some integrals involving the modified Lommel function of the first kind $t_{μ,ν}(x)$. In most cases, these inequalities are tight in certain limits. We also deduce a tight double inequality, involving the modified Lommel function $t_{μ,ν}(x)$, for a generalized hypergeometric function. The inequalities obtained in this paper generalise recent bounds for integrals involving the modified Struve function of the first kind.
Simple upper and lower bounds are established for the integral $\int_0^x\mathrm{e}^{-βu}u^νt_{μ,ν}(u)\,\mathrm{d}u$, where $x>0$, $0<β<1$, $μ+ν>-2$, $μ-ν\geq-3$, and $t_{μ,ν}(x)$ is the modified Lommel function of the first kind. Our bounds complement and improve on existing bounds for this integral, by either being sharper or increasing the range of validity. Our bounds also generalise recent bounds for an integral involving the modified Struve function of the first kind, and in some cases more a direct approach lead to sharper bounds when our general bounds are specialised to the modified St…
Integral representations and zeros of the Lommel function and the hypergeometric $_1F_2$ function
本文通过三角函数和超几何 $_2F_1$ 函数给出Lommel函数的积分表示，并利用Polya经典结果研究其零点分布，进而刻画超几何 $_1F_2$ 函数在特定参数下的零点分布。
Lommel functions, Padé approximants and hypergeometric functions
研究半整数参数下Lommel函数的多项式与三角函数显式组合，揭示多项式构成三角函数的Padé逼近，并给出整数参数下的积分表示与超几何函数闭式公式。
We obtain a new proof of Hurwitz's formula for the Hurwitz zeta function $ζ(s, a)$ beginning with Hermite's formula. The aim is to reveal a nice connection between $ζ(s, a)$ and a special case of the Lommel function $S_{μ, ν}(z)$. This connection is used to rephrase a modular-type transformation involving infinite series of Hurwitz zeta function in terms of those involving Lommel functions.
This paper offers a new and complete description of subnormal solutions of certain non-homogeneous second order periodic linear differential equations first studied by Gundersen and Steinbart in 1994. We have established a previously unknown relation that the general solutions (\textit{i.e.}, whether subnormal or not) of the DEs can be solved explicitly in terms of classical special functions, namely the Bessel, Lommel and Struve functions, which are important because of their numerous physical applications. In particular, we show that the subnormal solutions are written explicitly in terms of…
For the Bessel function \begin{equation} \label{bessel} J_ν(z) = \sum\limits_{k=0}^{\infty} \frac{(-1)^k \left( \frac{z}{2} \right)^{ν+2k}}{k! Γ(ν+1+k)} \end{equation} there exist several $q$-analogues. The oldest $q$-analogues of the Bessel function were introduced by F. H. Jackson at the beginning of this century, see M. E. H. Ismail \cite{Is1} for the appropriate references. Another $q$-analogue of the Bessel function has been introduced by W. Hahn in a special case and by H. Exton in full generality, see R. F. Swarttouw \cite{Sw1} for a historic overview. Here we concentrate on properties …
In this paper we consider some normalized Bessel, Struve and Lommel functions of the first kind, and by using the Euler-Rayleigh inequalities for the first positive zeros of combination of special functions we obtain tight lower and upper bounds for the radii of starlikeness of these functions. By considering two different normalization of Bessel and Struve functions we give some inequalities for the radii of convexity of the same functions. On the other hand, we show that the radii of univalence of some normalized Struve and Lommel functions are exactly the radii of starlikeness of the same f…
The modified Lommel function $t_{μ,ν}(x)$ is an important special function, but to date there has been little progress on the problem of obtaining functional inequalities for $t_{μ,ν}(x)$. In this paper, we advance the literature substantially by obtaining a simple two-sided inequality for the ratio $t_{μ,ν}(x)/t_{μ-1,ν-1}(x)$ in terms of the ratio $I_ν(x)/I_{ν-1}(x)$ of modified Bessel functions of the first kind, thereby allowing one to exploit the extensive literature on bounds for this ratio. We apply this result to obtain two-sided inequalities for the condition numbers $xt_{μ,ν}'(x)/t_{μ…
Tight lower and upper bounds for the radius of univalence of some normalized Bessel, Struve and Lommel functions of the first kind are obtained via Euler-Rayleigh inequalities. It is shown also that the radius of univalence of the Struve functions is greater than the corresponding radius of univalence of Bessel functions. Moreover, by using the idea of Kreyszig and Todd, and Wilf it is proved that the radii of univalence of some normalized Struve and Lommel functions are exactly the radii of starlikeness of the same functions. The Laguerre-Pólya class of entire functions plays an important rol…
In this paper our aim is to deduce some sufficient (and necessary) conditions for the close-to-convexity of some special functions and their derivatives, like Bessel functions, Struve functions, and a particular case of Lommel functions of the first kind, which can be expressed in terms of the hypergeometric function ${}_1F_2$. The key tool in our proofs is a result of Shah and Trimble about transcendental entire functions with univalent derivatives. Moreover, a known result of Pólya on entire functions, the infinite product representations and some results on zeros of Bessel, Struve and Lomme…
Geometric properties of the classical Lommel and Struve functions, both of the first kind, are studied. For each of them, there different normalizations are applied in such a way that the resulting functions are analytic in the unit disc of the complex plane. For each of the six functions we determine the radius of starlikeness precisely.
New necessary and sufficient conditions are given for the quantization of a class of periodic second order non-homogeneous ordinary differential equations in the complex plane in this paper. The problem is studied from the viewpoint of complex oscillation theory first developed by Bank and Laine (1982, 1983) and Gundersen and Steinbart (1994). We show that when a solution is complex non-oscillatory (finite exponent of convergence of zeros) then the solution, which can be written as special functions, must degenerate. This gives a necessary and sufficient condition when the Lommel function has …
We study generalized log-sine integrals at special values. At $π$ and multiples thereof explicit evaluations are obtained in terms of Nielsen polylogarithms at $\pm1$. For general arguments we present algorithmic evaluations involving Nielsen polylogarithms at related arguments. In particular, we consider log-sine integrals at $π/3$ which evaluate in terms of polylogarithms at the sixth root of unity. An implementation of our results for the computer algebra systems Mathematica and SAGE is provided.
The aim of this note is to provide a full space quadratic external field extension of a classical result of Marcel Riesz for the equilibrium measure on a ball with respect to Riesz s-kernels. We address the case s=d-3 for arbitrary dimension d, in particular the logarithmic kernel in dimension 3. The equilibrium measure for this full space external field problem turns out to be a radial arcsine distribution supported on a ball with a special radius. As a corollary, we obtain new integral identities involving special functions such as elliptic integrals and more generally hypergeometric functio…
The Hurwitz-type Euler zeta function is defined as a deformation of the Hurwitz zeta function: \begin{equation*} ζ_E(s,x)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+x)^s}. \end{equation*} In this paper, by using the method of Fourier expansions, we shall evaluate several integrals with integrands involving Hurwitz-type Euler zeta functions $ζ_E(s,x)$. Furthermore, the relations between the values of a class of the Hurwitz-type (or Lerch-type) Euler zeta functions at rational arguments have also been given.
Extension of Krein's special method for solving integral equations
该研究将求解积分方程的Krein特殊方法推广至积分方程组求解场景，推导了对应的解的推广公式，并通过实例验证了结果的有效性。
We show how the integral formula of Poisson for holomorphic functions on the right half plane can be used to quickly evaluate certain integrals from the Table of Gradshteyn and Ryzhik. In addition, we prove a version of this formula for principal value integrals and use it in several interesting cases.
We use tools from the multilinear oscillatory integral program developed by Christ, Li, Thiele, and Tao to treat special cases which are not covered by existing theory. Consideration of special cases leads to an extended class of examples for which $λ$-power decay holds.
A new method is presented for obtaining indefinite integrals of common special functions. The approach is based on a Lagrangian formulation of the general homogeneous linear ordinary differential equation of second order. A general integral is derived which involves an arbitrary function, and therefore yields an infinite number of indefinite integrals for any special function which obeys such a differential equation. Techniques are presented to obtain the more interesting integrals generated by such an approach, and many integrals, both previously known and completely new are derived using the…
In this paper, the authors establish a new type integral inequalities for differentiable s-convex functions in the second sense. By the well-known Hölder inequality and power mean inequality, they obtain some integral inequalities related to the s-convex functions and apply these inequalities to special means for positive real numbers.
By using Fubini theorem or Tonelli theorem, we find that the zeta function value at 2 is equal to a special integral. Furthermore, We find that this special integral is two times of another special integral. By using this fact we obtain the relationship between Genocchi numbers and Bernoulli numbers. And get some results about Bernoulli polynomials.
In this paper we generalize and specialize generating functions for classical orthogonal polynomials, namely Jacobi, Gegenbauer, Chebyshev and Legendre polynomials. We derive a generalization of the generating function for Gegenbauer polynomials through extension a two element sequence of generating functions for Jacobi polynomials. Specializations of generating functions are accomplished through the re-expression of Gauss hypergeometric functions in terms of less general functions. Definite integrals which correspond to the presented orthogonal polynomial series expansions are also given.
Special Solutions of $q$-Heun Equation by $q$-Integral Transformations
通过考虑多项式型解的 $q$-积分变换，得到了 $q$-Heun 方程表示为 $q$-超几何函数有限求和的特殊解。
Basics of Multiple Polyexponential Integrals
本文引入了定义为指数积分迭代积分的多重多指数积分，用于描述二阶常微分方程在不规则奇点附近局部解的微扰展开与渐近行为，并将其与未修饰和已修饰多重多指数函数相联系。
In this paper, we establish (presumably new type) integral inequalities for convex functions via the Hermite--Hadamard's inequalities. As applications, we apply these new inequalities to construct inequalities involving special means of real numbers, some error estimates for the formula midpoint are given. Finally, new inequalities for some special and $q-$special functions are also pointed out.
The well known table of Gradshteyn and Ryzhik contains indefinite and definite integrals of both elementary and special functions. We give proofs of several entries containing integrands with some combination of hyperbolic and trigonometric functions. In fact, we occasionally present an extension of such entries or else give alternative evaluations. We develop connections with special cases of special functions including the Hurwitz zeta function. Before concluding we mention new integrals coming from the investigation of certain elliptic functions.
We provide a general theorem for evaluating trigonometric Dirichlet series of the form $\sum_{n \geq 1} \frac{f (πn τ)}{n^s}$, where $f$ is an arbitrary product of the elementary trigonometric functions, $τ$ a real quadratic irrationality and $s$ an integer of the appropriate parity. This unifies a number of evaluations considered by many authors, including Lerch, Ramanujan and Berndt. Our approach is based on relating the series to combinations of derivatives of Eichler integrals and polylogarithms.
We consider some properties of integrals considered by Hardy and Koshliakov, and which have also been further extended recently by Dixit. We establish a new general integral formula from some observations about the digamma function. We also obtain lower and upper bounds for Hardy's integral through properties of the digamma function.
We show that the method of separation of variables gives a natural generalisation of integral relations for classical special functions of one variable. The approach is illustrated by giving a new proof of the ``quadratic'' integral relations for the continuous q-ultraspherical polynomials. The separating integral operator M expressed in terms of the Askey-Wilson operator is studied in detail: apart from writing down the characteristic (``separation'') equations it satisfies, we find its spectrum, eigenfunctions, inversion, invariants (invariant q-difference operators), and give its interpreta…
In this paper, we derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are h-convex and we point out the results for some special classes of functions. Some applications to special means of real numbers are also given.
Integrals of products of four modified Bessel functions
本文推导了四类修正贝塞尔函数乘积的积分公式，并推导出Airy函数乘积的新的定积分公式。
Analogous to the use of sums of squares of certain real-valued special functions to prove the reality of the zeros of the Bessel functions J_α(z) when α\ge -1, confluent hypergeometric functions {}_0F_1(c; z) when c > 0 or 0 > c > -1, Laguerre polynomials L_n^α(z) when α\ge -2, Jacobi polynomials P_n^{(α,β)}(z) when α\ge -1 and β\ge -1, and some other entire special functions considered in G. Gasper [Using sums of squares to prove that certain entire functions have only real zeros, in Fourier Analysis: Analytic and Geometric Aspects, W. O. Bray, P. S. Milojević and C. V. Stanojević, eds., Marc…
In this paper, we discuss a system of differential equations for the Fisher integral on the special orthogonal group. Especially, we explicitly give a set of linear differential operators which generates the annihilating ideal of the Fisher integral, and we prove that the annihilating ideal is a maximal left ideal of the ring of differential operators with polynomial coefficients. Our proof is given by a discussion concerned with an annihilating ideal of a Schwartz distribution associated with the Haar measure on the special orthogonal group. We also give differential operators annihilating th…
Solvability conditions for a class of Wiener-Hopf integral equations of 1-st kind
本文针对经典方法不适用的非正规第一类Wiener-Hopf积分方程，采用特殊分解方法，推导出其可解性条件。
A Lagrangian method is introduced recently for deriving indefinite integrals of special functions that satisfy homogeneous (nonhomogeneous) second-order linear differential equations. This paper extends this method to include indefinite Jackson $q$-integrals of special functions satisfying homogeneous (nonhomogeneous) second-order linear $q$-difference equations. Many $q$-integrals, both previously known and completely new, are derived using the method. We introduce samples of indefinite and definite $q$-integrals for Jackson's $q$-Bessel functions, $q$-hypergeometric functions, and some ortho…
This is a brief overview of the index hypergeometric transform (other terms for this integral operator are: Olevskii transform, Jacobi transform, generalized Mehler--Fock transform). We discuss applications of this transform to special functions and harmonic analysis. The text is an addendum to the Russian edition of the book by G.E.Andrews, R.Askey, and R.Roy, Special Functions, Encycl. of Math. Appl. 71, Cambridge Univ. Press, 1999.
In recent work (math.QA/0309252) on multivariate hypergeometric integrals, the author generalized a conjectural integral formula of van Diejen and Spiridonov to a ten parameter integral provably invariant under an action of the Weyl group E_7. In the present note, we consider the action of the affine Weyl group, or more precisely, the recurrences satisfied by special cases of the integral. These are of two flavors: linear recurrences that hold only up to dimension 6, and three families of bilinear recurrences that hold in arbitrary dimension, subject to a condition on the parameters. As a coro…
In this paper the authors investigate a power mean inequality for a special function which is defined by the complete elliptic integrals.
In the present paper we derive a unified new integral whose integrand contains products of Fox $H$-function and a general class of polynomials having general arguments. A large number of integrals involving various simpler functions follow as special cases of this integral.
In this paper, the author obtains new estimates on generalization of Hadamard, Ostrowski and Simpson type inequalities for Lipschitzian functions via Hadamard fractional integrals. Some applications to special means of positive reals numbers are also given.
In this paper we present a special formula for transforming integrals to series. The resulting series involves binomial transforms with the Taylor coefficients of the integrand. Five applications are provided for evaluating challenging integrals.
This paper deals with a Euler type integral operator involving k-Mittag-Leffler function defined by Gupta and Parihar [8]. Furthermore, some special cases are also taken into consideration.
Table in Gradshteyn and Ryzhik: Derivation of definite integrals of a Hyperbolic Function
本文利用围道积分和特殊函数的解析延拓，推导 Gradshteyn 与 Ryzhik 表中的双曲函数定积分及相关无穷和，并以对数函数乘三角函数的积分为例，得到 Catalan 常数和 π 等结果。
In the article it was shown the convergence of special integral of two dimensional Terry's problem. Main tools of the article are an investigation of real algebraic varieties and estimations of areas of algebraic surfaces.
We have recently established some integral inequalities for convex functions via the Hermite-Hadamard's inequalities. In continuation here, we also establish some interesting new integral inequalities for convex functions via the Hermite--Hadamard's inequalities and Jensen's integral inequality. Useful applications involving special means are also included.
The close form of some integrals involving recently developed generalized k-Struve functions is obtained. The outcome of these integrations is expressed in terms of generalized Wright functions. Several special cases are deduced which lead to some known results.
In this paper, we discuss the generalized integral formula involving Bessel-Struve kernel function $S_{α}\left( λz\right) $, which expressed in terms of generalized Wright functions. Many interesting special cases also obtained in this study.
In this paper, we establish new general inequality for convex functions. Then we apply this inequality to obtain the midpoint, trapezoid and averaged midpoint-trapezoid integral inequality. Also, some applications for special means of real numbers are provided.
We show that the evaluation of an integral considered by Boros and Moll is a special case of a convolution result about Student t-densities obtained by the authors in 2008.
In the paper it is considered the question on Convergence Exponent of the special integral of multidimensional Terry's problem. The question considered in the article consisted in investigation of singularities influence to the value of convergence exponent of the special integral
While investigating the properties of a galaxy model used in Stellar Dynamics, a curious integral identity was discovered. For a special value of a parameter, the identity reduces to a definite integral with a very simple symbolic value; but, quite surprisingly, all the consulted tables of integrals, and computer algebra systems, do not seem aware of this result. Here I show that this result is a special case ($n=0$ and $z=1$) of the following identity (established by elementary methods): $$ I_n(z)\equiv\int_0^1{{\rm K}(k) k\over (z+k^2)^{n+3/2}}dk = {(-2)^n\over (2n+1)!!} {d^n\over dz^n} {{\r…
A new generalization called $\mathtt{k}$-Struve function and its properties given by Nisar and saiful very recently. In this paper, we establish the pathway fractional integral representation of $\mathtt{k}$-Struve function. Many special cases also established to obtain the pathway integral representation of classical Struve function.
In this paper, we consider a new class of convex functions which is called $λ$-preinvex functions. We prove several Hermite-Hadamard type inequalities for differentiable $λ$-preinvex functions via Fractional Integrals. Some special cases are also discussed.
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.
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.
Persson and Strang (2003) evaluated the integral over [-1,1] of a squared odd degree Legendre polynomial divided by x^2 as being equal to 2. We consider a similar integral for orthogonal polynomials with respect to a general even orthogonality measure, with Gegenbauer and Hermite polynomials as explicit special cases. Next, after a quadratic transformation, we are led to the general nonsymmetric case, with Jacobi and Laguerre polynomials as explicit special cases. Examples of indefinite summation also occur in this context. The paper concludes with a generalization of the earlier results for H…
We review a special technique for evaluating challenging integrals by providing a number of examples. Many of our examples prove integrals from the popular table of Gradshteyn and Ryzhik.
Berndt-Type Integrals of Order Three and Series Associated with Jacobi Elliptic Functions
本文利用Jacobi椭圆函数的Fourier级数和Maclaurin级数展开及围道积分法，通过Gamma函数特殊值建立了六类双曲和与两族三阶Berndt型积分的显式求值，并给出了相关推论与示例。
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