Index transforms with the product of the associated Legendre functions

Index transforms with the product of the associated Legendre functions are introduced. Mapping properties are investigated in the Lebesgue spaces. Inversion formulas are proved. The results are applied to solve a boundary value problem in a wedge for a third order partial differential equation.     

On Legendre functions of imaginary degree and associated integral transforms

New integral representations, asymptotic formulas, and series expansions in powers of tanh(t/2) are obtained for the imaginary and real parts of the Legendre function P_iξ(cosht). Coefficients of these series expansions are orthogonal polynomials in the real variable ξ. A number of relations for these orthogonal polynomials are obtained on the basis of the generating function. Several inversion theorems are proven for the integral transforms involving the Legendre function of imaginary degree. In many cases it is preferable to employ these transforms, than Mehler-Fok transforms, since conditions placed on functions are less restrictive.     

Asymptotic expansions of integrals of two Bessel functions via the generalized hypergeometric and Meijer functions

Asymptotic expansions of certain finite and infinite integrals involving products of two Bessel functions of the first kind are obtained by using the generalized hypergeometric and Meijer functions. The Bessel functions involved are of arbitrary (generally different) orders, but of the same argument containing a parameter which tends to infinity. These types of integrals arise in various contexts, including wave scattering and crystallography, and are of general mathematical interest being related to the Riemann—Liouville and Hankel integrals. The results complete the asymptotic expansions derived previously by two different methods — a straightforward approach and the Mellin-transform technique. These asymptotic expansions supply practical algorithms for computing the integrals. The leading terms explicitly provide valuable analytical insight into the high-frequency behavior of the solutions to the wave-scattering problems.     

Linearization formula for the product of associated q-ultraspherical polynomials

We obtain an explicit formula for the linearization coefficient of the product of two associated q-ultraspherical polynomials in terms of a multiple of a balanced, terminating very-well-poised ₁₀φ₉ series. We also discuss the nonnegativity properties of the coefficients as well as some special cases. 2000 Mathematics Subject Classification Primary—33D45; Secondary—33D8 This work was supported in part by an NSERC grant A6197.     

Solution of nonlinear weakly singular Volterra integral equations using the fractional‐order Legendre functions and pseudospectral method

In this article, our main goal is to render an idea to convert a nonlinear weakly singular Volterra integral equation to a non‐singular one by new fractional‐order Legendre functions. The fractional‐order Legendre functions are generated by change of variable on well‐known shifted Legendre polynomials. We consider a general form of singular Volterra integral equation of the second kind. Then the fractional Legendre–Gauss–Lobatto quadratures formula eliminates the singularity of the kernel of the integral equation. Finally, the Legendre pseudospectral method reduces the solution of this problem to the solution of a system of algebraic equations. This method also can be utilized on fractional differential equations as well. The comparison of results of the presented method and other numerical solutions shows the efficiency and accuracy of this method. Also, the obtained maximum error between the results and exact solutions shows that using the present method leads to accurate results and fast convergence for solving nonlinear weakly singular Volterra integral equations. Copyright © 2015 John Wiley & Sons, Ltd.     

A simple formula for an analogue of conditional Wiener integrals and its applications

Let denote the space of real-valued continuous functions on the interval and for a partition of , let be given by .

In this paper, with the conditioning function , we derive a simple formula for conditional expectations of functions defined on which is a probability space and a generalization of Wiener space. As applications of the formula, we evaluate the conditional expectation of functions of the form

for and derive a translation theorem for the conditional expectation of integrable functions defined on the space .

     

Boundedness in Lebesgue spaces for commutators of multilinear singular integrals and RBMO functions with non-doubling measures

The boundedness in Lebesgue spaces for commutators generated by multilinear singular integrals and RBMO(μ) functions of Tolsa with non-doubling measures is obtained, provided that‖μ‖=∞and multilinear singular integrals are bounded from L1(μ)×L1(μ)to L1/2,∞(μ).     

Unified treatment for the evaluation of generalized complete and incomplete gamma functions

The present work applies the binomial expansion theorems to evaluate the generalized complete and incomplete gamma functions arising in the wave scattering and diffraction theory. A simple and efficient algorithm for the calculation of these functions is developed. Some numerical results are presented for significant mapping examples and they are briefly discussed. The formulas obtained are numerically stable for all values of parameters occurring in generalized complete and incomplete gamma functions.     

Resolution of singularities in two dimensions and the stability of integrals

For a small disk D centered at the origin in R², a smooth real-valued function S(x,y) on D, and a positive epsilon, we consider the measure of the points in D where |S(x,y)|<?, as well as oscillatory integral analogues. Specifically, we consider the effect of perturbing S(x,y) on these quantities. Besides being of intrinsic interest, these questions are important in the analysis of Fourier transforms of surface-supported measures. Complex and higher-dimensional analogues of these questions are also connected to various issues in algebraic and complex geometry. For real-analytic S(x,y), this question has been investigated for example by Karpushkin, using versal deformation theory, and by Phong–Stein–Sturm, who developed a method often referred to as the method of algebraic estimates.In this paper, we show how the use of resolution of singularities algorithms in two dimensions, along with some one-dimensional Van der Corput-type lemmas, provides another method for dealing with such questions. As a result, we prove new estimates and theorems for these and related quantities. Furthermore, since these algorithms apply to all smooth functions, the theorems will hold for all smooth functions as opposed to the earlier real-analytic results.     

The equiconvergence of expansions in eigenfunctions and associated functions of an integral operator with involution

In this paper we study an integral operator with involution. We solve the problem on the exact inversion of this operator, we obtain and study the integro-differential system for the Fredholm resolvent and, finally, we prove the theorem on the equiconvergence of expansions in eigenfunctions and associated functions, in the usual trigonometric system.     

A simple formula for an analogue of conditional wiener integrals and its applications II

Let C[0, T] denote the space of real-valued continuous functions on the interval [0, T] with an analogue w _ϕ of Wiener measure and for a partition 0 = t ₀ < t ₁ < ... < t _n < t _n+1 = T of [0, T], let X _n : C[0, T] → ℝ ⁿ⁺¹ and X ⁿ⁺¹: C[0, T] → ℝ ⁿ⁺² be given by X _n (x) = (x(t ₀), x(t ₁), ..., x(t _n )) and X _n+1(x) = (x(t ₀), x(t ₁), ..., x(t _n+1)), respectively. In this paper, using a simple formula for the conditional w _ϕ-integral of functions on C[0, T] with the conditioning function X _n+1, we derive a simple formula for the conditional w _ϕ-integral of the functions with the conditioning function X _n . As applications of the formula with the function X _n , we evaluate the conditional w _ϕ-integral of the functions of the form F _m (x) = ∫₀ ^T (x(t)) ^m for x ∈ C[0, T] and for any positive integer m. Moreover, with the conditioning X _n , we evaluate the conditional w _ϕ-integral of the functions in a Banach algebra which is an analogue of the Cameron and Storvick’s Banach algebra . Finally, we derive the conditional analytic Feynman w _ϕ-integrals of the functions in .      

Some applications of the Gamma and polygamma functions involving convolutions of the Rayleigh functions,multiple Euler sums and log‐sine integrals

The Gamma function and its n th logarithmic derivatives (that is, the polygamma or the psi‐functions) have found many interesting and useful applications in a variety of subjects in pure and applied mathematics. Here we mainly apply these functions to treat convolutions of the Rayleigh functions by recalling a general identity expressing a certain class of series as psi‐functions and to evaluate a class of log‐sine integrals in an algorithmic way. We also evaluate some Euler sums and give much simpler psi‐function expressions for some known parameterized multiple sums (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Upper bounds for the associated number of zeros of Abelian integrals for two classes of quadratic reversible centers of genus one

In this paper, by using the method of Picard-Fuchs equation and Riccati equation, we study the upper bounds for the associated number of zeros of Abelian integrals for two classes of quadratic reversible centers of genus one under any polynomial perturbations of degree $n$, and obtain that their upper bounds are $3n-3$ ($n\geq 2$) and $18\left[\frac{n}{2}\right]+3\left[\frac{n-1}{2}\right]$ ($n\geq 4$) respectively, both of the two upper bounds linearly depend on $n$.     

Reduction and transformation formulas for the Appell and related functions in two variables

In many seemingly diverse areas of applications, reduction, summation, and transformation formulas for various families of hypergeometric functions in one, two, and more variables are potentially useful, especially in situations when these hypergeometric functions are involved in solutions of mathematical, physical, and engineering problems that can be modeled by (for example) ordinary and partial differential equations. The main object of this article is to investigate a number of reductions and transformations for the Appell functions F₁,F₂,F₃, and F₄ in two variables and the corresponding (substantially more general) double‐series identities. In particular, we observe that a certain reduction formula for the Appell function F₃ derived recently by Prajapati et al., together with other related results, were obtained more than four decades earlier by Srivastava. We give a new simple derivation of the previously mentioned Srivastava's formula 12 . We also present a brief account of several other related results that are closely associated with the Appell and other higher‐order hypergeometric functions in two variables. Copyright © 2017 John Wiley & Sons, Ltd.     

On a product formula for the Conley–Zehnder index of symplectic paths and its applications

Using invariance by fixed-endpoints homotopies and a generalized notion of symplectic Cayley transform, we prove a product formula for the Conley–Zehnder index of continuous paths with arbitrary endpoints in the symplectic group. We discuss two applications of the formula, to the metaplectic group and to periodic solutions of Hamiltonian systems.