相关论文: More integrals of products of Airy functions
The problem of evaluation of higher derivatives of Airy functions in a closed form is investigated. General expressions for the polynomials which have arisen in explicit formulae for these derivatives are given in terms of particular values…
We evaluate definite integrals involving the product of four modified Bessel functions of the first and second kind and a power function. We provide general formulas expressed in terms of the Meijer $G$-function and generalized…
We present a systematic study of higher-order Airy-type differential equations providing the explicit form of the solutions, deriving their power series expansions and a probabilistic interpretation. Under suitable convergence hypotheses,…
An integral representation for matrix Airy function is presented
We derive a product formula for the multiple stochastic integrals with respect to Levy process. The idea is to use exponential vectors and the polarization technique which greatly simplify the argument.
We present a simplified integral of functions of several variables. Although less general than the Riemann integral, most functions of practical interest are still integrable. On the other hand, the basic integral theorems can be obtained…
Integral representations for a complete set of linearly independent products of two solutions of the Airy equation whose arguments differ by $z_0$ are obtained using the Laplace contour integral method. This generalizes similar integral…
The Mellin transform of quartic products of shifted Airy functions is evaluated in a closed form. Some particular cases expressed in terms of the logarithm function and complete elliptic integrals special values are presented.
The Airy integral is a well-known contour integral solution of Airy's equation which has several applications and which has been used for mathematical illustrations due to its interesting properties. We present and derive properties of two…
Integrals occurring in Thomas-Fermi theory which contains the logarithm of the Airy function Ai'(x) have been obtained in terms of analytical expressions.
The Airy process is characterized by its finite-dimensional distribution functions. We show that each finite-dimensional distribution function is expressible in terms of a solution to a system of differential equations.
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…
In this paper, we study the formulae for a product of two product Euler polynomials. From this study, we derive some formulae for the integral of the product of two or more Ruler polynomials.
In this brief note the operatorial methods are applied to the study of the Airy function and its generalizations.
In this paper we evaluate sums and integrals of products of Fubini polynomials and have new explicit formulas for Fubini polynomials and numbers. As a consequence of these results new explicit formulas for p-Bernoulli numbers and…
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…
The derivatives with respect to order {\nu} 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…
A refinement of the Hardy inequality has been presented by use of superquadratic function.
We compute the integrals of products of Hermite functions using the generating functions. The precise asymptotics of 4 Hermite functions are presented below. This estimate is relevant for the corresponding cubic nonlinear equation.
In this paper, we derive a formula on the integral of products of the higher-order Euler polynomials. By the same way, similar relations are obtained for $l$ higher-order Bernoulli polynomials and $r$ higher-order Euler polynomials.…