The Umbral operator and the integration involving generalized Bessel-type functions (original) (raw)

Certain new unified integrals associated with the product of generalized Bessel functions

Communications in Numerical Analysis, 2016

Our focus to presenting two very general integral formulas whose integrands are the integrand given in the Oberhettingers integral formula and a finite product of the generalized Bessel function of the first kind, which are expressed in terms of the generalized Lauricella functions. Among a large number of interesting and potentially useful special cases of our main results, some integral formulas involving such elementary functions are also considered.

Certain unified integrals associated with Bessel functions

Boundary Value Problems, 2013

A remarkably large number of integral formulas involving a variety of special functions have been developed by many authors. Very recently, Ali gave three interesting unified integrals involving the hypergeometric function 2 F 1 . Using Ali's method, in this paper, we present two generalized integral formulas involving the Bessel function of the first kind J ν (z), which are expressed in terms of the generalized (Wright) hypergeometric functions. Some interesting special cases of our main results are also considered. MSC: Primary 33B20; 33C20; secondary 33B15; 33C05

Certain New Integral Formulas Involving the Generalized Bessel Functions

Bulletin of the Korean Mathematical Society, 2014

A remarkably large number of integral formulas involving a variety of special functions have been developed by many authors. Also many integral formulas involving various Bessel functions have been presented. Very recently, Choi and Agarwal derived two generalized integral formulas associated with the Bessel function Jν (z) of the first kind, which are expressed in terms of the generalized (Wright) hypergeometric functions. In the present sequel to Choi and Agarwal's work, here, in this paper, we establish two new integral formulas involving the generalized Bessel functions, which are also expressed in terms of the generalized (Wright) hypergeometric functions. Some interesting special cases of our two main results are presented. We also point out that the results presented here, being of general character, are easily reducible to yield many diverse new and known integral formulas involving simpler functions.

Certain unified integrals involving a product of Bessel functions of the first kind

A remarkably large number of integrals involving a product of certain combinations of Bessel functions of several kinds as well as Bessel functions, themselves, have been investigated by many authors. Motivated the works of both Garg and Mittal and Ali, very recently, Choi and Agarwal gave two interesting unified integrals involving the Bessel function of the first kind Jν (z). In the present sequel to the aforementioned investigations and some of the earlier works listed in the reference, we present two generalized integral formulas involving a product of Bessel functions of the first kind, which are expressed in terms of the generalized Lauricella series due to Srivastava and Daoust. Some interesting special cases and (potential) usefulness of our main results are also considered and remarked, respectively.

On a new class of integrals involving Bessel functions of the first kind

Communications in Numerical Analysis, 2014

In recent years, several integral formulas involving a variety of special functions have been developed by many authors. Also many integral formulas containing the Bessel function J ν (z) have been presented. Very recently, Rakha et al. presented some generalized integral formulas involving the hypergeometric functions. In this sequel, here, we aim at establishing two generalized integral formulas involving a Bessel functions of the first kind, which are expressed in terms of the generalized Wright hypergeometric function. Some interesting special cases of our main results are also considered.

Some integrals involving Bessel functions

Arxiv preprint math/9307213, 1993

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.

On Generalized Integrals and Ramanujan-Jacobi Special Functions

In this article we consider new generalized functions for evaluating integrals and roots of functions. The construction of these generalized functions is based on Rogers-Ramanujan continued fraction, the Ramanujan-Dedekind eta, the elliptic singular modulus and other similar functions. We also provide modular equations of these new generalized functions and remark some interesting properties.

C A ] 6 M ay 2 01 8 Generalizations of Ramanujan ’ s integral associated with infinite Fourier cosine transforms in terms of hypergeometric functions and its applications

2018

In this paper, we obtain analytical solution of an unsolved integral RC(m,n) of Srinivasa Ramanujan [Mess. Math., XLIV, 75-86, 1915], using hypergeometric approach, Mellin transforms, Infinite Fourier cosine transforms, Infinite series decomposition identity and some algebraic properties of Pochhammer’s symbol. Also we have given some generalizations of the Ramanujan’s integral RC(m,n) in the form of integrals I ∗ C(υ,b,c,λ ,y),JC(υ,b,c,λ ,y),KC(υ,b,c, λ ,y),IC(υ,b,λ ,y) and solved it in terms of ordinary hypergeometric functions 2F3, with suitable convergence conditions. Moreover as applications of Ramanujan’s integral RC(m,n), the new nine infinite summation formulas associated with hypergeometric functions 0F1, 1F2 and 2F3 are obtained. 2010 AMS Classification: 33C20; 42A38; 33E20; 33C60.