Euler Type Integrals and Integrals in Terms of Extended Beta Function

We derive the evaluations of certain integrals of Euler type involving generalized hypergeometric series. Further, we establish a theorem on extended beta function, which provides evaluation of certain integrals in terms of extended beta function and certain special polynomials. The possibility of extending some of the derived results to multivariable case is also investigated. 1. Introduction Euler generalized the factorial function from the domain of natural numbers to the gamma function defined over the right half of the complex plane. This led Legendre (in 1811) to decompose the gamma function into the incomplete gamma functions, and [1], which are obtained from (1) by replacing the upper and lower limits by , respectively. The closed-form solutions to a considerable number of problems in applied mathematics, astrophysics, nuclear physics, statistics, and engineering can be expressed in terms of incomplete gamma functions. These functions develop singularities at the negative integers. Chaudhry and Zubair [2] extended the domain of these functions to the entire complex plane by inserting a regularization factor in the integrand of (1). For Re , this factor clearly removes the singularity coming from the limit . For , this factor becomes unity, and thus we get the original gamma function. We note the following relation [3, page 20 (1.2)]: where is the modified Bessel function of the second kind of order (or Macdonald’s function) [1]. The relationships between the generalized gamma and Macdonald functions could not have been apparent in the original gamma function. These generalized gamma functions proved very useful in diverse engineering and physical problems; see, for example, [2, 3] and references therein. We note that Riemann’s zeta function defined by the series [1, page 85 (2.98)] is useful in proving convergence or divergence of other series by means of comparison test. Zeta function is closely related to the logarithm of the gamma function and to the polygamma functions. The regularizer also proved very useful in extending the domain of Riemann’s zeta function, thereby providing relationships that could not have been obtained with the original zeta function. In view of the fact that the regularization factor is useful in extending the domain of the gamma and zeta functions. The domain of other special functions could be usefully extended in a similar manner. In particular, Euler’s beta function has a close relationship to gamma function, that is, and can be usefully extendable. We first recall the basic Euler integral which defines the beta