Journal of Mathematics

Journal of Mathematics / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 686324 | https://doi.org/10.1155/2014/686324

Subuhi Khan, Mustafa Walid Al-Saad, "Euler Type Integrals and Integrals in Terms of Extended Beta Function", Journal of Mathematics, vol. 2014, Article ID 686324, 12 pages, 2014. https://doi.org/10.1155/2014/686324

Euler Type Integrals and Integrals in Terms of Extended Beta Function

Academic Editor: Alfredo Peris
Received15 Feb 2014
Accepted12 May 2014
Published12 Jun 2014

Abstract

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 function as follows:

Keeping in view the fact that for the - symmetry to be preserved there must be symmetry of the integrand in and , Chaudhry et al. [3] introduced an extension of Euler’s beta function in the following form: which for reduces to the original beta function.

The transformations and in (7) yield the following integral representations for the extended beta function (EBF) [3, page 22 (2.9), (2.11)]: respectively.

The EBF is extremely useful in the sense that most of the properties of the beta function carry over naturally and simply for it. This extension is also important due to the fact that this function is related to other special functions for particular values of the variables.

We note the following connections [3]: where denotes the Whittaker function [1] and denotes the second form of solutions of Kummer’s equation [4].

Also, we note that where erfc( ) denotes the complementary error function [4].

Explicit evaluations of some integrals of Euler type for some particular functions , specially in the symmetric case , are derived in [5]. These evaluations are related to various reduction formulae for hypergeometric functions represented by such integrals. These formulae generalize the evaluations of some symmetric Euler integrals implied by the following result due to Pitman [6].

If a standard Brownian bridge is sampled at time , time and at independent random times with uniform distribution on , then the broken line approximation to the bridge obtained from these values has a total variation whose mean square is .

Motivated and inspired by the work of Ismail and Pitman [5] and Chaudhry et al. [3], in this paper, we derive the evaluations of certain Euler type integrals and some integrals in terms of EBF . In Section 2, we obtain the evaluations of certain integrals of the following type: where denotes the generalized hypergeometric series [4], for particular functions and . In Section 3, we establish a theorem on EBF and apply it to obtain evaluations of certain integrals in terms of EBF . Finally, we give some concluding remarks in Section 4.

2. Euler Type Integrals

We derive the evaluations of certain integrals of the following type:

We evaluate the integrals of type (15) for particular functions and by considering the following cases.

Case 1. Taking , , , and in (15) and using [5, page 962 (7)] where denotes the Appell function [4], in the r.h.s. after expanding and simplifying, we find the following integral:

Case 2. Taking , , , and in (15) and using [7, page 303 (1)] we find the following integral:

Case 3. Taking , , , and and proceeding as in Case 2, we find the following integral:

Case 4. Taking , , , and in (15), where is the complete elliptic integral of the first kind [4, page 35 (32)], defined by and using [7, page 265 (4)] we find the following integral:

Case 5. Taking , , , and in (15), where is the complete elliptic integral of the second kind [4, page 35 (33)], defined by and using [7, page 280 (3)] we find the following integral:

Case 6. Taking , , , and in (15) and using [7, page 173 (151)] we find the following integral:

Case 7. Taking , , , and in (15) and using [7, page 185 (1)] where denotes the error function [4], we find the following integral: Now, we establish a theorem and apply it to obtain the evaluations of certain integrals in terms of extended beta function.

3. Integrals in Terms of Extended Beta Function

Consider an -variable generating function which possesses a formal (not necessarily convergent for ) power series expansion in such that [4, page 80 (8)] where each member of the generated set is independent of and the coefficient set may contain the parameters of the set but is independent of and .

Theorem 1. Let the generating function defined by (31) be such that remains uniformly convergent for , , and . Then where is the extended beta function defined by (7).

Proof. Applying the definition of given in (31) in the l.h.s. of (32), we get which by using (9) yields the r.h.s. of (32).

Corollary 2. With definition (31) and notations as in Theorem 1, one has

Proof. Taking and in (32), we get (34).

Corollary 3. With definition (31) and notations as in Theorem 1, one has

Proof. Taking and in (32), we get (35).

Corollary 4. With definition (31) and notations as in Theorem 1, one has

Proof. Taking and in (32) and making use of relation (10), we get (36).
Now, we apply Theorem 1 to derive the evaluations of certain Euler type integrals in terms of EBF . We consider the following cases.
Case 1. Consider the generating function of the multivariable Hermite polynomials [8, page 602 (20,21)]
Applying Theorem 1 to generating function (37), we find the following integral:
Taking in (38), we get where are the 3-variable Hermite polynomials (3VHP) [9].
Again, taking in (38), we get where are the 2-variable Hermite-Kampé de-Fériet polynomials (2VHKdFP) [10].
Next, replacing by , by and taking in (38) and using the relation , we get where are the Gould-Hopper generalized Hermite polynomials (GHGHP) [11].
Further, replacing by and taking , in (38) and using the relation , we get where are the generalized Hermite polynomials [12].
Furthermore, taking in (42) and using the relation , we get where are the ordinary Hermite polynomials [1].
Case 2. Consider the generating function of the 2-variable Laguerre polynomials [13] which can also be expressed as where denotes the 0th-order Tricomi function [4].
Applying Theorem 1 to generating functions (44) and (45), we find the following integrals: respectively.
Taking in (46) and (47) and using the relation , we get respectively, where are the ordinary Laguerre polynomials [1].
Again, taking in (46) and (47) and using the relation , we get respectively.
Next, taking in (46) and (47) and using the relation , we get respectively. Replacing by in (53), it reduces to (50).
Case 3. Consider the generating function of the Hermite-Appell polynomials [14, page 759 (2.3)]
Applying Theorem 1 to generating function (54), we find the following integral:
Taking and replacing by in (55) and using the relation [14, page 760 (2.6)], we get where are the Appell polynomials [4, 15].
Case 4. Consider the generating function of the Laguerre-Appell polynomials [16] which can also be expressed as follows:
Applying Theorem 1 to generating functions (57) and (58), we find the following integrals: respectively.
Taking and replacing by in (59) and (60) and using the relation [16, page 9 (2.8)], we get (56) and