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On an Integral Involving Bessel Polynomials and -Function of Two Variables and Its Application

Harmendra Kumar Mandia*, Yashwant Singh*

Department of Mathematics, Seth Motilal (P.G.) College, Jhunjhunu, Rajasthan, IndiaDepartment of Mathematics,Government College, Kaladera, Jaipur, Rajasthan , India

Abstract: This paper deals with the evaluation of an integral involving product of Bessel polynomials and -function of two variables. By making use of this integral the solution of the time-domain synthesis problem is investigated.

Keywords: -function of two variables, Bessel polynomials, Mellin-Barnes type integral, Time-domain synthesis problem,H-function of two variables

Received July 11, 2014; Revised August 01, 2014; Accepted August 15, 2014

Citation: Harmendra Kumar Mandia, Yashwant Singh, On an Integral Involving Bessel Polynomials and -Function of Two Variables and Its Application. Turkish Journal of Analysis and Number Theory, 2 (4). 130-133. January 2014. doi:10.12691/tjant-2-4-5

1. Introduction

The object of this paper is to evaluate an integral involving Bessel polynomials and the -function of two variables due to Singh and Mandia [8], and to apply it in obtaining a particular solution of the classical problem known as the ‘time-domain synthesis problem’, occurring in the electric network theory. On specializing the parameters, the -function of two variables may be reduced to almost all elementary functions and special functions appearing in applied Mathematics Erdelyi, A. et. al. ([2], p.215-222). The special solution derived in the paper is of general character and hence may encompass several cases of interest.

The -function of two variables will be defined and represented by Singh and Mandia [8] in the following manner:

[picture] | (1.1)

Where

[picture] | (1.2)

[picture] | (1.3)

[picture] | (1.4)

Where and are not equal to zero (real or complex), and an empty product is interpreted as unity are non-negative integers such that . All the are complex parameters. (not all zero simultaneously), similarly (not all zero simultaneously). The exponents can take on non-negative values.

The contour is in -plane and runs from to . The poles of lie to the right and the poles of to the left of the contour. For not an integer, the poles of gamma functions of the numerator in (1.3) are converted to the branch points.

The contour is in -plane and runs from to . The poles of lie to the right and the poles of to the left of the contour. For not an integer, the poles of gamma functions of the numerator in (1.4) are converted to the branch points.

The functions defined in (1.1) is an analytic function of and , if

[picture] | (1.5)

[picture] | (1.6)

The integral in (1.1) converges under the following set of conditions:

[picture] | (1.7)

[picture] | (1.8)

[picture] | (1.9)

The behavior of the -function of two variables for small values of follows as:

[picture] | (1.10)

Where

[picture] | (1.11)

For large value of ,

[picture] | (1.12)

Where

[picture] | (1.13)

Provided that and .

If we take in (1.1), the -function of two variables reduces to -function of two variables due to [7].

The following results are needed in the analysis that follows:

Bessel polynomials are defined as

[picture] | (1.14)

Orthogonality property of Bessel polynomials is derived by Exton ([4], p.215, (14)):

[picture] | (1.15)

Where .

The integral defined by Bajpai et.al. [1] is also required:

[picture] | (1.16)

Where .

2. Integral

The integral to be evaluated is

[picture] | (2.1)

Where

For , and conditions (1.7), (1.8) and (1.9) are also satisfied.

Proof: To establish (2.1), express the -function of two variables in its integrand as a Mellin-Barnes type integral (1.1) and interchange the order of integration which is permissible due to the absolute convergence of the integrals involved in the process, we obtain

Now evaluating the inner integral with the help of (1.16), it becomes

Which on applying (1.1), yields the desired result (2.1).

Special Case: If we take in (1.1), the -function of two variables reduces to -function of two variables due to [7], and we get

[picture] | (2.2)

Provided all condition are satisfied given in (2.1).

3. Solution of the Time-Domain Synthesis Problem of Signals

The classical time-domain synthesis problem occurring in electric network theory is as follows ([4], p. 139):

Given an electrical signal described by a real valued conventional function on , construct an electrical network consisting of finite number of components and which are all fixed, linear and positive, such that output of , resulting from a delta-function approximates on in some sense.

In order to obtain a solution of this problem, we expand the function into a convergent series:

[picture] | (3.1)

Or real-valued function . Let every partial sum

[picture] | (3.2)

Possesses the two properties

(i) for

(ii) The Laplace transform 0f is a rational function having a zero as and all its poles in the left-hand -plane, except possibly for a simple pole at the origin.

After choosing in (3.2) sufficiently large whatever approximation criterion is being used, an orthogonal series expansion may be employed. The Bessel polynomial transformation and (1.15) yields as immediate solution in the following form:

Where

[picture] | (3.3)

Where .

The function is continuous and of bounded variation in the open interval.

4. Particular Solution of the Problem

The particular solution of the problem is:

[picture] | (4.1)

Where and result (1.7), (1.8) and (1.9) are also holds.

Proof: Let us consider

[picture] | (4.2)

Equation (4.2) is valid, since is continuous and of bounded variation in the open interval .

Multiplying both sides of (4.2) by and integrating with respect to fromto , we get

Now using (2.1) and (1.15), we obtain

[picture] | (4.3)

On account of the most general character of the result (4.2) due to presence of the -function of two variables, numerous special cases can be derived but further sake of brevity those are not presented here.

References

[1]   | Bajpai, S.D. and Al-Hawaj, A.Y.; Application of Bessel polynomials involving generalized hypergeometric functions, J.Indian Acad. Math., vol.13 (1), (1991), 1-5.
[2]   | Erdelyi, A. et. al.; Higher Transcendental Functions, vol.1, McGraw-Hill, New York, 1953.
[3]   | Erdelyi, A. et. al.; Tables of Integral Transforms, vol.2, McGraw-Hill, New York, 1954.
[4]   | Exton, H.; Handbook of Hypergeometric Integrals, ELLIS Harwood Ltd., Chichester, 1978.
[5]   | Inayat-Hussain, A.A.; New properties of hypergeometric series derivable from Feynman integrals: II A generalization of the H-function, J. Phys. A. Math. Gen. 20 (1987).
[6]   | Mathai, A.M. and Saxena, R.K.; Lecture Notes in Maths. 348, Generalized Hypergeometric Functions With Applications in Statistics and Physical Sciences, Springer-Verlag, Berlin, 1973.
[7]   | Mittal, P.K. and Gupta, K.C.; An integral involving generalized function of two variables. Proc. Indian Acad. Sci. Sect. A( 75), (1961), 67-73.
[8]   | Singh,Y. and Mandia, H. ; A study of -function of two variables, International Journal of Innovative research in science, engineering and technology,Vol.2,(9),(2013), 4914-4921.