A Note on Certain Laplace Transforms of Convolution-Type Integrals Involving Product of Two Generalized Hypergeometric Functions

Rathie, Arjun Kumar; Geum, Young Hee; Kim, Hwajoon

doi:https://doi.org/10.1155/2021/8827275

Mathematical Problems in Engineering

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Various Approaches for Generalized Integral Transforms

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Research Article | Open Access

Volume 2021 | Article ID 8827275 | https://doi.org/10.1155/2021/8827275

A Note on Certain Laplace Transforms of Convolution-Type Integrals Involving Product of Two Generalized Hypergeometric Functions

Arjun Kumar Rathie,¹Young Hee Geum,²and Hwajoon Kim³

Academic Editor: Xue-Bo Chen

Received30 Sept 2020

Accepted22 May 2021

Published30 May 2021

Abstract

The aim of this research paper is to provide as many as forty-five attractive Laplace transforms of convolution type related to the product of generalized hypergeometric functions. These are achieved by employing summation theorems for the series _pF_p−1 (for , and 5) available in the literature. The obtained research result is to provide an easier method than the existing method.

1. Introduction and Results’ Required

The theory of hypergeometric and generalized hypergeometric functions [1–3] are fundamental in the field of mathematics, engineering mathematics, and mathematical physics. Most of the commonly used functions that occur in the analysis are special cases or limiting cases of and .

It is well known that and defined bywhere and . Applications related to the detailed content can be found in [4–7].

A natural generalization of this function can be represented by

In the theory of hypergeometric and generalized hypergeometric functions, the following classical summation theorems for the series (for , and 5) play an important role [3, 4].

Gauss’ summation theorem [8]:which provided .

Gauss’s second summation theorem [9]:

Bailey’s summation theorem [8]:

Kummer’s summation theorem [8]:

Watson summation theorem [8]:which provided .

Dixon’s summation theorem [9]:which provided .

Whipple’s summation theorem [9]:which provided .

Second Whipple’s summation theorem [1]:

Dougall’s summation theorem [1]:

In addition to this, we have the following general result of the Laplace transform of convolution-type integrals involving the product of two generalized hypergeometric functions available in the literature, see [6, 7]:

The aim of this note is to demonstrate how one can easily obtain in all forty-five Laplace transforms of convolution type related to the product of two generalized hypergeometric functions from the general result (12) by employing various summation theorems (3) to (11). The results obtained earlier by Milovanovic̃ et al. [1, 2] follow special cases of our main findings.

2. A Note on Certain Laplace Transforms of Convolution-Type Integrals Involving Product of Two Generalized Hypergeometric Functions

In this section, we shall establish in all forty-five Laplace transforms of convolution type related to the product of two generalized hypergeometric functions mentioned in the following theorems. All delta values that appear in this section are shown in Section 1.

Theorem 1. For , , , , and , the following result holds true:

Theorem 2. For , , , and , the following result holds true:

Theorem 3. For , , , and , the following result holds true:

Theorem 4. For , , , and , the following result holds true:

Theorem 5. For , , , , and , the following result holds true:

Theorem 6. For , , , , and , the following result holds true:

Theorem 7. For , , , and , the following result holds true:

Theorem 8. For , , , and , the following result holds true:

Theorem 9. For , , , , and , the following result holds true:

Theorem 10. For , , and , the following result holds true:

Theorem 11. For , , and , the following result holds true:

Theorem 12. For , , and , the following result holds true:

Theorem 13. For , , , and , the following result holds true:

Theorem 14. For , , , and , the following result holds true:

Theorem 15. For , , and , the following result holds true:

Theorem 16. For , , and , the following result holds true:

Theorem 17. For , , , and , the following result holds true:

Theorem 18. For , , and , the following result holds true:

Theorem 19. For , , and , the following result holds true:

Theorem 20. For , , , and , the following result holds true:

Theorem 21. For , , , and , the following result holds true:

Theorem 22. For , , and , the following result holds true:

Theorem 23. For , , and , the following result holds true:

Theorem 24. For , , , and , the following result holds true:

Theorem 25. For , , and , the following result holds true:

Theorem 26. For , , , and , the following result holds true:

Theorem 27. For , , , and , the following result holds true:

Theorem 28. For , , and , the following result holds true:

Theorem 29. For , , and , the following result holds true:

Theorem 30. For , , , and , the following result holds true:

Theorem 31. For , , , , and , the following result holds true:

Theorem 32. For , , , , and , the following result holds true:

Theorem 33. For , , , and , the following result holds true:

Theorem 34. For , , , and , the following result holds true:

Theorem 35. For , , , , and , the following result holds true:

Theorem 36. For , , , , and , the following result holds true:

Theorem 37. For , , , and , the following result holds true:

Theorem 38. For , , , and , the following result holds true:

Theorem 39. For , , , , and , the following result holds true:

Theorem 40. For , , and , the following result holds true:

Theorem 41. For , , and , the following result holds true:

Theorem 42. For , , , and , the following result holds true:

Theorem 43. For , , and , the following result holds true:

Theorem 44. For , , , and , the following result holds true:

Theorem 45. For , , , , and , the following result holds true: