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 pFp−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:
Proof. The proofs of results (13) to (57) asserted in Theorems 1–45 are quite straight forward. For example, in order to prove result (13) asserted in Theorem 1, we proceed as follows. In the general result (9), if we get , , , and , then it takes the following form:We now observe that the twice appearing on the right-hand side of (58) can be calculated by Gauss’ summation theorem (3), and we easily arrive at the desired result (13). This completes the proof of result (13) asserted by Theorem 1. The remaining results (14)–(57) can be established in a similar way. We would like to leave the details to the interested readers.
3. Corollaries
In this section, we shall mention several known results of our main findings:(a)In Theorems 1, 2, 4, 40, 42, and 44, if we take and , we get the known results due to Milovanovic̃ et al. (Theorems 2.1, 2, and 2.4 in [6] and Corollary 1.3 and 1.5 in [7], respectively).(b)In Theorems 3 and 11, if we take and , we get the known results due to Milovanovic̃ et al. (Theorems 2.3 in [6] and Corollary 4 in [7], respectively).(c)In Theorems 5 to 8, 13–16, and 26–29, if we take and , we get the known results due to Milovanovic̃ et al. (Theorems 2.5 to 2.10, 2.14 to 2.16, 2.23, 2.25, and 2.29 in [6] respectively).(d)In Theorems 20–23, if we take and , we get the known results due to Milovanovic̃ et al. (Theorems 2.11 to 2.13 and 2.27 in [6], respectively).(e)In Theorems 31–34, 36–38, 40, 41, and 43, if we take and , we get the known results due to Milovanovic̃ et al. (Theorems 2.17 to 2.22, 2.31, 2.33, 2.35, and 2.37 [6], respectively).(f)In Theorems 9, 17, 30, 35, 39, 42, and 44, if we take and , we get the known results due to Milovanovic̃ et al. (Theorems 2.24, 2.26, 2.30, 2.32, 2.34, 2.36, and 2.38 in [6], respectively).(g)In Theorem 24, if we take and , we get the known result due to Milovanovic̃ et al. (Theorem 2.28 in [6]).(h)In Theorem 45, if we take and , we get the known result due to Milovanovic̃ et al. (Theorem 2.39 in [6]).(i)In Theorem 18, if we take and , we get a known result due to Milovanovic̃ et al. (Corollary 2 in [7]).(j)In Theorem 19, if we take and , we get a known result due to Milovanovic̃ et al. (Corollary 6 in [7]).
4. Conclusion
In this note, our aim is to demonstrate how one can easily obtain in all forty-five attractive Laplace transforms of convolution type related to the generalized hypergeometric functions from the general result recorded in [5].
Applications related to engineering will be addressed in the next study. It is hoped that the results could be of potential use in the areas of engineering mathematics and mathematical physics.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.