# Article ID: MTJPAM-D-20-00026

## Title: Generalizations and Applications of Srinivasa Ramanujan’s Integral RS(m, n) via Hypergeometric Approach and Integral Transforms

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-20-00026; Volume 3 / Issue 3 / Year 2021 (Special Issue), Pages 216-226

Document Type: Research Paper

aDepartment of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia (Central University), New Delhi, 110025, India

bDepartment of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia (Central University), New Delhi, 110025, India — Post Graduate Department of Mathematics, Govt. Degree College Boys Baramulla, University of Kashmir, Kashmir, 193502, India

Received: 9 September 2020, Accepted: 20 February 2021, Published: 25 April 2021.

Full Text: PDF

Abstract

In this paper, we obtain analytical solution of an unsolved integral $\textbf{R}_{S}(m,n)$ of Srinivasa Ramanujan, using hypergeometric approach, Mellin transforms, Infinite Fourier sine transforms, Infinite series decomposition identity and some algebraic properties of Pochhammer’s symbol. Also we have given some generalizations of the Ramanujan’s integral $\textbf{R}_{S}(m,n)$ in the form of integrals $\textbf{I}^{*}_{S}(\upsilon,b,c,\lambda,y)$, $\textbf{J}_{S}(\upsilon,b,c,\lambda,y)$, $\textbf{K}_{S}(\upsilon,b,c, \lambda,y)$, $\textbf{I}_{S}(\upsilon,b,\lambda,y)$ and solved them in terms of ordinary hypergeometric functions ${}_{2}F_{3}$, with suitable convergence conditions. Moreover as applications of Ramanujan’s integral $\textbf{R}_{S}(m,n)$, the new three infinite summation formulas associated with hypergeometric functions ${}_{1}F_{2}$ and ${}_{0}F_{1}$ ( or cosine, sine and Lommel functions) are obtained.

Keywords: Generalized hypergeometric function, Infinite Fourier sine transforms, Ramanujan’s integrals, Fox-Wright psi hypergeometric function, Mellin transforms, Series decomposition identity, Bounded sequence

References:
1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover publications, Newyork, 1972.
2. R. P. Agarwal, Resonance of Ramanujan’s Mathematics, Vol. I, New Age International (p) Limited Publisher, New Delhi, 1996.
3. R. P. Agarwal, Resonance of Ramanujan’s Mathematics. Vol. II, New Age International (p) Limited Publisher, New Delhi, 1996. R. P. Agarwal, Resonance of Ramanujan’s Mathematics. Vol. III, New Age International (p) Limited Publisher, New Delhi, 1999.
4. T. Amdeberhan, L. A. Medina and V. H. Moll, The integrals in Gradhteyn and Ryzhik. Part 5: Some trigonometric integrals, Scientia Series A: Mathematical Sciences, 15, 47-60, 2007. G. E. Andrews and B. C. Berndt, Ramanujan’s Lost Notebook. Part I, Springer-Verlag, New York, 2005.
5. G. E. Andrews and B. C. Berndt, Ramanujan’s Lost Notebook. Part II, Springer-Verlag, New York, 2009.
6. G. E. Andrews and B. C. Berndt, Ramanujan’s Lost Notebook. Part III, Springer-Verlag, New York, 2012.
7. G. E. Andrews and B. C. Berndt, Ramanujan’s Lost Notebook. Part IV, Springer-Verlag, New York, 2013.
8. L. C. Andrews and B. K. Shivamoggi, Integral Transforms for Engineers, Prentice-Hall of India, New Delhi, 2003. W. N. Bailey, Generalized Hypergeometric Series, Cambridge Math. Tract No. 32, Cambridge Univ. press, Cambridge; Reprinted by Stechert-Hafner, New York, 1935.
9. R. J. Beerends, H. G. ter Morsche, J. C. Van den Berg and E. M. Van de Vri, Fourier and Laplace Transforms, Translated from Dutch by R.J. Beerends, Cambridge University Press, 2003.
10. B. C. Berndt, Ramanujan’s Notebooks, Part I, Springer-Verlag, New York, 1985.
11. B. C. Berndt, Ramanujan’s Notebooks, Part II, Springer-Verlag, New York, 1989. B. C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991.
12. B. C. Berndt, Ramanujan’s Notebooks, Part IV, Springer-Verlag, New York, 1994.
13. B. C. Berndt, Ramanujan’s Notebooks, Part V, Springer-Verlag, New York, 1998.
14. B. C. Berndt, Integrals associated with Ramanujan and elliptic functions, Ramanujan J. 1, 2016.
15. B. C. Berndt and A. Straub, Certain Integrals Arising from Ramanujan’s Notebooks, Symmetry Integrability Geom. Methods Appl. (SIGMA) 11 (83), 2015.
16. G. A. Campbell and R. M. Foster, Fourier Integrals for Practical Applications, Van Nostrand, New york, 1948.
17. B. C. Carlson, Some extensions of Lardner’s relations between 0F3 and Bessel functions, SIAM J. Math. Anal. 1, 232-242, 1970.
18. H. S. Carslaw, Introduction to the Theory of Fourier’s Series and Integrals, Macmillan and co., limited st. Martin’s street, London, 1921. V. A. Ditkin and A. P. Prudnikov, Integral Transforms and Operational Calculus, Pergamon Press, Oxford, London, Frankfurt, 1965. A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York, Toronto and London, 1953.
19. A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, Vol. 1, McGraw-Hill, New york, Toronto and London, 1954.
20. G. H. Hardy, P .V. S. Aiyar and B. M. Wilson, Collected Papers of Srinivasa Ramanujan, First published by Cambridge University press, Cambridge, 1927; Reprinted by Chelsea, New york, 1962; Reprinted by the American Mathematical society, Providence, Rhode Island, 2000. A. A. Kilbas and M. Saigo, H-Transforms: Theory and Applications (Analytical Methods and Special Functions), CRC Press Company, Boca Raton, London, New York, Washington, D.C., Vol. 9, 2004. A. A. Kilbas, M. Saigo and J. J. Trujillo, On the generalized Wright function, Fract. Calc. Appl. Anal. 5 (4), 2002, 437-460.
21. T. J. Lardner, Relations between 0F3 and Bessel functions, SIAM Rev. 11, 69-72, 1969. T. M. MacRobert, Integrals involving a modified Bessel function of the second kind and an E-function, Proc. Glasgow Math. Assoc. 2, 93-96, 1954. O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions: Theory and algorithmic Tables, Ellis Horwood Ltd. John Wiley, 1983.
22. J. L. Meyer, A generalization of an integral of Ramanujan, Ramanujan J. 14, 79-88, 2007.
23. K. S. Nisar, R. S. Mondal, P. Agarwal and M. Al-Dhaifallah, The Umbral operator and the integration involving generalized Bessel-type functions, Open Math. De Gruyter open, 13 (1), 2015.
24. F. Oberhettinger, Tables of Bessel Transforms, Springer-Verlag, Berlin, Heidelberg, New York, 1972.
25. F. Oberhettinger, Tables of Mellin Transforms, Springer-Verlag, Berlin, Heidelberg, New York, 1974.
26. F. Oberhettinger, Tables of Fourier Transforms and Fourier Transforms of Distributions, Springer Verlag, Berlin, 1990.
27. F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010.
28. A. P. Prudnikov, Y. A. Brychkov and O. I. Marichev, Integrals and Series: Volume 1: Elementary Functions, Taylor and Francis, 1986.
29. A. P. Prudnikov, Y. A. Brychkov and O. I. Marichev, Integrals and Series: Volume 2: Special Functions, Taylor and Francis, 1986.
30. A. P. Prudnikov, Y. A. Brychkov and O. I. Marichev, Integrals and Series: Volume 3: More special functions, Gordon and Breach Science Publishers, 1990.
31. A. P. Prudnikov, Y. A. Brychkov and O. I. Marichev, Integrals and Series: Volume 4: Direct Laplace transforms, Gordon and Breach Science Publishers, 1992.
32. A. P. Prudnikov, Y. A. Brychkov and O. I. Marichev, Integrals and Series: Volume 5: Inverse Laplace transforms, Gordon and Breach Science Publishers, 1992. M. I. Qureshi and S. A. Dar, Evaluation of some definite integrals of Ramanujan, using hypergeometric approach, Palest. J. Math. 6 (1), 1-3, 2017.
33. M. I. Qureshi and S. A. Dar, Generalizations of Ramanujan’s integral associated with infinite Fourier cosine transforms in terms of hypergeometric functions and its applications, Communicated for publication.
34. M. I. Qureshi and I. H. Khan, Ramanujan integrals and other definite integrals associated with Gaussian hypergeometric functions, South East Asian J. Math. Math. Sci. 4 (1), 39-52, 2005. M. I. Qureshi, K. A. Quraishi and R. Pal, A class of hypergeometric generalizations of an integral of Srinivasa Ramanujan, Asian J. Current Engineering Math. 2 (3), 190-194, 2013. M. I. Qureshi, K. A. Quraishi and R. Pal, Some applications of celebrated master theorem of Ramanujan, British Journal of Mathematics and Computer Science 4 (20), 2862-2871, 2014.
35. S. Ramanujan, Some definite integrals connected with Gauss’s sums, Mess. Math. XLIV, 75-86, 1915.
36. S. Ramanujan, Some definite integrals, J. Indian Math. Soc. 11, 81-87, 1915.
37. S. Ramanujan’s, Notebooks Vol. 1, Tata Institute of Fundamental Research, Bombay, 1957.
38. S. Ramanujan’s, Notebooks Vol. 2, Tata Institute of Fundamental Research, Bombay, 1957.
39. S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.
40. B. L. Sharma, A formula for hypergeometric series and its application, An. Univ. Timisoara Ser. Sti. Mat. 12, 145-154, 1974. I. N. Sneddon, Fourier Transforms, McGraw Hill Book Company, Inc, Newyork, 1951.
41. I. N. Sneddon, The Use of Integral Transforms, McGraw Hill Book Company, Inc, Newyork, 1972.
42. H. M. Srivastava, A note on certain identities involving generalized hypergeometric series, Nederl. Akad. Wetensch. Proc. Ser. A 82=Indag. Math. 41, 191-201, 1979.
43. H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester, U.K.), John Wiley and Sons, New york, Chichester, Brisbane and Toronto, 1984.
44. H. M. Srivastava, M. I. Qureshi, A. Singh and A. Arora, A family of hypergeometric integrals associated with Ramanujan’s integral formula, Adv. Stud. Contemp. Math., Kyungshang 18 (2), 113-125, 2009.
45. H. M. Srivastava, A. Tassaddiq, G. Rahman, S. N. Kottakkaran and I. Khan, A new extension of the t-Gauss hypergeometric function and its associated properties, Mathematics, 7 (10), 996, 2019.
46. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press, Second Edition, 1948.
47. N. Wiener, The Fourier Integral and Certain of its Applications, Dover publications, New York, 1951.
48. E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, J. London Math. Soc. 10, 286-293, 1935.
49. E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, Proc. London Math. Soc. 2 (46), 389-408, 1940.