Some Extensions and Generalizations of Kümmer's Third Summation Theorem
DOI:
https://doi.org/10.54172/mjsc.v37i4.910Keywords:
Hypergeometric functions, Kümmer's third summation theorem, Hypergeometric summation theoremsAbstract
The motive of this research paper is to obtain explicit forms of certain extensions and generalizations of Kümmer's third summation theorem, which have not previously appeared in the literature, by using the summation theorem given by Rakha and Rathie (2011). The results derived in this paper are interesting and may be beneficial.
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Andrews, G. E., Askey, R., Roy, R., Roy, R., & Askey, R. (1999). Special functions (Vol. 71). Cambridge university press Cambridge. DOI: https://doi.org/10.1017/CBO9781107325937
Arora, A., & Singh, R. (2008). Salahuddin, Development of a family of summation formulae of half argument using Gauss and Bailey theorems. Journal of Rajasthan Academy of Physical Sciences, 7, 335-342.
Awad, M. M., Koepf, W., Mohammed, A. O., Rakha, M. A., & Rathie, A. K. (2021). A Study of Extensions of Classical Summation Theorems for the Series $$ _ {3} F_ {2} $$3 F 2 and $$ _ {4} F_ {3} $$4 F 3 with Applications. Results in Mathematics, 76(2), 1-19. DOI: https://doi.org/10.1007/s00025-021-01367-9
Bailey, W. (1953). Generalized hypergeometric series, Cambridge Math: Tracts.
Carlson, B. C. (1977). Special functions of applied mathematics.
Clausen, T. (1828). Über die Falle, wenn die Reihe von der Form y= 1+... etc. ein Quadrat von der Form z= 1... etc. hat. J. Reine Angew. Math, 3, 89-91. DOI: https://doi.org/10.1515/crll.1828.3.89
Erdélyi, A., Magnus, W., Oberhettinger, F., & Tricomi, F. (1955). Higher Transcendental Functions, Vol. III, McGraw-Hill Book Company, New York, Toronto and London, 1955.
Goursat, E. (1883). : Mémoire sur les fonctions hypergéométriques d'ordre supérieur. Ann. Sci. École Norm. Sup.(Ser 2) 12, 261-286; 395-430. DOI: https://doi.org/10.24033/asens.225
Kim, Y. S., Rakha, M. A., & Rathie, A. K. (2010). Extensions of Certain Classical Summation Theorems for the Series F21, F32, and F43 with Applications in Ramanujan's Summations. Int. J. Math. Math. Sci., 2010, 309503:309501-309503:309526.
Kim, Y. S., Rathie, A. K., & Paris, R. (2013). Some summation formulas for the hypergeometric series r+ 2Fr+ 1 (1 2). Hacettepe Journal of Mathematics and Statistics, 42(3), 281-287.
Koepf, W., Kim, I., & Rathie, A. K. (2019). On a new class of Laplace-type integrals involving generalized hypergeometric functions. Axioms, 8(3), 87. DOI: https://doi.org/10.3390/axioms8030087
Kummer, E. (1836). Uber die hypergeometrische Reihe $1+frac {alpha {cdot}beta}{1 {cdot}gamma} x+frac {alpha (alpha+ 1)beta (beta+ 1)}{1 {cdot} 2 {gamma}{cdot}(gamma+ 1)} x^{2}+{cdots} $. J. Reine Angew Math, 15, 39-83.
Miller, A. R. (2005). A summation formula for Clausen's series 3F2 (1) with an application to Goursat's function 2F2 (x). Journal of Physics A: Mathematical and General, 38(16), 3541. DOI: https://doi.org/10.1088/0305-4470/38/16/005
Prudnikov, A. P., Brychkov, I. U. A., Bryčkov, J. A., & Marichev, O. I. (1986). Integrals and series: special functions (Vol. 2). CRC press.
Qureshi, M., & Baboo, M. (2016). Some Unified And Generalized Kummer’s First Summation Theorems With Applications In Laplace Transform. Asia Pacific Journal of Mathematics, 3(1), 10-23.
Qureshi, M., & Khan, M. K. (2020). Some Quadratic Transformations and Reduction Formulas associated with Hypergeometric Functions. Applications and Applied Mathematics: An International Journal (AAM), 15(3), 6.
Rainville, E. (1971). Special Function, McMillan, New York (1960): Reprinted by Chelsea Publishing Company, Bronx, New York.
Rakha, M. A., & Rathie, A. K. (2011). Generalizations of classical summation theorems for the series 2 F 1 and 3 F 2 with applications. Integral Transforms and Special Functions, 22(11), 823-840. DOI: https://doi.org/10.1080/10652469.2010.549487
Slater, L. J. (1966). Generalized hypergeometric functions. Cambridge university press.
Srivastava, H., & Manocha, H. (1984). Treatise on generating functions. John Wiley & Sons, Inc., 605 Third Ave., New York, Ny 10158, USA, 1984, 500.
Srivastava, H. M., & Choi, J. (2011). Zeta and q-Zeta functions and associated series and integrals. Elsevier. DOI: https://doi.org/10.1016/B978-0-12-385218-2.00002-5
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