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On a new class of Laplace-type integrals involving generalized hypergeometric functions. (English) Zbl 1432.33004

Summary: In the theory of generalized hypergeometric functions, classical summation theorems for the series \(_2 F_1\), \(_3 F_2\), \(_4 F_3\), \(_5 F_4\) and \(_7 F_6\) play a key role. Very recently, Masjed-Jamei and Koepf established generalizations of the above-mentioned summation theorems. Inspired by their work, the main objective of the paper is to provide a new class of Laplace-type integrals involving generalized hypergeometric functions \(_p F_p\) for \(p = 2, 3, 4, 5\) and 7 in the most general forms. Several new and known cases have also been obtained as special cases of our main findings.

MSC:

33C20 Generalized hypergeometric series, \({}_pF_q\)
33C05 Classical hypergeometric functions, \({}_2F_1\)
33C90 Applications of hypergeometric functions

Software:

SumTools
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andrews, G.E.; Askey, R.; Roy, R.; Special Functions; Encyclopedia of Mathematics and Its Applications: Cambridge, UK 1999; Volume Volume 71 . · Zbl 0920.33001
[2] Bailey, W.N.; ; Generalized Hypergeometric Series: Cambridge, UK 1935; . · Zbl 0011.02303
[3] Oberhettinger, F.; Badi, L.; ; Tables of Laplace Transforms: Berlin, Germany 1973; . · Zbl 0285.65079
[4] Rainville, E.D.; ; Special Functions: New York, NY, USA 1960; . · Zbl 0092.06503
[5] Koepf, W.; ; Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities: London, UK 2014; . · Zbl 1296.33002
[6] Bromwich, T.J.; ; An Introduction to the Theory of Infinite Series: New York, NY, USA 1948; .
[7] Knopp, K.; ; Theory and Applications of Infinite Series: New York, NY, USA 1990; .
[8] Luke, Y.L.; ; The Special Functions and Their Approximations: New York, NY, USA 1969; Volume Volume 1 . · Zbl 0193.01701
[9] Bailey, W.N.; Products of Generalized Hypergeometric Series; Proc. Lond. Math. Soc.: 1928; Volume 28 ,242-254. · JFM 54.0392.04
[10] Prudnikov, A.P.; Brychkov, Y.A.; Marichev, O.I.; ; More Special Functions: Amsterdam, The Netherlands 1990; Volume Volume 3 . · Zbl 0967.00503
[11] Masjed-Jamei, M.; Koepf, W.; Some Summation Theorems for Generalized Hypergeometric Functions; Axioms: 2018; Volume 7 . · Zbl 1407.42015
[12] Kim, Y.S.; Rakha, M.A.; Rathie, A.K.; Extensions of Certain Classical Summation Theorems for the Series 2F1, 3F2 and 4F3 with Applications in Ramanujan’s Summations; Int. J. Math. Math. Sci.: 2010; Volume 2010 ,309503. · Zbl 1210.33012
[13] Lavoie, J.L.; Grondin, F.; Rathie, A.K.; Generalizations of Watson’s theorem on the sum of a 3F2; Indian J. Math.: 1992; Volume 34 ,23-32. · Zbl 0793.33005
[14] Lavoie, J.L.; Grondin, F.; Rathie, A.K.; Arora, K.; Generalizations of Dixon’s Theorem on the sum of a 3F2(1); Math. Comp.: 1994; Volume 62 ,267-276. · Zbl 0793.33006
[15] Lavoie, J.L.; Grondin, F.; Rathie, A.K.; Generalizations of Whipple’s theorem on the sum of a 3F2; J. Comput. Appl. Math.: 1996; Volume 72 ,293-300. · Zbl 0853.33005
[16] Rakha, M.A.; Rathie, A.K.; Generalizations of classical summation theorems for the series 2F1 and 3F2 with applications; Integral Transform. Spec. Funct.: 2011; Volume 22 ,823-840. · Zbl 1241.33006
[17] Davis, B.; ; Integral Transforms and Their Applications: New York, NY, USA 2002; . · Zbl 0996.44001
[18] Doetsch, G.; ; Introduction to the Theory and Applications of the Laplace Transformation: New York, NY, USA 1974; . · Zbl 0278.44001
[19] Erdelyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F.G.; ; Tables of Integral Transforms: New York, NY, USA 1954; Volume Volume I-II . · Zbl 0055.36401
[20] Slater, L.J.; ; Confluent Hypergeometric Functions: Cambridge, UK 1960; . · Zbl 0086.27502
[21] Deepthi, P.; Prajapati, J.C.; Rathie, A.K.; New Laplace transforms of the 2F2 hypergeometric function; J. Fract. Calc. Appl.: 2017; Volume 8 ,150-155. · Zbl 1488.44002
[22] Parmar, R.K.; Purohit, S.D.; Certain integral transforms and fractional integral formulas for the extended hypergeometric functions; TWMS J. Appl. Eng. Math.: 2017; Volume 7 ,74-81. · Zbl 1376.44001
[23] Kim, Y.S.; Rathie, A.K.; Civijovic, D.; New Laplace transforms of Kummer’s confluent hypergeometric functions; Math. Comput. Model.: 2012; Volume 55 ,1068-1071. · Zbl 1255.44002
[24] Kim, Y.S.; Rathie, A.K.; Lee, C.H.; New Laplace transforms for the generalized hypergeometric functions 2F2; Honam Math. J.: 2015; Volume 37 ,245-252. · Zbl 1318.33007
[25] Jun, S.; Kim, I.; Rathie, A.K.; On a new class of Eulerian’s type integrals involving generalized hypergeometric functions; Aust. J. Math. Anal. Appl.: 2019; Volume 16 ,10. · Zbl 1409.33004
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