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Yasmin, Ghazala; Muhyi, Abdulghani

Some new results for Humbert’s double hypergeometric series \(\psi_2\) and \(\phi_2\). (English) Zbl 1414.33015

Palest. J. Math. 8, No. 2, 146-158 (2019).
Summary: This article aims to establish new expressions of Humbert’s functions \(\psi_2\) and \(\phi_2\) with the help of the generalization of Kummer’s summation theorem. Some special cases are also obtained. Further, we present some interesting integrals involving the Humbert’s functions, which are expressed in terms of generalized (Wright) hypergeometric function. Also, some special cases were considered as an application of the presented integral formulas.

MSC:

33C70 Other hypergeometric functions and integrals in several variables
33B15 Gamma, beta and polygamma functions
33C20 Generalized hypergeometric series, \({}_pF_q\)

Keywords:

gamma function; generalized hypergeometric function; Kummer’s theorem; generalized (Wright) hypergeometric function; Humbert functions; Lavoie-Trottier integral formula
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References:

[1] W. N. Bailey, Generalized Hypergeometric Series, Cambridge: Cambridge Univ. Press, (1935), Reprinted by Stechert Hafner, New York, (1964). · JFM 61.0406.01
[2] Yu. A. Brychkov, Handbook of Special Functions, Derivatives, Integrals, Series and Other Formulas, CRC Press, Taylar and Francis Group, A Chapman and Hall Book, (2008). · Zbl 1158.33001
[3] J. L. Burchnall and T. W. Chaundy, Expansions of Appell’s double hypergeometric functions, Quart. J. Math. (Oxford Ser.)11, 249-270 (1940). · Zbl 0025.16301
[4] J. Choi, Contiguous Extensions of Dixon’s Theorem on the Sum of a3F2, J. Inequal. Appl. 2010, Article ID 589618 (2010). · Zbl 1185.33007
[5] J. Choi and A. Hasanov, Applications of the operator H(α, β) to the Humbert double hypergeometric functions, Comput. Math. Appl. 61, 663-671 (2011). · Zbl 1217.33011
[6] J. Choi and A. K. Rathie, Certain new generating relations for products of two Laguerre polynomials, Commun. Korean Math. Soc.30 (3), 191-200 (2015). · Zbl 1331.33017
[7] J. Choi and A. K. Rathie, Certain summation formulas for Humbert’s double hypergeometric series ψ2 and φ2, Korean Math. Soc. 30 (4), 439-445 (2015). · Zbl 1329.33005
[8] J. Choi and A. K. Rathie, Generalizations of two summation formulas for the generalized hypergeometric function of higher order due to Exton, Commun. Korean Math. Soc. 25 (3), 385-389 (2010). · Zbl 1210.33011
[9] J. Choi and A. K. Rathie, Reducibility of Certain Kampé De Fériet Function with an Application to Generating relations for products of two Laguerre polynomials, Filomat 30:7, 2059-2066 (2016). · Zbl 1474.33015
[10] J. L. Lavoie, F. Grondin and A. K. Rathie, Generalizations of Whipple’s theorem on the sum of a3F2, J. Comput. Appl. Math.72 (2), 293-300 (1996). · Zbl 0853.33005
[11] J. L. Lavoie and G. Trottier, On the sum of certain Appell’s series, Ganita 20 (1), 31-32 (1969). · Zbl 0208.08201
[12] G. Lohofer, Theory of an electromagnetically deviated metal sphere.1:absorbed power, SIAM J. Appl. Math.49, 567-581 (1989). · Zbl 0692.35093
[13] V. V. Manako, A connection formula between double hypergeometric series ψ2and φ3, Int. Trans. Spec. Fun.23 (7), 503-508 (2012). · Zbl 1257.33005
[14] A. M. Mathai and R. K. Saxena, Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences, Springer-Verlag, Berlin, Heidelberg, New York, (1973). · Zbl 0272.33001
[15] A. W. Niukkanen, Generalised hypergeometric seriesNF (x1, ..., xN) arising in physical and quantum chemical applications, J. Phys. A: Math. Gen. 16, 1813-1825 (1983). · Zbl 0527.33001
[16] E. D. Rainville, Special Functions, Macmillan Company, New York, (1960). · Zbl 0092.06503
[17] A. K. Rathie, On representation of Humbert’s double hypergeometric series φ2in a series of Gauss’s2F1 function, arXiv: 1312.0064v1 (2016).
[18] N. Shekhawat, J. Choi, A. K. Rathie and Om Prakash, On a new class of series identities, Honam Math. J.37(3), 339-352 (2015). · Zbl 1325.33002
[19] H. M. Srivastava and B. R. K. Kashyap, Special Functions in Queuing Theory and Related Stochastic Processes, Academic Prees, New York, London, San Francisco, (1982). · Zbl 0492.60089
[20] H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, (1984). · Zbl 0535.33001
[21] Wei et al., Certain transformations for multiple hypergeometric functions, Advances in Difference Equations, 2013:360 (2013). · Zbl 1347.33017
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