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The incomplete Lauricella functions of several variables and associated properties and formulas. (English) Zbl 1400.33024
Summary: Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava et al., Integral Transforms Spec. Funct. 23, No. 9, 659–683 (2012; Zbl 1254.33004)] and the second Appell function [A. Çetinkaya, Appl. Math. Comput. 219, No. 15, 8332–8337 (2013; Zbl 1318.33001)], we introduce here the incomplete Lauricella functions \(\gamma^{(n)}_A\) and \(\Gamma^{(n)}_A\) of \(n\) variables. We then systematically investigate several properties of each of these incomplete Lauricella functions including, for example, their various integral representations, finite summation formulas, transformation and derivative formulas, and so on. We provide relevant connections of some of the special cases of the main results presented here with known identities. Several potential areas of application of the incomplete hypergeometric functions in one and more variables are also pointed out.
33C65 Appell, Horn and Lauricella functions
33B15 Gamma, beta and polygamma functions
33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)
33C20 Generalized hypergeometric series, \({}_pF_q\)
DLMF; Equator
Full Text: DOI
[1] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formu- las, graphs, and mathematical tables, Tenth Printing, National Bureau of Standards, Applied Mathematics Series 55, Washington, D.C., 1972; Reprinted by Dover Publications, New York, 1965.
[2] L. C. Andrews, Special functions for engineers and applied mathematicians, Macmillan Company, New York, 1985.
[3] A. Annamalai and C. Tellambura, Cauchy-Schwarz bound on the generalized Marcum Q-function with applications, Wireless Commun. Mob. Comput., 1(2001), 243–253.
[4] W. N. Bailey, Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics 32, Stechert-Hafner, Inc., New York, 1964.
[5] B. C. Carlson, Special functions of applied mathematics, Academic Press, New York, San Francisco and London, 1977. · Zbl 0394.33001
[6] A. C¸ etinkaya, The incomplete second Appell hypergeometric functions, Appl. Math. Comput., 219(2013), 8332–8337. · Zbl 1318.33001
[7] M. A. Chaudhry, A. Qadir, H. M. Srivastava and R. B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 159(2004), 589–602. · Zbl 1067.33001
[8] M. A. Chaudhry and S. M. Zubair, On a class of incomplete gamma functions with applications, Chapman and Hall (CRC Press Company), Boca Raton, FL, 2002. · Zbl 0977.33001
[9] A. Erd´elyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, Vols. I, II, McGraw-Hill Book Company, New York, Toronto and London, 1953. · Zbl 0051.30303
[10] A. A. Kilbas, H. M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematical Studies 204, Elsevier (NorthHolland) Science Publishers, Amsterdam, London and New York, 2006. · Zbl 1092.45003
[11] S.-D. Lin, H. M. Srivastava and J.-C. Yao, Some classes of generating relations as- sociated with a family of the generalized Gauss type hypergeometric functions, Appl. Math. Inf. Sci. 9(2015), 1731–1738.
[12] Y. L. Luke, Mathematical functions and their approximations, Academic Press, New York, San Francisco and London, 1975. · Zbl 0318.33001
[13] M.-J. Luo and R. K. Raina, Extended generalized hypergeometric functions and their applications, Bull. Math. Anal. Appl., 5(4)(2013), 65–77. · Zbl 1314.33013
[14] W. Magnus,F. Oberhettinger and R. P. Soni,Formulas and theorems for the special functions of mathematical physics,Third Enlarged edition,Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Ber¨ucksichtingung der Anwendungsgebiete, Bd. 5, Springer-Verlag, Berlin, Heidelberg and New York, 1966. · Zbl 0143.08502
[15] J. I. Marcum, A statistical theory of target detection by pulsed radar, Trans. IRE, IT-6(1960), 59–267.
[16] K. B. Oldham, J. Myland and J. Spanier, An atlas of functions, With Equator, the atlas function calculator, Second edition, With 1 CD-ROM (Windows), Springer, New York, 2009. · Zbl 1167.65001
[17] F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of mathematical functions, [With 1 CD-ROM (Windows, Macintosh and UNIX)], US Department of Commerce, National Institute of Standards and Technology, Washington, D.C., 2010; Cambridge University Press, Cambridge, London and New York, 2010.
[18] E. ¨Ozergin, M. A. ¨Ozarslan and A. Altın, Extension of gamma, beta and hypergeo- metric functions, J. Comput. Appl. Math., 235(2011), 4601–4610. · Zbl 1218.33002
[19] P. A. Padmanabham and H. M. Srivastava, Summation formulas associated with the (r) Lauricella function FA, Appl. Math. Lett., 13(2000), 65–70. · Zbl 0940.33006
[20] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and series, Vol.II, Gordon and Breach Science Publishers, New York, 1988. · Zbl 0733.00005
[21] E. D. Rainville, Special functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971. · Zbl 0092.06503
[22] M. K. Simon and M.-S. Alouini, Digital communication over fading channels: a uni- fied approach to performance analysis, John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 2000.
[23] L. J. Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, London and New York, 1966. · Zbl 0135.28101
[24] H. M. Srivastava, On a summation formula for the Appell function F2, Proc. Cambridge Philos. Soc., 63(1967), 1087–1089. · Zbl 0166.32203
[25] H. M. Srivastava, Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Inform. Sci., 5(2011), 390–444.
[26] R. Srivastava, Some properties of a family of incomplete hypergeometric functions, Russian J. Math. Phys., 20(2013), 121–128. · Zbl 1279.33006
[27] R. Srivastava, Some generalizations of Pochhammer’s symbol and their associated families of hypergeometric functions and hypergeometric polynomials, Appl. Math. Inf. Sci., 7(2013), 2195–2206.
[28] R. Srivastava, Some classes of generating functions associated with a certain family of extended and generalized hypergeometric functions, Appl. Math. Comput., 243(2014), 132–137. · Zbl 1335.33005
[29] H. M. Srivastava, A. C¸ etinkaya and ˙I. O. Kıymaz, A certain generalized Pochham- mer symbol and its applications to hypergeometric functions, Appl. Math. Comput., 226(2014), 484–491.
[30] H. M. Srivastava, M. A. Chaudhry and R. P. Agarwal, The incomplete Pochham- mer symbols and their applications to hypergeometric and related functions, Integral Transforms Spec. Funct., 23(2012), 659–683. · Zbl 1254.33004
[31] R. Srivastava and N. E. Cho, Generating functions for a certain class of incomplete hypergeometric polynomials, Appl. Math. Comput., 219(2012), 3219–3225. · Zbl 1309.33009
[32] R. Srivastava and N. E. Cho, Some extended Pochhammer symbols and their ap- plications involving generalized hypergeometric polynomials, Appl. Math. Comput., 234(2014), 277–285. · Zbl 1302.33007
[33] H. M. Srivastava and J. Choi, Series associated with the zeta and related functions, Kluwer Acedemic Publishers, Dordrecht, Boston and London, 2001. · Zbl 1014.33001
[34] H. M. Srivastava and J. Choi, Zeta and q-zeta functions and associated series and integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012. · Zbl 1239.33002
[35] H. M. Srivastava and P. W. Karlsson, Multiple Gaussian hypergeometric series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985. · Zbl 0552.33001
[36] H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984. · Zbl 0535.33001
[37] N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathe- matical Physics, John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1996.
[38] G. N. Watson, A Treatise on the Theory of Bessel Functions, Second edition, Cambridge University Press, Cambridge, London and New York, 1944. · Zbl 0063.08184
[39] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis: An Introduc- tion to the General Theory of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendental Functions, Fourth edition, Cambridge University Press, Cambridge, London and New York, 1963.
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