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A unified elliptic-type integral and associated recurrence relations. (English) Zbl 1282.33031

Summary: Elliptic-type integrals have their importance or potential in certain problems in radiation physics and nuclear technology. A number of earlier works on the subject contains several interesting unifications and generalizations of some significant families of elliptic-type integrals. Beside explicit representations of certain families of unified elliptic-type integrals, contour integral representation, various recurrence relations are derived in the present investigation. The results derived in this article are of manifold generality and provide extensions and unification to a large number of known results established earlier.

MSC:

33E05 Elliptic functions and integrals
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[1] P. Appell and J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques; Polynômes d’Hermite, Gauthier-Villars, Paris (1926).
[2] M. J. Berger and J. C. Lamkin, Sample calculation of gamma ray penetration into shelters, Contribution of sky shine and roof contamination, J. Res. N.B.S., 60 (1958), 109–116.
[3] V. B. L. Chaurasia and S. C. Pandey, On certain generalized families of unified elliptic-type integrals pertaining to Euler integrals and generating functions, Ren-diconti del Circolo Matematico di Palermo, 58 (2009), 69–86. · Zbl 1176.33017 · doi:10.1007/s12215-009-0008-0
[4] V. B. L. Chaurasia and S. C. Pandey, Unified elliptic-type integrals and asymptotic formulas, Demonstratio Mathematica, 41(3) (2008), 531–541. · Zbl 1157.26302
[5] L. F. Epstein and J. H. Hubbell, Evaluation of a generalized elliptic-type integral, J. Res. N.B.S., 67 (1963), 1–17. · Zbl 0114.06402
[6] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, 1 McGraw-Hill, New York (1953). · Zbl 0051.30303
[7] J. D. Evans, J. H. Hubbell and V. D. Evans, Exact series solution to the Epstein-Hubbell generalized elliptic-type integral using complex variable residue theory, Appl. Math., 53 (1993), 173–189. · Zbl 0767.33015
[8] H. Exton, Handbook of hypergeometric integrals, John Wiley and Sons, New York (1978). · Zbl 0377.33001
[9] M. L. Glasser and S. L. Kalla, Recursion relations for a class of generalized elliptic-type integrals, Rev. Tec. Ing. Univ. Zulia, 12 (1989), 47–50. · Zbl 0681.33002
[10] J. H. Hubbell, R. L. Bach and R. J. Herbold, Radiation field from a circular disk source, J. Res. N.B.S., 65 (1961), 249–264.
[11] S. L. Kalla, Results on generalized elliptic-type integrals, Mathematical structure – computational mathematics – mathematical modelling 2, Publ. House Bulgar. Acad. Sci., Sofia, (1984), 216–219.
[12] S. L. Kalla, The Hubbell rectangular source integral and its generalizations, Radiat. Phys. Chem., 41 (1993), 775–781. · doi:10.1016/0969-806X(93)90325-O
[13] S. L. Kalla, S. Conde and J. H. Hubbell, Some results on generalized elliptic-type integrals, Appl. Anal., 22 (1986), 273–287. · Zbl 0588.33001 · doi:10.1080/00036818608839623
[14] S. L. Kalla and B. N. Al-Saqabi, On a generalized elliptic-type integral, Rev. Bra. Fis., 16 (1986), 145–156.
[15] S. L. Kalla, C. Leubner and J. H. Hubbell, Further results on generalized elliptic-type integrals, Appl. Anal., 25 (1987), 269–274. · Zbl 0606.33003 · doi:10.1080/00036818708839690
[16] S. L. Kalla and V. K. Tuan, Asymptotic formulas for generalized Elliptic-type integrals, Comput. Math. Appl., 32 (1996), 49–55. · Zbl 0891.33010 · doi:10.1016/0898-1221(96)00124-1
[17] E. L. Kaplan, Multiple elliptic integrals, J. Math. and Phys., 29 (1950), 69–75. · Zbl 0037.32402
[18] B. N. Al-Saqabi, A generalization of elliptic-type integrals, Hadronic J., 10 (1987), 331–337.
[19] A. Al-Zamel, V. K. Tuan and S. L. Kalla, Generalized Elliptic-type integrals and asymptotic formulas, Appl. Math. Comput., 114 (2000), 13–25. · Zbl 1049.33017 · doi:10.1016/S0096-3003(99)00092-2
[20] J. Matera, L. Galue and S. L. Kalla, Asymptotic expansions for some elliptic-type integrals, Raj. Acad. Phy. Sci., 1(2) (2002), 71–82. · Zbl 1059.33030
[21] A. M. Mathai and R. K. Saxena, The H-Function with Application in Statistics and other Disciplines, Halsted Press, New York (1978). · Zbl 0382.33001
[22] E. D. Rainville, Special Functions, Reprinted by Chelsea Pub. Co., Bronx, New York (1971). · Zbl 0231.33001
[23] M. Salman, Generalized elliptic-type integrals and their representations, Appl. Math. Comput., 181(2) (2006), 1249–1256. · Zbl 1102.33016 · doi:10.1016/j.amc.2006.02.025
[24] R. K. Saxena, S. L. Kalla and J. H. Hubbell, Asymptotic expansion of a unified Elliptic-type integrals, Math. Balkanica, 15 (2001), 387–396. · Zbl 1082.33504
[25] R. K. Saxena and S. L. Kalla, A new method for evaluating Epstein-Hubbell generalized elliptic-type integrals, Int. J. Appl. Math., 2 (2000), 732–742. · Zbl 1142.33321
[26] R. K. Saxena and M. A. Pathan, Asymptotic formulas for unified Elliptic-type integrals, Demonstratio Mathematica, 36(3) (2003), 579–589. · Zbl 1082.33008
[27] R. N. Siddiqi, On a class of generalized elliptic-type integrals, Rev. Brasileira Fis., 19 (1989), 137–147.
[28] H. M. Srivastava and S. Bromberg, Some families of generalized elliptic-type integrals, Math. Comput. Modelling, 21(3) (1995), 29–38. · Zbl 0820.33010 · doi:10.1016/0895-7177(94)00212-7
[29] H. M. Srivastava and R. N. Siddiqi, A unified presentation of certain families of elliptic-type integrals related to radiation field problems, Radiat. Phys. Chem., 46 (1995), 303–315. · doi:10.1016/0969-806X(94)00174-I
[30] H. M. Srivastava and P. W. Karlsson, Multiple Gaussian hypergeometric series, John Wiley and Sons, New York (1985). · Zbl 0552.33001
[31] H. M. Srivastava and H. L. Manocha, A Treatise on generating functions, Ellis Hor-wood Ltd., Chichester (1985). · Zbl 0535.33001
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