Volume 51 | Number 2 | Year 2017 | Article Id. IJMTT-V51P519 | DOI : https://doi.org/10.14445/22315373/IJMTT-V51P519

Elliptic-type integrals have their importance and potential in certain problems in radiation physics and nuclear technology [4,5,7,10,15,17,22,23]. A number of earlier works on the subject contains several interesting unifications and generalizations of some significant families of elliptic-type integrals. The present paper is intended to obtain certain new theorems on generating functions. The results obtained in this paper are of manifold generality and basic in nature. Beside deriving various known and new elliptic-type integrals and their generalizations these theorems can be used to evaluate various Euler-type integrals involving a number of generating functions.

[1] B.N. Al-Saqabi, A generalization of elliptic-type integral, Hadronic J., 10(1987), 331-337.

[2] A. Al-Zamel, V.K. Tuan and S.L. Kalla, Generalized elliptic-type integrals and asymptotic formulas, Appl. Math. Comput., 114 (2000), 13-25.

[3] F.Y. Ayant, An integral associated with the Aleph-functions of several variables. International Journal of Mathematics Trends and Technology (IJMTT), 31(3) (2016), 142-154.

[4] M.L. 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.

[5] J. Bjorkberg and G. Kristensson, Electromagnetic scattering by a perfectly conducting elliptic disk, Canad. J. Phys., 65(1987), 723-734.

[6] V.B.L. Chaurasia and V. Gill, Generalized Elliptic-type integrals and generating functions with Aleph-function, Gen. Math. Notes. , 14(1) (2013), 21-34.

[7] V.B.L. Chaurasia and D. Kumar, Solution of the time-fractional Navier-Stokes equation, Gen. Math. Notes, 4(2) (2011), 49-59.

[8] V.B.L. Chaurasia and R.C. Meghwal, Unified presentation of certain families of elliptic-type integrals related to Euler integrals and generating functions, Tamkang Journal of Mathematics, 43(4) (2012), 00-00.

[9] V.B.L. Chaurasia and S.C. Pandey, Unified elliptic-type integrals and asymptotic formulas, Demonstratio Mathematica, 41(3) (2008), 531-541.

[10] V.B.L. Chaurasia and J. Singh, Application of sumudu transform in fractional kinetic equations, Gen. Math. Notes, 2(1) (2011), 86-95.

[11] V.B.L. Chaurasia and Y. Singh, Generalized elliptic-type integrals and generating functions, Demonstratio Mathematica, 47(1) (2014), 310-323.

[12] L.F. Epstein and J.H. Hubbell, Evaluation of a generalized elliptic-type integral, J. Res. N.B.S., 67(1963), 1-17.

[13] 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. Comp., 53(1993), 173-189.

[14] 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.

[15] 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.

[16] S.K. Kalla, Results on generalized elliptic-type integrals, mathematical structure computational mathematicsmathematical modeling, Sofia: Publ. House, Bulgar. Acad. Sci., 2(1984), 216-219.

[17] S.K. Kalla, The Hubbell rectangular source integral and its generalizations, Radiat. Phys. Chem., 41(1993), 775- 781.

[18] S.L. Kalla and B. Al-Saqabi, On a generalized elliptic-type integral, Rev. Bra. Fis., 16(1986), 145-156.

[19] S.L. Kalla, S. Conde and J.H. Hubbell, Some results on generalized elliptic-type integrals, Appl. Anal., 22(1986), 273-287.

[20] S.L. Kalla, C. Leubner and J.H. Hubbell, Further results on generalized elliptic type integrals, Appl. Anal., 25(1987), 269-274.

[21] S.L. Kalla and V.K. Tuan, Asymptotic formulas for generalized elliptic-type integrals, Comput. Math. Appl., 32(1996), 49-55.

[22] E.L. Kaplan, Multiple elliptic integrals, J. Math. And Phys., 29(1950), 69-75.

[23] P. Klinga and S.M. Khanna, Dose rate to the inner ear during Mosebauer experiments, Phys. Med. Biol., 28(1983), 359-366.

[24] J. Matera, L. Galue and S.L. Kalla, Asymptotic expansions for some elliptic-type integrals, Raj. Acad. Phy. Sci., 1(2) (2002), 71-82.

[25] M. Salman, Generalized elliptic-type integrals and their representations, Appl. Math. Comput., 181(2) (2006), 1249-1256.

[26] R.K. Saxena and M.A. Pathan, Asymptotic formulas for unified Elliptic-type integrals, Demonstratio Mathematica, 36(3) (2003), 579-589.

[27] R.K. Saxena and S.L. Kalla, Asymptotic formulas for unified Elliptic-type integrals, Int. Tran. Spec. Funct., 15(4) (2004), 359-368.

[28] R.K. Saxena, S.L. Kalla and J.H. Hubbell, Asymptotic expansion of a unified Elliptic-type integrals, Math. Balkanica, 15 (2001), 387-396.

[29] 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.

[30] K. Sharma K. On the integral representation and applications of the generalized function of two variables , InternationalJournal of Mathematical Engineering and Sciences ,3(1) ( 2014 ) , 1-13.

[31] C.K. Sharma C.K.and S.S. Ahmad, On the multivariable I-function. Acta ciencia Indica Math , 20(2) (1994), 113- 116.

[32] C.K. Sharma C.K. and mishra P.L. On the I-function of two variables and its properties. Acta Ciencia Indica Math , 17 (1991), 667-672.

[33] R.N. Siddiqui, On a class of generalized elliptic-type integrals, Rev. Brasileira Fis., 19(1989), 137-147.

[34] H.M. Srivastava and H.L. Manocha, A Treatise on Generating Functions, Chichester: Ellis Horwood Ltd., (1985).

[35] 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.

[36] H.M. Srivastava and S. Bromberg, Some families of generalized elliptic-type integrals, Math. Comput. Modelling, 21(3) (1995), 29-38.

[37] H.M.Srivastava H.M. and R. Panda, Some expansion theorems and generating relations for the H-function of severalcomplex variables. Comment. Math. Univ. St. Paul. 24(1975),119-137.

[38] H.M. Srivastava and R. Panda, Some bilateral generating function for a class of generalized hypergeometric polynomials. J Reine Angew Math 283/284, 1976, 265-274.

[39] N. Südland, B. Baumann and T.F. Nonnenmacher, Who knows about the Aleph ( )-function?, ℵ Fract. Calc. Appl. Anal., 1(4) (1998), 401-402.

[40] N. Südland, B. Baumann and T.F. Nonnenmacher, Fractional driftless FokkerPlanck equation with power law diffusion coefficients, In V.G. Gangha, E.W. Mayr and W.G. Vorozhtsov (Eds.), Computer Algebra in ScientificComputing(CASC Konstanz 2001), Springer, Berlin, (2001).

Frederic Ayant, Vinod Gill, "Generalized Elliptic-Type Integrals and Generating Functions with Multivariable Aleph-Function," *International Journal of Mathematics Trends and Technology (IJMTT)*, vol. 51, no. 2, pp. 149-161, 2017. *Crossref*, https://doi.org/10.14445/22315373/IJMTT-V51P519