FINITE INTEGRAL INVOLVING INCOMPLETE ALEPH-FUNCTIONS, GENERALIZED EXTENDED HURWITZ'S ZETA FUNCTION, INCOMPLETE GAMMA FUNCTION, AND ELLIPTIC INTEGRALS OF THE FIRST KIND
Abstract
In this study, we derive a general finite integral that incorporates the generalized
Hurwitz-Lerch Zeta function of two variables, the incomplete Gamma function,
elliptic integrals of the first kind, and incomplete Aleph-functions. The paper concludes
with a discussion of several corollaries and observations.
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References
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[2] F.Y. Ayant and D. Kumar, A uni ed study of Fourier series involving the Aleph-function and the
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Ganita Sandesh 11 (1997), 99{108.
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Chem. Biol. Phys. Sci., Sect. C 6(3) (2016), 1116{1131.
[13] D. Kumar, F.Y. Ayant and M.S. Bajrolia, Improper integrals involving the incomplete Alephfunctions,
Sohag J. Math. 11(2), (2024), 23{27.
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Mittag-Leer function and incomplete I{functions, Res. Math. 9(1), (2022), 1{7.
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the basic analogue of multivariable Aleph-function, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci.
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[16] D. Kumar, F.Y. Ayant, A. Singh and P.K. Banerji, Finite integral formula involving Aleph-function
and generalized Mittag-Leer function, Probl. Anal. Issues Anal. 9(27)(1) (2020), 96{109.
[17] D. Kumar, F.Y. Ayant, D.L. Suthar, P. Nirwan and M. Kumari, Boros integral involving the class
of polynomials and incomplete @{functions, Proc. Indian Natl. Sci. Acad. 90(2) (2024), 1{8.
[18] D. Kumar, F.Y. Ayant and F. Ucar, Integral involving Aleph-function and the generalized incomplete
hypergeometric function, TWMS J. App. and Eng. Math. 10(3) (2020), 650{656.
[19] D. Kumar, D. Baleanu, F.Y. Ayant and N. Sudland, On transformation involving basic analogue to
the Aleph-function of two variables, Fractal Fract. 6(2): 71 (2022), 1{8.
[20] D. Kumar and J. Daiya, Fractional calculus pertaining to generalized H-functions, Glob. J. Sci.
Front. Res. F. Math. Decis. Sci. 14(3) (2014), 25{36.
[21] D. Kumar, R.K. Gupta, B.S. Shaktawat and J. Choi, Generalized fractional calculus formulas involving
the product of Aleph-function and Srivastava polynomials, Proc. Jangjeon Math. Soc. 20(4)
(2017), 701{717.
[22] D. Kumar, R.K. Saxena and J. Ram, Finite integral formulas involving Aleph function, Bol. Soc.
Parana. Mat. 36(1) (2018), 177{193.
[23] D.M. Lee, A.K. Rathie, R.K. Parmar and Y.S. Kim, Generalization of extended Beta function,
hypergeometric and con
uent hypergeometric functions, Honam Math. J. 33 (2011), 187{206.
[24] E. Ozergin, M.A. Ozarslan and A. Altin, Extension of Gamma, Beta and hypergeometric functions,
J. Comput. Appl. Math. 235 (16) (2011), 4601{4610.
[25] R.K. Parmar, J. Choi and S.D. Purohit, Further generalization of the extended Hurwitz's-Lerch
Zeta functions, Bol. Soc. Paran. Mat. 37(1) (2019), 177{190.
[26] J. Ram and D. Kumar, Generalized fractional integration of the @ {function, J. Rajasthan Acad.
Phys. Sci. 10(4) (2011), 373{382.
[27] V.P. Saxena, The I-function, Anamaya Publishers, New Delhi, 2008.
[28] L.J. Slater, Generalized hypergeometric functions, Cambridge University Press, 1966.
[29] H.M. Srivastava, M.A. Chaudhary and R.P. Agarwal, The incomplete Pochhammer symbols and
their applications to hypergeometric and related functions, Integral Transforms Spec. Funct. 23
(2012), 659{683.
[30] H.M. Srivastava, R.K. Saxena and R.K. Parmar, Some families of the incomplete H-function and
the incomplete H
-functions and associated integrals transforms and operators of fractional calculus
with applications, Russ. J. Math. Phys. 25 (2018), 116{138.
[31] N. Sudland, N.B. Baumann and T.F. Nonnenmacher, Open problem: who knows about the Alephfunctions?,
Fract. Calc. Appl. Anal. 1(4) (1998), 401{402.
[32] D.L. Suthar, S. Agarwal and D. Kumar, Certain integrals involving the product of Gaussian hypergeometric
function and Aleph function, Honam Math. J. 41 (1) (2019), 1{17.
[33] E.T. Whittaker and G.N. Watson, A course of modern analysis, Cambridge [Eng.], The University
Press, The Macmillan Company, New York, 1943.
[2] F.Y. Ayant and D. Kumar, A uni ed study of Fourier series involving the Aleph-function and the
Kampe de Feriet's function, Int. J. Math. Trend. Technol. 35(1) (2016), 40{48.
[3] M.K. Bansal and D. Kumar, On the integral operators pertaining to a family of incomplete I-
functions, AIMS Math. 5(2) (2020), 1247-1259.
[4] M.K. Bansal, D. Kumar, I. Khan, J. Singh and K.S. Nisar, Certain uni ed integrals associated with
product of M-series and incomplete H-functions, Mathematics 7(12):1191 (2019), 1-11.
[5] M.K. Bansal, D. Kumar, K.S. Nisar and J. Singh, Certain fractional calculus and integral transform
results of incomplete @-functions with applications, Math. Methods Appl. Sci. 43(8) (2020), 5602{
5614.
[6] S. Batra and P. Rai, A further extension of generalized Hurwitz's-Lerch zeta function of two variables,
Int. J. Pure Appl. Math. 119(18) (2018), 1551{1556.
[7] S. Batra and P. Rai, A further generalization of extended Hurwitz's-Lerch zeta function of two
variables, Adv. Stud. Contemp. Math. 30(3) (2020), 391{399.
[8] Y.A. Brychkov, Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas
(1st ed.), Chapman and Hall/CRC, New York, 2008.
[9] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol.
I, McGraw-Hill Book Inc., New York, Toronto and London, 1953.
[10] L. Euler, De progressionibus transcendenti bus, Novi Comment. Acad. Sci. Imp. Petropol. 5 (1729),
61{95.
[11] S.P. Goyal and R.K. Ladha, On the generalization Zeta function and generalized Lambert's function,
Ganita Sandesh 11 (1997), 99{108.
[12] D. Kumar, Generalized fractional di erintegral operators of the Aleph-function of two variables, J.
Chem. Biol. Phys. Sci., Sect. C 6(3) (2016), 1116{1131.
[13] D. Kumar, F.Y. Ayant and M.S. Bajrolia, Improper integrals involving the incomplete Alephfunctions,
Sohag J. Math. 11(2), (2024), 23{27.
[14] D. Kumar, F.Y. Ayant, P. Nirwan and D.L. Suthar, Boros integral involving the generalized multiindex
Mittag-Leer function and incomplete I{functions, Res. Math. 9(1), (2022), 1{7.
[15] D. Kumar, F.Y. Ayant, K.S. Nisar and D.L. Suthar, On fractional q{integral operators involving
the basic analogue of multivariable Aleph-function, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci.
93(2) (2023), 211{218.
[16] D. Kumar, F.Y. Ayant, A. Singh and P.K. Banerji, Finite integral formula involving Aleph-function
and generalized Mittag-Leer function, Probl. Anal. Issues Anal. 9(27)(1) (2020), 96{109.
[17] D. Kumar, F.Y. Ayant, D.L. Suthar, P. Nirwan and M. Kumari, Boros integral involving the class
of polynomials and incomplete @{functions, Proc. Indian Natl. Sci. Acad. 90(2) (2024), 1{8.
[18] D. Kumar, F.Y. Ayant and F. Ucar, Integral involving Aleph-function and the generalized incomplete
hypergeometric function, TWMS J. App. and Eng. Math. 10(3) (2020), 650{656.
[19] D. Kumar, D. Baleanu, F.Y. Ayant and N. Sudland, On transformation involving basic analogue to
the Aleph-function of two variables, Fractal Fract. 6(2): 71 (2022), 1{8.
[20] D. Kumar and J. Daiya, Fractional calculus pertaining to generalized H-functions, Glob. J. Sci.
Front. Res. F. Math. Decis. Sci. 14(3) (2014), 25{36.
[21] D. Kumar, R.K. Gupta, B.S. Shaktawat and J. Choi, Generalized fractional calculus formulas involving
the product of Aleph-function and Srivastava polynomials, Proc. Jangjeon Math. Soc. 20(4)
(2017), 701{717.
[22] D. Kumar, R.K. Saxena and J. Ram, Finite integral formulas involving Aleph function, Bol. Soc.
Parana. Mat. 36(1) (2018), 177{193.
[23] D.M. Lee, A.K. Rathie, R.K. Parmar and Y.S. Kim, Generalization of extended Beta function,
hypergeometric and con
uent hypergeometric functions, Honam Math. J. 33 (2011), 187{206.
[24] E. Ozergin, M.A. Ozarslan and A. Altin, Extension of Gamma, Beta and hypergeometric functions,
J. Comput. Appl. Math. 235 (16) (2011), 4601{4610.
[25] R.K. Parmar, J. Choi and S.D. Purohit, Further generalization of the extended Hurwitz's-Lerch
Zeta functions, Bol. Soc. Paran. Mat. 37(1) (2019), 177{190.
[26] J. Ram and D. Kumar, Generalized fractional integration of the @ {function, J. Rajasthan Acad.
Phys. Sci. 10(4) (2011), 373{382.
[27] V.P. Saxena, The I-function, Anamaya Publishers, New Delhi, 2008.
[28] L.J. Slater, Generalized hypergeometric functions, Cambridge University Press, 1966.
[29] H.M. Srivastava, M.A. Chaudhary and R.P. Agarwal, The incomplete Pochhammer symbols and
their applications to hypergeometric and related functions, Integral Transforms Spec. Funct. 23
(2012), 659{683.
[30] H.M. Srivastava, R.K. Saxena and R.K. Parmar, Some families of the incomplete H-function and
the incomplete H
-functions and associated integrals transforms and operators of fractional calculus
with applications, Russ. J. Math. Phys. 25 (2018), 116{138.
[31] N. Sudland, N.B. Baumann and T.F. Nonnenmacher, Open problem: who knows about the Alephfunctions?,
Fract. Calc. Appl. Anal. 1(4) (1998), 401{402.
[32] D.L. Suthar, S. Agarwal and D. Kumar, Certain integrals involving the product of Gaussian hypergeometric
function and Aleph function, Honam Math. J. 41 (1) (2019), 1{17.
[33] E.T. Whittaker and G.N. Watson, A course of modern analysis, Cambridge [Eng.], The University
Press, The Macmillan Company, New York, 1943.
Published
2025-12-29
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