Fractional calculus pertaining to multivariable Aleph-function
DOI:
https://doi.org/10.5269/bspm.42941Abstract
In this paper we study a pair of unied and extended fractional integral operator involving the multivariable Aleph-function, Aleph-function and general class of polynomials. During this study, we establish ve theorems pertaining to Mellin transforms of these operators. Furthers, some properties of these operators have also been investigated. On account of the general nature of the functions involved herein, a large number of (known and new) fractional integral operators involved simpler functions can also be obtained . We will quote the particular case concerning the multivariable I-function dened by Sharma and Ahmad [20] and the I-function of one variable dened by Saxena [13].
References
1. F. Y. Ayant and D. Kumar, A unified study of Fourier series involving the Aleph-function and the Kampe de Feriet's function, International Journal of Mathematics Trends and Technology, 35(1) (2016), 40-48. https://doi.org/10.14445/22315373/IJMTT-V35P507
2. F. Y. Ayant and D. Kumar, Certain finite double integrals involving the hypergeometric function and Aleph-function, International Journal of Mathematics Trends and Technology, 35(1) (2016), 49-55. https://doi.org/10.14445/22315373/IJMTT-V35P508
3. F. Y. Ayant and D. Kumar, Generating relations and multivariable Aleph-function, Analysis, 38(3) (2018), 137-143. https://doi.org/10.1515/anly-2017-0054
4. F. Y. Ayant and D. Kumar, Fredholm type integral equation with special functions, Acta Universitatis Sapientiae Mathematica, 10(1) (2018), 5-17. https://doi.org/10.2478/ausm-2018-0001
5. J. Choi, J. Daiya, D. Kumar and R. K. Saxena, Fractional differentiation of the product of Appell function F3 and multivariable H-function, Commun. Korean Math. Soc., 31(1) (2016), 115-12 https://doi.org/10.4134/CKMS.2016.31.1.115
6. J. Daiya, J. Ram and D. Kumar, The multivariable H-function and the general class of Srivastava polynomials involving the generalized Mellin-Barnes contour integrals, FILOMAT Journal, 30(6) (2016), 1457-1464. https://doi.org/10.2298/FIL1606457D
7. G. Dorrego and D. Kumar, A generalization of the kinetic equation using the Prabhakar-type operators, Honam Mathematical J., 39(3) (2017), 401-416.
8. A. Erdelyi, On some functional transformations, Univ. Politec. Torino, Rend. Sem. Mat., 10 (1950), 217-234.
9. D. Kumar, Generalized fractional differintegral operators of the Aleph-function of two variables, Journal of Chemical, Biological and Physical Sciences, Section C, 6(3) (2016), 1116-1131.
10. D. Kumar, F. Y. Ayant and J. Choi, Application of product of the multivariable A-function and the multivariable Srivastava polynomials, East Asian Math. J., 34(3) (2018), 295-303.
11. D. Kumar and J. Choi, Certain generalized fractional differentiation of the product of two ℵ-functions associated with the Appell function F3, Applied Mathematical Sciences, 10(4) (2016), 187-196. https://doi.org/10.12988/ams.2016.511709
12. D. Kumar and J. Daiya, Fractional calculus pertaining to generalized H-functions, Global Journal of Science Frontier Research: F, Mathematics and Decision Sciences, 14(3) (2014), 25-36.
13. D. Kumar, S. D. Purohit and J. Choi, Generalized fractional integrals involving product of multivariable H-function and a general class of polynomials, J. Nonlinear Sci. Appl., 9 (2016), 8-21. https://doi.org/10.22436/jnsa.009.01.02
14. D. Kumar, S. D. Purohit, A. Secer and A. Atangana, On generalized fractional kinetic equations involving generalized Bessel function of the first kind, Mathematical Problems in Engineering, Article ID 289387, 2015, (2015), 7 pp. https://doi.org/10.1155/2015/289387
15. D. Kumar, R. K. Saxena and J. Ram, Finite integral formulas involving Aleph function, Boletim da Sociedade Paranaense de Matematica, 36(1) (2018), 177-193. https://doi.org/10.5269/bspm.v36i1.28123
16. E. R. Love, Some integral equations involving hypergeometric functions, J. Edinburgh Math. Soc. 3(15) (1967), 169-198. https://doi.org/10.1017/S0013091500011706
17. S. D. Purohit, Solutions of fractional partial differential equations of quantum mechanics, Adva. Appl. Math. Mech. 5 (2013), 639-651. https://doi.org/10.4208/aamm.12-m1298
18. S. D. Purohit and S.L. Kalla, On fractional partial differential equations related to quantum mechanics, J. Phys. A: Math. Theor., 44(4) (2011), 8 pp. https://doi.org/10.1088/1751-8113/44/4/045202
19. M. Saigo, R. K. Saxena and J. Ram, On the fractional calculus operator associated with the H-function, Ganita Sandesh, 6(1) (1992), 36-47.
20. V. P. Saxena, The I-function, Anamaya Publishers, New Delhi, 2008.
21. R. K. Saxena and V.S. Kiryakova, On relation between the two-dimensional H-transforms in terms of Erd'elyi-Kober operators, Math. Balkanica, 6 (1992), 133-140.
22. R. K. Saxena and D. Kumar, Generalized fractional calculus of the Aleph function involving a general class of polynomials, Acta Mathematica Scientia, 35(5) (2015), 1095-1110. https://doi.org/10.1016/S0252-9602(15)30042-4
23. R. K. Saxena and R. K. Kumbhat, Fractional integration operators of two variables, Proc. Indian Acad. Sci., 78 (1973), 177-186. https://doi.org/10.1007/BF03049477
24. R. K. Saxena and R. K. Kumbhat, Integral operators involving H-function, Indian J. of Pure Appl. Math., 5 (1974), 1-6.
25. R. K. Saxena and R. K. Kumbhat, Some properties of generalized Kober operators, Vijnana Parishad Anusandhan Patrika, 18 (1975), 139-150.
26. R. K. Saxena, J. Ram and D. Kumar, Generalized fractional integral of the product of two Aleph-functions, Applications and Applied Mathematics, 8(2) (2013), 631-646.
27. C. K. Sharma and S.S. Ahmad, On the multivariable I-function, Acta ciencia Indica Math., 20(2) (1994), 113-116.
28. H. M. Srivastava, A contour integral involving Fox's H-function, Indian J. Math., 14 (1972), 1-6.
29. H. M. Srivastava and R. Panda, Some bilateral generating functions for a class of generalized hypergeometric polynomials, J. Reine, Angew, Math., 283/284 (1976), 265-274. https://doi.org/10.1515/crll.1976.283-284.265
30. N. Sudland, B. Baulmann and T. F. Nonnenmacher, Open problem: who knows about the Aleph (ℵ)-function?, Fract. Calc. Appl. Anal., 1(4), (1998), 401-402.
2. F. Y. Ayant and D. Kumar, Certain finite double integrals involving the hypergeometric function and Aleph-function, International Journal of Mathematics Trends and Technology, 35(1) (2016), 49-55. https://doi.org/10.14445/22315373/IJMTT-V35P508
3. F. Y. Ayant and D. Kumar, Generating relations and multivariable Aleph-function, Analysis, 38(3) (2018), 137-143. https://doi.org/10.1515/anly-2017-0054
4. F. Y. Ayant and D. Kumar, Fredholm type integral equation with special functions, Acta Universitatis Sapientiae Mathematica, 10(1) (2018), 5-17. https://doi.org/10.2478/ausm-2018-0001
5. J. Choi, J. Daiya, D. Kumar and R. K. Saxena, Fractional differentiation of the product of Appell function F3 and multivariable H-function, Commun. Korean Math. Soc., 31(1) (2016), 115-12 https://doi.org/10.4134/CKMS.2016.31.1.115
6. J. Daiya, J. Ram and D. Kumar, The multivariable H-function and the general class of Srivastava polynomials involving the generalized Mellin-Barnes contour integrals, FILOMAT Journal, 30(6) (2016), 1457-1464. https://doi.org/10.2298/FIL1606457D
7. G. Dorrego and D. Kumar, A generalization of the kinetic equation using the Prabhakar-type operators, Honam Mathematical J., 39(3) (2017), 401-416.
8. A. Erdelyi, On some functional transformations, Univ. Politec. Torino, Rend. Sem. Mat., 10 (1950), 217-234.
9. D. Kumar, Generalized fractional differintegral operators of the Aleph-function of two variables, Journal of Chemical, Biological and Physical Sciences, Section C, 6(3) (2016), 1116-1131.
10. D. Kumar, F. Y. Ayant and J. Choi, Application of product of the multivariable A-function and the multivariable Srivastava polynomials, East Asian Math. J., 34(3) (2018), 295-303.
11. D. Kumar and J. Choi, Certain generalized fractional differentiation of the product of two ℵ-functions associated with the Appell function F3, Applied Mathematical Sciences, 10(4) (2016), 187-196. https://doi.org/10.12988/ams.2016.511709
12. D. Kumar and J. Daiya, Fractional calculus pertaining to generalized H-functions, Global Journal of Science Frontier Research: F, Mathematics and Decision Sciences, 14(3) (2014), 25-36.
13. D. Kumar, S. D. Purohit and J. Choi, Generalized fractional integrals involving product of multivariable H-function and a general class of polynomials, J. Nonlinear Sci. Appl., 9 (2016), 8-21. https://doi.org/10.22436/jnsa.009.01.02
14. D. Kumar, S. D. Purohit, A. Secer and A. Atangana, On generalized fractional kinetic equations involving generalized Bessel function of the first kind, Mathematical Problems in Engineering, Article ID 289387, 2015, (2015), 7 pp. https://doi.org/10.1155/2015/289387
15. D. Kumar, R. K. Saxena and J. Ram, Finite integral formulas involving Aleph function, Boletim da Sociedade Paranaense de Matematica, 36(1) (2018), 177-193. https://doi.org/10.5269/bspm.v36i1.28123
16. E. R. Love, Some integral equations involving hypergeometric functions, J. Edinburgh Math. Soc. 3(15) (1967), 169-198. https://doi.org/10.1017/S0013091500011706
17. S. D. Purohit, Solutions of fractional partial differential equations of quantum mechanics, Adva. Appl. Math. Mech. 5 (2013), 639-651. https://doi.org/10.4208/aamm.12-m1298
18. S. D. Purohit and S.L. Kalla, On fractional partial differential equations related to quantum mechanics, J. Phys. A: Math. Theor., 44(4) (2011), 8 pp. https://doi.org/10.1088/1751-8113/44/4/045202
19. M. Saigo, R. K. Saxena and J. Ram, On the fractional calculus operator associated with the H-function, Ganita Sandesh, 6(1) (1992), 36-47.
20. V. P. Saxena, The I-function, Anamaya Publishers, New Delhi, 2008.
21. R. K. Saxena and V.S. Kiryakova, On relation between the two-dimensional H-transforms in terms of Erd'elyi-Kober operators, Math. Balkanica, 6 (1992), 133-140.
22. R. K. Saxena and D. Kumar, Generalized fractional calculus of the Aleph function involving a general class of polynomials, Acta Mathematica Scientia, 35(5) (2015), 1095-1110. https://doi.org/10.1016/S0252-9602(15)30042-4
23. R. K. Saxena and R. K. Kumbhat, Fractional integration operators of two variables, Proc. Indian Acad. Sci., 78 (1973), 177-186. https://doi.org/10.1007/BF03049477
24. R. K. Saxena and R. K. Kumbhat, Integral operators involving H-function, Indian J. of Pure Appl. Math., 5 (1974), 1-6.
25. R. K. Saxena and R. K. Kumbhat, Some properties of generalized Kober operators, Vijnana Parishad Anusandhan Patrika, 18 (1975), 139-150.
26. R. K. Saxena, J. Ram and D. Kumar, Generalized fractional integral of the product of two Aleph-functions, Applications and Applied Mathematics, 8(2) (2013), 631-646.
27. C. K. Sharma and S.S. Ahmad, On the multivariable I-function, Acta ciencia Indica Math., 20(2) (1994), 113-116.
28. H. M. Srivastava, A contour integral involving Fox's H-function, Indian J. Math., 14 (1972), 1-6.
29. H. M. Srivastava and R. Panda, Some bilateral generating functions for a class of generalized hypergeometric polynomials, J. Reine, Angew, Math., 283/284 (1976), 265-274. https://doi.org/10.1515/crll.1976.283-284.265
30. N. Sudland, B. Baulmann and T. F. Nonnenmacher, Open problem: who knows about the Aleph (ℵ)-function?, Fract. Calc. Appl. Anal., 1(4), (1998), 401-402.
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