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Original Article
Introduction to Dominating Sequential Hypergeometric Elementary Functions
Dharmendra Kumar Yadav1
Shivjee Yadav2
1 2 University Department of Mathematics, Lalit Narayan Mithila University, Darbhanga, Bihar, India.
Published Online: January-February 2026
Pages: 65-77
Cite this article
↗ https://www.doi.org/10.59256/ijrtmr.20260601009References
1. Bailey, W. N. (1964). Generalized Hypergeometric Series, Stechert Hafner Service Agency, New York and London.
2. Chaundy, T. W. (1943). An extension of hypergeometric functions (1), Quart. J. Math., Oxford Ser., 14: 55-78.
3. Chaudhary, M. K. & Yadav, D. K. (2024). Study of Hypergeometric Functions in Context of Nonelementary Integrals. Journal of Harbin
Engineering University, 45(11), 442 – 451.
4. Chaudhary, M. K. & Yadav, D. K. (2025). Reintegral Calculus: Research in Integral Calculus, Volume-II, Notion Press, 111 – 119.5. Cherry, G. W. (1985). Integration in finite terms with special functions: the error function. Journal of Symbolic Computation, 1(3), 283-
302.
6. Cherry, G. W. (1986). Integration in finite terms with special functions: the logarithmic integral. SIAM Journal on Computing, 15(1), 1-
21.
7. Courant R. & Robbins H. (2021). What is Mathematics? An Elementary Approach to Ideas and Methods, 2nd India Edition, Oxford
University Press, India, 272-275.
8. Du, Z., Eleftheriou, M., Moreira, J. E. & Yap, C. (2002). Hypergeometric Functions in Exact Geometric Computation, Electronic Notes
in Theoretical Computer Science, 66 (1), 1-12.
9. Gale, A. S. & Watkeys, C. W. (1923). Elementary Functions and Applications, Henry Holt and Company, New York.
10. Hannah, J. P. (2013). Identities for the Gamma and Hypergeometric Functions: An Overview from Euler to the Present (Master Degree
Thesis), University of the Witwatersrand, Johannesberg.
11. Kasper, T. (1980). Integration in finite terms: the Liouville theory. ACM Sigsam Bulletin, 14(4), 2-8.
12. Katz, V. J. (2019). A History of Mathematics, 3rd Edition, Pearson India Education, 156.
13. Marchisotto, E. A., & Zakeri, G. A. (1994). An invitation to integration in finite terms. The College Mathematics Journal, 25(4), 295-308.
14. Muller, N. M. (2006). Elementary Functions: Algorithms and Implementation. Birkhauser, Boston.
15. Risch, R. H. (1969). The problem of integration in finite terms. Transactions of the American Mathematical Society, 139, 167-189.
16. Risch, R. H. (1970). The solution of the problem of integration in finite terms.
17. Risch, R. H. (2022). On the integration of elementary functions which are built up using algebraic operations. In Integration in Finite
Terms: Fundamental Sources (200-216). Cham: Springer International Publishing.
18. Ritt, J. F. (2022). Integration in Finite Terms Liouville’s Theory of Elementary Methods. In Integration in Finite Terms: Fundamental
Sources (31-134). Cham: Springer International Publishing.
19. Rosenlicht, M. (1972). Integration in finite terms. The American Mathematical Monthly, 79(9), 963-972.
20. Sao, G. S. (2021). Special Functions, 3rd Revised Edition, Shree Shiksha Sahitya Prakashan, Meerut, 1-3, 40-45.
21. Sharma, J. N. & Gupta, R. K. (2020). Special Functions, Krishna Prakashan Media (P) Ltd., Meerut, 34th Edition, 70-72.
22. Vygodsky, M. (1987). Mathematical Hand Book: Elementary Mathematics, Mir Publishers, Moscow, 377-378.
23. Wikipedia contributors. (2025, July 26). Closed - form expression. In Wikipedia, The Free Encyclopedia. Retrieved 15:58, July 29, 2025,
from https://en.wikipedia.org/w/index.php?title=Closed-form_expression&oldid=1302649400.
24. Wikipedia contributors. (2025, April 10). Confluent hypergeometric function. In Wikipedia, The Free Encyclopedia. Retrieved 15:59,
July 29, 2025, from https://en.wikipedia.org/w/index.php?title=Confluent_hypergeometric_function&oldid=1284850626.
25. Wikipedia contributors. (2025, July 13). Elementary function. In Wikipedia, The Free Encyclopedia. Retrieved 15:59, July 29, 2025,
from https://en.wikipedia.org/w/index.php?title=Elementary_function&oldid=1300225591.
26. Wikipedia contributors. (2025, May 22). Function (mathematics). In Wikipedia, The Free Encyclopedia. Retrieved 16:00, July 29, 2025,
from https://en.wikipedia.org/w/index.php?title=Function_(mathematics)&oldid=1291690153.
27. Wikipedia contributors. (2025, July 29). Hypergeometric function. In Wikipedia, The Free Encyclopedia. Retrieved 16:01, July 29, 2025,
from https://en.wikipedia.org/w/index.php?title=Hypergeometric_function&oldid=1303082072.
28. Yadav, D. K., Sen, D. K. & Chauhan, R. (2009). Introduction of a Dominating Function, International J. of Math. Sci. & Engg. Appls.
(IJMSEA), 3(3), 121 – 132.
29. Yadav, D. K. & Sen, D. K. (2012). Dominating Sequential Functions, International J. of Math. Sci. & Engg. Appls. (IJMSEA), 6(1), 391
– 401.
30. Yadav, D. K. (2015). Early basic foundations of modern integral calculus, International Journal of Education and Science Research
Review, 2(2), 37 – 44.
31. Yadav, D. K. & Sen, D. K. (2017). Dominating Sequential Functions: Superset of Elementary Functions, GRIN Verlag Publishing,
Germany, 23 – 32.
32. Yadav, S. & Yadav, D. K. (2024). Review of Nonelementary Integrals in Context of Hypergeometric Functions. Journal of Computational
Analysis and Applications, 33 (7), 1706 – 1723.
33. Yadav, D. K. & Sen, D. K. (2025). Reintegral Calculus: Research in Integral Calculus, Volume-I, Notion Press, 135 – 146.
34. Yadav, S. & Yadav, D. K. (2025a). Review of Dominating Elementary Functions and their Representation in terms of Hypergeometric
Functions, Journal of Harbin Engineering University, 46(8), 1-8.
35. Yadav, S. & Yadav, D. K. (2025b). Dominating Sequential Functions in Context of Nonelementary and Hypergeometric Functions,
Journal of Harbin Engineering University, 46(9), 55-62.
36. Yadav, D. K. & Yadav, S. (2025c). Classical Nonelementary Integrals and Dominating Sequential Functions, Asian Journal of
Mathematics, 21(10), 129-143.
2. Chaundy, T. W. (1943). An extension of hypergeometric functions (1), Quart. J. Math., Oxford Ser., 14: 55-78.
3. Chaudhary, M. K. & Yadav, D. K. (2024). Study of Hypergeometric Functions in Context of Nonelementary Integrals. Journal of Harbin
Engineering University, 45(11), 442 – 451.
4. Chaudhary, M. K. & Yadav, D. K. (2025). Reintegral Calculus: Research in Integral Calculus, Volume-II, Notion Press, 111 – 119.5. Cherry, G. W. (1985). Integration in finite terms with special functions: the error function. Journal of Symbolic Computation, 1(3), 283-
302.
6. Cherry, G. W. (1986). Integration in finite terms with special functions: the logarithmic integral. SIAM Journal on Computing, 15(1), 1-
21.
7. Courant R. & Robbins H. (2021). What is Mathematics? An Elementary Approach to Ideas and Methods, 2nd India Edition, Oxford
University Press, India, 272-275.
8. Du, Z., Eleftheriou, M., Moreira, J. E. & Yap, C. (2002). Hypergeometric Functions in Exact Geometric Computation, Electronic Notes
in Theoretical Computer Science, 66 (1), 1-12.
9. Gale, A. S. & Watkeys, C. W. (1923). Elementary Functions and Applications, Henry Holt and Company, New York.
10. Hannah, J. P. (2013). Identities for the Gamma and Hypergeometric Functions: An Overview from Euler to the Present (Master Degree
Thesis), University of the Witwatersrand, Johannesberg.
11. Kasper, T. (1980). Integration in finite terms: the Liouville theory. ACM Sigsam Bulletin, 14(4), 2-8.
12. Katz, V. J. (2019). A History of Mathematics, 3rd Edition, Pearson India Education, 156.
13. Marchisotto, E. A., & Zakeri, G. A. (1994). An invitation to integration in finite terms. The College Mathematics Journal, 25(4), 295-308.
14. Muller, N. M. (2006). Elementary Functions: Algorithms and Implementation. Birkhauser, Boston.
15. Risch, R. H. (1969). The problem of integration in finite terms. Transactions of the American Mathematical Society, 139, 167-189.
16. Risch, R. H. (1970). The solution of the problem of integration in finite terms.
17. Risch, R. H. (2022). On the integration of elementary functions which are built up using algebraic operations. In Integration in Finite
Terms: Fundamental Sources (200-216). Cham: Springer International Publishing.
18. Ritt, J. F. (2022). Integration in Finite Terms Liouville’s Theory of Elementary Methods. In Integration in Finite Terms: Fundamental
Sources (31-134). Cham: Springer International Publishing.
19. Rosenlicht, M. (1972). Integration in finite terms. The American Mathematical Monthly, 79(9), 963-972.
20. Sao, G. S. (2021). Special Functions, 3rd Revised Edition, Shree Shiksha Sahitya Prakashan, Meerut, 1-3, 40-45.
21. Sharma, J. N. & Gupta, R. K. (2020). Special Functions, Krishna Prakashan Media (P) Ltd., Meerut, 34th Edition, 70-72.
22. Vygodsky, M. (1987). Mathematical Hand Book: Elementary Mathematics, Mir Publishers, Moscow, 377-378.
23. Wikipedia contributors. (2025, July 26). Closed - form expression. In Wikipedia, The Free Encyclopedia. Retrieved 15:58, July 29, 2025,
from https://en.wikipedia.org/w/index.php?title=Closed-form_expression&oldid=1302649400.
24. Wikipedia contributors. (2025, April 10). Confluent hypergeometric function. In Wikipedia, The Free Encyclopedia. Retrieved 15:59,
July 29, 2025, from https://en.wikipedia.org/w/index.php?title=Confluent_hypergeometric_function&oldid=1284850626.
25. Wikipedia contributors. (2025, July 13). Elementary function. In Wikipedia, The Free Encyclopedia. Retrieved 15:59, July 29, 2025,
from https://en.wikipedia.org/w/index.php?title=Elementary_function&oldid=1300225591.
26. Wikipedia contributors. (2025, May 22). Function (mathematics). In Wikipedia, The Free Encyclopedia. Retrieved 16:00, July 29, 2025,
from https://en.wikipedia.org/w/index.php?title=Function_(mathematics)&oldid=1291690153.
27. Wikipedia contributors. (2025, July 29). Hypergeometric function. In Wikipedia, The Free Encyclopedia. Retrieved 16:01, July 29, 2025,
from https://en.wikipedia.org/w/index.php?title=Hypergeometric_function&oldid=1303082072.
28. Yadav, D. K., Sen, D. K. & Chauhan, R. (2009). Introduction of a Dominating Function, International J. of Math. Sci. & Engg. Appls.
(IJMSEA), 3(3), 121 – 132.
29. Yadav, D. K. & Sen, D. K. (2012). Dominating Sequential Functions, International J. of Math. Sci. & Engg. Appls. (IJMSEA), 6(1), 391
– 401.
30. Yadav, D. K. (2015). Early basic foundations of modern integral calculus, International Journal of Education and Science Research
Review, 2(2), 37 – 44.
31. Yadav, D. K. & Sen, D. K. (2017). Dominating Sequential Functions: Superset of Elementary Functions, GRIN Verlag Publishing,
Germany, 23 – 32.
32. Yadav, S. & Yadav, D. K. (2024). Review of Nonelementary Integrals in Context of Hypergeometric Functions. Journal of Computational
Analysis and Applications, 33 (7), 1706 – 1723.
33. Yadav, D. K. & Sen, D. K. (2025). Reintegral Calculus: Research in Integral Calculus, Volume-I, Notion Press, 135 – 146.
34. Yadav, S. & Yadav, D. K. (2025a). Review of Dominating Elementary Functions and their Representation in terms of Hypergeometric
Functions, Journal of Harbin Engineering University, 46(8), 1-8.
35. Yadav, S. & Yadav, D. K. (2025b). Dominating Sequential Functions in Context of Nonelementary and Hypergeometric Functions,
Journal of Harbin Engineering University, 46(9), 55-62.
36. Yadav, D. K. & Yadav, S. (2025c). Classical Nonelementary Integrals and Dominating Sequential Functions, Asian Journal of
Mathematics, 21(10), 129-143.
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