Ramanujan Summation for Classic Combinatorial
Problem

Dr. R. Sivaraman

doi:https://doi.org/10.14445/22315373/IJMTT-V67I8P510

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 67 | Issue 8 | Year 2021 | Article Id. IJMTT-V67I8P510 | DOI : https://doi.org/10.14445/22315373/IJMTT-V67I8P510

Ramanujan Summation for Classic Combinatorial Problem

Dr. R. Sivaraman

Citation :

Dr. R. Sivaraman, "Ramanujan Summation for Classic Combinatorial Problem," International Journal of Mathematics Trends and Technology (IJMTT), vol. 67, no. 8, pp. 82-87, 2021. Crossref, https://doi.org/10.14445/22315373/IJMTT-V67I8P510

Abstract

The concept of Ramanujan Summation has been dealt with several forms in recent decades. In this paper, I will define Ramanujan summation evaluated through a definite integral and using this, I had computed the Ramanujan summation for the divergent series whose terms represent the maximum number of regions formed by considering n points in the circumference of a circle which are joined by chords. This classic geometric problem along with Ramanujan summation method has produced an interesting and new result which is derived in detail in this paper.

Keywords

Ramanujan Summation, Regions in a Circle, Binomial Coefficients, Newton’s Forward Interpolation Formula, Pascal’s Identity.

References

[1] R. Sivaraman, Understanding Ramanujan Summation, International Journal of Advanced Science and Technology, 29(7) (2020) 1472 – 1485.
[2] R. Sivaraman,Sum of powers of natural numbers, AUT AUT Research Journal, 11(4) (2020) 353 – 359.
[3] S. Ramanujan, Manuscript Book 1 of Srinvasa Ramanujan, First Notebook, Chapter VIII, 66 – 68.
[4] Bruce C. Berndt, Ramanujan’s Notebooks Part II, Springer, Corrected Second Edition, (1999).
[5] G.H. Hardy, J.E. Littlewood, Contributions to the theory of Riemann zeta-function and the theory of distribution of primes, Acta Arithmetica, 41(1) (1916) 119 – 196.
[6] S. Plouffe , Identities inspired by Ramanujan Notebooks II , part 1, July 21 (1998), and part 2, April (2006).
[7] Bruce C. Berndt, An Unpublished Manuscript of Ramanujan on Infinite Series Identities, Illinois University, American Mathematical Society publication
[8] R. Sivaraman, Remembering Ramanujan, Advances in Mathematics: Scientific Journal, 9(1) (2020) 489–506.
[9] R. Sivaraman, Bernoulli Polynomials and Ramanujan Summation, Proceedings of First International Conference on Mathematical Modeling and Computational Science, Advances in Intelligent Systems and Computing, 1292 (2021) 475 – 484.
[10] B. Candelpergher, H. Gopalakrishna Gadiyar, R. Padma, Ramanujan Summation and the Exponential Generating Function, Cornell University, (2009).