Metallic Ratios and Ramanujan Summation

International Journal of Mathematics Trends and Technology (IJMTT)
© 2021 by IJMTT Journal
Volume-67 Issue-8
Year of Publication : 2021
Authors : Dr. R. Sivaraman


MLA Style: Dr. R. Sivaraman"Metallic Ratios and Ramanujan Summation" International Journal of Mathematics Trends and Technology 67.8 (2021):133-141. 

APA Style: Dr. R. Sivaraman(2021). Metallic Ratios and Ramanujan Summation  International Journal of Mathematics Trends and Technology, 67(8), 133-141.

Metallic Ratios are class of numbers which are irrationals. The well known Golden Ratio and Silver Ratio are special cases of sequences of Metallic Ratios. In this paper, after introducing Metallic Ratios formally, I have proved some interesting inequalities for powers of metallic ratios whose lower and upper bounds will be connected to Ramanujan Summation method leading to very interesting and new results. In this paper, I had extended the concept of Ramanujan Summation technique to the bounds of powers of metallic ratios. In Ramanujan Summation Method, Ramanujan showed that Ramanujan Sum of all even powers of positive integers is always zero. Similar to this, I had proved that Ramanaujan Summation of all even powers of lower bounds of Metallic ratios of order k is always -1/2. The result regarding computation of upper bounds of Metallic ratios through two previous lower bounds has been established. This result enables us to compute the Ramanujan Summation of Upper bounds of Metallic Ratios in terms of the corresponding Lower bounds. Further, I had shown that the Ramanujan Summation of Upper bounds of Metallic ratios of order 2r + 1 are equal to Upper bounds of Metallic ratios of order 2r + 2. In the final section, the computation of Ramanujan Summation values of lower and upper bounds for first eight powers of metallic powers were carried out. These values verify the theorems proved in this paper. The whole idea of assigning Ramanujan Summation to bounds of Metallic Ratios is very new and so the results obtained in this paper provide great insights and opens great scope towards understanding the behavior of metallic ratios.


[1] R. Sivaraman, Understanding Ramanujan Summation, International Journal of Advanced Science and Technology (IJAST), 29(7) (2020) 1472 – 1485.
[2] R. Sivaraman, Remembering Ramanujan, Advances in Mathematics: Scientific Journal, 9(1) (2020) 489–506.
[3] 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) Springer Nature, 475 – 484.
[4] Juan B. Gil and Aaron Worley, Generalized Metallic Means, Fibonacci Quarterly, 57(1) (2019) 45-50.
[5] Dann Passoja, Reflections on the Gold, Silver and Metallic Ratios, (2015).
[6] K. Hare, H. Prodinger, and J. Shallit, Three series for the generalized golden mean, Fibonacci Quart. 52(4) (2014) 307–313.
[7] R. Sivaraman, Exploring Metallic Ratios, Mathematics and Statistics, Horizon Research Publications, 8(4) (2020) 388 – 391.
[8] Krcadinac V., A new generalization of the golden ratio. Fibonacci Quarterly, 2006;44(4):335–340.
[9] R. Sivaraman, Relation between Terms of Sequences and Integral Powers of Metallic Ratios, Turkish Journal of Physiotherapy and Rehabilitation, Volume 32, Issue 2, 2021,1308 – 1311.
[10] R. Sivaraman, Expressing Numbers in terms of Golden, Silver and Bronze Ratios, Turkish Journal of Computer and Mathematics Education, Vol. 12, No. 2, (2021), 2876 – 2880.

Keywords : Recurrence Relation, Powers of Metallic Ratios, Lower and Upper Bounds, Mathematical Induction, Ramanujan Summation.