Karaji's L-Summing Method and the Generalized Ibn al-Haytham Recursive Formula

Year 2021, , 122 - 128, 30.06.2021

Hossein Teimoori Faal

https://doi.org/10.47000/tjmcs.717074

Abstract

In this paper, we first review two folkloric summation techniques
due to two Islamic medieval scholars Karaji and Ibn al-Haytham by emphasizing on their higher-dimensional extensions. We also include several interesting examples of algebraic identities obtained
form the generalized Ibn al-Haytham formulas and Karaji's
L-summing method. Finally, we
will also show the interplay between these two different methods
of summation.

Keywords

Karaji, the generalized Ibn al-Haytham formula, summation techniques, sums of powers, harmonic numbers

References

• [1] Greene, D. H., Knuth, D. E., Mathematics for the Analysis of Algorithms (3rd Edition), Birkhauser, Boston, 1990.
• [2] Hassani, M., Identities by L-summing method, Int. J. Math. Comput. Sci., 1(2006), 165–172.
• [3] Katz, V. J., Ideas of calculus in Islam and India, Math. Mag., 68(1995), 163–174.
• [4] Teimoori, H., A Generalization of Ibn Al-Haytham recursive formula for sums of powers, Turk. J. Math. Comput. Sci., 1(2018), 165–172.
• [5] Teimoori, H. A Systematic way to prove some classes of integer sequences identities, (in preparation).
• [6] Woepcke, F., Extrait du Fakhr, Traite D’algebre par Abou Bekr Mohammed Ben Alhacan Alkarkhi, Reprint of the 1853 original, Hildesheim, 1982.
• [7] Zeilberger, D., The method of creative telescoping, J. Symbolic Comput., 11(1991), 195–204.