Mathematics and Statistics Vol. 9(3), pp. 209 - 217
DOI: 10.13189/ms.2021.090301
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On Some Properties of Leibniz's Triangle

R. Sivaraman *
Department of Mathematics, D. G. Vaishnav College, Chennai, India


One of the Greatest mathematicians of all time, Gotfried Leibniz, introduced amusing triangular array of numbers called Leibniz's Harmonic triangle similar to that of Pascal's triangle but with different properties. I had introduced entries of Leibniz's triangle through Beta Integrals. In this paper, I have proved that the Beta Integral assumption is exactly same as that of entries obtained through Pascal's triangle. The Beta Integral formulation leads us to establish several significant properties related to Leibniz's triangle in quite elegant way. I have shown that the sum of alternating terms in any row of Leibniz's triangle is either zero or a Harmonic number. A separate section is devoted in this paper to prove interesting results regarding centralized Leibniz's triangle numbers including obtaining a closed expression, the asymptotic behavior of successive centralized Leibniz's triangle numbers, connection between centralized Leibniz's triangle numbers and Catalan numbers as well as centralized binomial coefficients, convergence of series whose terms are centralized Leibniz's triangle numbers. All the results discussed in this section are new and proved for the first time. Finally, I have proved two exceedingly important theorems namely Infinite Hockey Stick theorem and Infinite Triangle Sum theorem. Though these two theorems were known in literature, the way of proving them using Beta Integral formulation is quite new and makes the proof short and elegant. Thus, by simple re-formulation of entries of Leibniz's triangle through Beta Integrals, I have proved existing as well as new theorems in much compact way. These ideas will throw a new light upon understanding the fabulous Leibniz's number triangle.

Leibniz's Triangle, Pascal's Triangle, Harmonic Numbers, Centralized Leibniz's Triangle numbers, Inverted Infinite Hockey Stick Property, Infinite Triangle Sum Property

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] R. Sivaraman , "On Some Properties of Leibniz's Triangle," Mathematics and Statistics, Vol. 9, No. 3, pp. 209 - 217, 2021. DOI: 10.13189/ms.2021.090301.

(b). APA Format:
R. Sivaraman (2021). On Some Properties of Leibniz's Triangle. Mathematics and Statistics, 9(3), 209 - 217. DOI: 10.13189/ms.2021.090301.