On Some Properties of Leibniz's Triangle

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.

Abstract 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.

Introduction
The great German polymath Gotfried Wilhelm Leibniz, who was one of the discoverers of Calculus introduced a number triangle containing unit fractions which satisfy certain recurrence relation condition. Leibniz's triangle described by Leibniz in 1673 contains several interesting properties which were explored by many later mathematicians. During the time Leibniz introduced his harmonic triangle, another mathematician Pietro Mengoli discussed the same version of the triangle for finding quadratures(areas) of a semi-circles with different diameters. But Leibniz defined the elements of the triangle recursively and so the triangle got the name Leibniz's triangle. We may also call Leibniz's triangle as Harmonic Triangle as the outer diagonal of Leibniz's triangle are occupied by Harmonic numbers. In this paper, I will introduce the elements of Leibniz's triangle in a different perspective through Beta Integrals and discuss several properties through this new definition.

Construction of Leibniz's Triangle
The Pascal's triangle consists of entries which are binomial coefficients of the form ( , ) where 0 nm . The Pascal's triangle is displayed in Figure 1. Leibniz's triangle exhibit many exciting properties similar to that of Pascal's triangle. The upcoming sections focus on exhibiting such properties. For doing this, I will introduce entries of Leibniz's triangle in a completely different way.

The Entries
, mn L of Leibniz's Triangle in Figure 3 is Defined by the Integral   Using the relationship between Beta and Gamma Integrals and noting that ( 1) ! k k    for all whole numbers k, we have Lemma 1, thus establishes the fact that the entries of Leibniz's triangle obtained through Pascal's triangle are exactly the same as entries of the reformulation as defined in (4.1).

Properties of Leibniz's Triangle
In view of (4.2), having identified the entries of Leibniz's triangle as the Beta integral, in this section I prove the following interesting and basic properties concerning Leibniz's triangle.

Symmetric Property
The entries of Leibniz's triangle are symmetric with respect to vertical line through the centre i.e. ,, m n n m LL  (5.1) Proof: By (2.3) we know that Beta Integral is symmetric with respect to its indices m and n. Hence, from (4.2), we get , ,

Harmonic Numbers Property
The outermost diagonal entries in the Leibniz's triangle are Harmonic numbers of the form 1 , Hence from (5.2), we see that the outmost diagonal entries of the Leibniz's triangle are Harmonic numbers of the form 1 , 0,1, 2,3, 4,... 1 k k  

Recurrence Relation
The entries of the Leibniz's triangle satisfy the recurrence relation given by This provides (5.3) and completes the proof. Many sources about Leibniz's triangle use the recurrence relation property in (5.3) to generate subsequent entries of Leibniz's triangle.

Infinite Sum Property
The sum of reciprocals of all triangular numbers is 2, i.e.
This completes the proof.

Sum of Alternating Terms
The

Centralized Leibniz's Triangle Numbers
We define numbers of the form , mm L as centralized Leibniz's triangle numbers. Since m + m = 2m is always even, we find from Figure 3 Thus the sum of centralized Leibniz's triangle numbers converges to 2 33  . This proves (6.4) and completes the proof.

Summations in Leibniz's Triangle
In Pascal's triangle there is a wonderful property concerning summing entries in a particular diagonal called Hockey Stick Property. There is a similar version in Leibniz's triangle producing gallery of infinite series summations which I will prove below.

Inverted Infinite Hockey Stick Property
We first look at the two figures 4 and 5 displayed below. In Figures 4 and 5 we see that if we add the numbers in the infinite rectangular strip the sum would be a number indicated in the circle located at North-West direction.
Thus from Figures 4 and 5 we see that We thus see upon applying the recurrence relation property repeatedly, only the first term in the right hand side remains while all other terms cancel out mutually. This kind of summation is often referred as Telescopic Summation. Using the same principle we can select any infinite rectangular strip of numbers in the Leibniz's triangle whose sum would be a number just located above at the north-west corner of the strip. I refer this property as Inverted Infinite Hockey Stick Property. I now provide the general proof of this property.

Infinite Triangle Sum Property
In Pascal's Triangle, we notice a Rhombus Shape as displayed in Figure 6.
As shown in Figure 6, if we add all the numbers inside the rhombus their sum would be 64 = 84 -20, which is the difference of the numbers located just vertically below the rhombus shown in circle and to the right of the number 10. This property of rhombus shape sum holds true for any rhombus taken in Pascal's triangle. Is there a similar property that can be observed in Leibniz's triangle? Figure  7 answers this question.
From Figure 7, we notice that if we add all the numbers inside the triangle, we get their sum as 1 4 .
a number located just above the vertex of the infinite triangle. Like Pascal's triangle, this infinite triangle sum property also holds true for any triangle considered in Leibniz's triangle. I will now formally prove this fact. Similarly, the sum of all the numbers in the infinite triangle shown in Figure 8

Conclusions
Usually, the entries of Leibniz's triangle would either be identified through Recursion or obtained from Pascal's triangle as explained in section 3. But in this paper, I identified the entries of Leibniz's triangle as Beta Integral values which is not done before in any paper. I proved that the Beta Integral assumption is exactly same as that of entries obtained through Pascal's triangle in section 4 through equation (4.4). This equivalence made me establish so many identities in subsequent sections, that too very elegantly.
In section 5, I have proved five results in which the first four are commonly known, but the proofs are given with respect to integrals, which were not done before. The fifth result of section 5, namely sum of alternating terms of entries of Leibniz's triangle is not so well known, again proved easily by using integrals. The six results proved in section 6 are all relatively new and not much known in the literature about them. Hence, they are completely new results of this paper. In particular the proof of sum of all centralized Leibniz's triangle numbers is convergent provides a new dimension to existing properties.
In section 7, I had proved Infinite Hockey Stick Theorem, which is done so easily. As far as I knew there is no such easy way of proving as I did for that theorem. Further the Infinite Hockey Stick Theorem provides us a way of constructing gallery of several infinite series with unit fractions whose sum can be readily obtained by the equation (7.3). Similarly, in section 8, I had proved Infinite triangle sum property whose proof like previous one was done in a single line. Illustrations and Figures were provided for better understanding wherever necessary. Figures 4,5,6,7,8 were incorporated from [11]. This paper, thus contain almost all the known properties as well as unknown properties of the not so well known Leibniz's triangle. Most importantly by observing that the entries are Beta Integral values, I could prove almost every result in this paper quite beautifully and in short. This proves that by thinking little differently we can see deeper truths in a more elegant way.