Number Theory/Egreteau's nabla triangle

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Egreteau's triangle is a different expansion of the binomial coefficients, which allows their use for negative and positive numbers at the same time; together with the subtraction rule, this in return allows for the creation of completely new nomial triangles. The indicates that negative binomial coefficients and the subtraction rule is employed. In contrary to Egreteau's and Pascal's triangle, this binomial triangle starts with a positive and a negative entry.


Definition

The entry in the th row and th column of Egreteau's triangle is denoted by:



We set:



for . Further, Egreteau's triangle fulfills the subtraction rule:



for all , . With the stated binomial coefficients the triangle can be visualized as follows:



which continues with , , ; equivalent to the binomial coefficients are their values, which are as follows:



Characteristics

With the established binomial coefficients and rule, Egreteau's triangle has some analogous characteristics as those triangles we have already encountered. The most obvious one is:



for and:



for . This yields:


for .


Theorem

Let . Then we have:



For and the sum equals obviously to zero.

Proof

This is a direct consequence of:



which creates an alternating pattern of the sign within the triangle. Those values then annihilate each other. Therefore by induction follows automatically the made statement.


Theorem

Let and be the difference operator. Then we have:


Proof

This is a direct consequence of:



By induction follows automatically the made statement.


Remark: representation of Egreteau's triangle

All nomial triangles refer to polynomials. For Egreteau's triangle we have: