# P nomial triangle

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${\displaystyle p-}$nomial triangles, where ${\displaystyle 1 < p \in \mathbb{N}}$, are triangular arranged arrays of specified binomial coefficients. Although for ${\displaystyle p = 1}$ such triangles can be constructed as well, they are usually not considered. The most common cases are for ${\displaystyle p = 2}$, which are known as binomial ${\displaystyle -}$ and for ${\displaystyle p = 3}$, which are called trinomial triangles; less known are so called monomial triangles where ${\displaystyle p = 1}$.

The rows of a ${\displaystyle p-}$nomial triangle are conventionally enumerated starting with row ${\displaystyle n = 0}$ at the top (the zeroth row). The entries in each row are numbered from the left beginning with ${\displaystyle k = 0}$ and are usually staggered relative to the numbers in the adjacent rows. The ${\displaystyle p-}$nomial triangle is constructed in the following manner: in row zero (the topmost row), there is a unique entry, which is conventionally set equal to zero or one, but it can take any arbitrary (complex number) value; the same is true for the first row. Those are usually called the starting values, whereby their amount are determined by ${\displaystyle p}$. Each entry of each subsequent row then is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as zero. Thus, each entry after the starting values depended on the previous.

Although ${\displaystyle p-}$nomial triangles seem similar to alternating ${\displaystyle k-}$forms, they are not the same.

Positive valued ${\displaystyle p-}$nomial triangles represent ${\displaystyle p-}$nomial distributions, which can be visualized by graphs when plotted. Therefore, ${\displaystyle p-}$nomial triangles connect number theory with combinatorics and probability calculus.

## Definition

Let ${\displaystyle 1 < p \in \mathbb{N}}$ and ${\displaystyle n}$, ${\displaystyle k_{i} \in \mathbb{Z}}$ for all ${\displaystyle i}$. Then we denote with:

${\displaystyle T_{\operatorname{com}}^{p}}$

the set of all possible ${\displaystyle p-}$nomial triangles. An element, respectively a ${\displaystyle p-}$nomial triangle, ${\displaystyle Q^{p}}$ of the set ${\displaystyle T_{\operatorname{com}}^{p}}$ we denote with:

${\displaystyle Q^{p} := \displaystyle\binom{\pm n}{\pm k_{1}\text{, } \pm k_{2}, \dots \text{, } \pm k_{p}}_{\operatorname{com}_{nk_{i}}^{p}} \in \mathbb{C}}$

whereby for ${\displaystyle p = 2}$ we have the binomial coefficients. The ${\displaystyle p-}$nomial unit triangle we denote with:

${\displaystyle U^{p} := \displaystyle\binom{\pm n}{\pm k_{1}\text{, } \pm k_{2}, \dots \text{, } \pm k_{p}} = 1}$

and with:

${\displaystyle O^{p} := \displaystyle\binom{\pm n}{\pm k_{1}\text{, } \pm k_{2}, \dots \text{, } \pm k_{p}} = 0}$

we denote the ${\displaystyle p-}$nomial zero triangle for all ${\displaystyle n}$ and ${\displaystyle k_{i}}$, whereby ${\displaystyle k_{i}}$ is a well defined sequence sufficing the condition:

${\displaystyle \displaystyle\sum\limits_{i = 1}^{p}~\vert k_{i} \vert = n}$

## Remark: monomial triangles

We point out, that it is of course possible to construct triangles with ${\displaystyle p = 1}$. They represent the simple powers of:

${\displaystyle x^{n}_{1} = \displaystyle\sum\limits_{k = 0}^{n}~\displaystyle\frac{1}{2^{n}}~\binom{n}{k}~x^{n - k}x^{k}}$

whereby the right hand side follows after the binomial theorem. We only renounce such triangles here; nonetheless, the proofs hold for ${\displaystyle p = 1}$.

## Theorem: Operations on ${\displaystyle p-}$nomial triangles

Let ${\displaystyle A}$ and ${\displaystyle B}$ be both some ${\displaystyle p-}$nomial triangles, whereby ${\displaystyle 1 < p \in \mathbb{N}}$. Then the operations:

${\displaystyle \begin{array}{lll} A \bigoplus B := C & \wedge & A \ominus B := D \\ & & \\ A \bigotimes B := E & \wedge & A \oslash B := F\\ \end{array}}$

are well defined, whereby each operation is taken row by row, respectively ${\displaystyle p-}$nomial coefficient by ${\displaystyle p-}$nomial coefficient, the ${\displaystyle p-}$nomial triangle ${\displaystyle C}$, ${\displaystyle D}$ and ${\displaystyle E}$ exists, and ${\displaystyle F}$ only exists if no entry in ${\displaystyle B}$ equals zero.

#### Proof

It is a enough to give an example for each operation. Then the rest follows by induction. Let now ${\displaystyle A}$ be Egreteau's ${\displaystyle -}$ and ${\displaystyle B}$ equal Pascal's triangle, which we denote with ${\displaystyle E}$, respectively with ${\displaystyle P}$.

#### Example: ${\displaystyle [E \oplus P]}$

For ${\displaystyle E \bigoplus P = C}$ we can write also:

${\displaystyle \displaystyle\binom{n}{k}_{E} \bigoplus \displaystyle\binom{n}{k} = \displaystyle\binom{n}{k}_{C}}$

and for ${\displaystyle C}$ we have/this results to:

${\displaystyle \begin{array}{ccccccccccccc} & & & & & & 1 & & & & & & \\ & & & & & & & & & & & & \\ & & & & & 0 & & 0 & & & & & \\ & & & & & & & & & & & & \\ & & & & 0 & & 2 & & 0 & & & & \\ & & & & & & & & & & & & \\ & & & 0 & & 4 & & 2 & & 0 & & & \\ & & & & & & & & & & & & \\ & & 0 & & 6 & & 4 & & 4 & & 0 & & \\ & & & & & & & & & & & & \\ & 0 & & 8 & & 6 & & 12 & & 4 & & 0 & \\ & & & & & & & & & & & & \\ \dots & & \dots & & \dots & & \dots & & \dots & & \dots & & \dots\\ \end{array}}$

#### Example: ${\displaystyle [E \ominus P]}$

For ${\displaystyle E \ominus P = D}$ we can write also:

${\displaystyle \displaystyle\binom{n}{k}_{E} \ominus \displaystyle\binom{n}{k} = \displaystyle\binom{n}{k}_{D}}$

and for ${\displaystyle D}$ we have/this results to:

${\displaystyle \begin{array}{ccccccccccccc} & & & & & & -1 & & & & & & \\ & & & & & & & & & & & & \\ & & & & & -2 & & -2 & & & & & \\ & & & & & & & & & & & & \\ & & & & -2 & & -2 & & -2 & & & & \\ & & & & & & & & & & & & \\ & & & -2 & & -2 & & -4 & & -2 & & & \\ & & & & & & & & & & & & \\ & & -2 & & -2 & & -8 & & -4 & & -2 & & \\ & & & & & & & & & & & & \\ & -2 & & -2 & & -14 & & -8 & & -6 & & -2 & \\ & & & & & & & & & & & & \\ \dots & & \dots & & \dots & & \dots & & \dots & & \dots & & \dots\\ \end{array}}$

#### Example: ${\displaystyle [E \otimes P]}$

For ${\displaystyle E \bigotimes P = G}$ we can write also:

${\displaystyle \displaystyle\binom{n}{k}_{E} \bigotimes \displaystyle\binom{n}{k} = \displaystyle\binom{n}{k}_{G}}$

and for ${\displaystyle G}$ we have/this results to:

${\displaystyle \begin{array}{ccccccccccccc} & & & & & & 0 & & & & & & \\ & & & & & & & & & & & & \\ & & & & & -1 & & -1 & & & & & \\ & & & & & & & & & & & & \\ & & & & -1 & & 0 & & -1 & & & & \\ & & & & & & & & & & & & \\ & & & -1 & & 3 & & -3 & & -1 & & & \\ & & & & & & & & & & & & \\ & & -1 & & 8 & & -12 & & 0 & & -1 & & \\ & & & & & & & & & & & & \\ & -1 & & 15 & & -40 & & 20 & & -5 & & -1 & \\ & & & & & & & & & & & & \\ \dots & & \dots & & \dots & & \dots & & \dots & & \dots & & \dots\\ \end{array}}$

#### Example: ${\displaystyle [E \oslash P]}$

For ${\displaystyle E \oslash P = F}$ we can write also:

${\displaystyle \displaystyle\binom{n}{k}_{E} \oslash \displaystyle\binom{n}{k} = \displaystyle\binom{n}{k}_{F}}$

and for ${\displaystyle F}$ we have/this results to:

${\displaystyle \begin{array}{ccccccccccccc} & & & & & & 0 & & & & & & \\ & & & & & & & & & & & & \\ & & & & & -1 & & -1 & & & & & \\ & & & & & & & & & & & & \\ & & & & -1 & & 0 & & -1 & & & & \\ & & & & & & & & & & & & \\ & & & -1 & & \displaystyle\frac{1}{3} & & -\displaystyle\frac{1}{3} & & -1 & & & \\ & & & & & & & & & & & & \\ & & -1 & & \displaystyle\frac{1}{2} & & -\displaystyle\frac{1}{3} & & 0 & & -1 & & \\ & & & & & & & & & & & & \\ & -1 & & \displaystyle\frac{3}{5} & & -\displaystyle\frac{2}{5} & & \displaystyle\frac{1}{5} & & -\displaystyle\frac{1}{5} & & -1 & \\ & & & & & & & & & & & & \\ \dots & & \dots & & \dots & & \dots & & \dots & & \dots & & \dots\\ \end{array}}$

It is obvious, that the four operations ${\displaystyle \oplus}$, ${\displaystyle \ominus}$, ${\displaystyle \otimes}$ and ${\displaystyle \oslash}$ hold on ${\displaystyle p-}$nomial triangles with ${\displaystyle \mathbb{N} \ni p > 1}$ as only the amount of entries in a row increase.

## Remark: implication Theorem Operations on ${\displaystyle p-}$nomial triangles

We specifically point out, that the theorem, establishing the operations on ${\displaystyle p-}$nomial triangles, is a much stronger proposition, than a defintion would have been.

## Remark: deduction Theorem Operations on ${\displaystyle p-}$nomial triangles

We can see now, that:

${\displaystyle \begin{array}{ll} E \bigoplus P = C \subset T_{\operatorname{com}}^{p} & E \ominus P = D \subset T_{\operatorname{com}}^{p}\\ & \\ E \bigotimes P = G \subset T_{\operatorname{com}}^{p} & E \oslash P = F \subset T_{\operatorname{com}}^{p}\\ \end{array}}$

each outcome produces a triangle that is a subset of ${\displaystyle T_{\operatorname{com}}^{p}}$ as we have only specified the ${\displaystyle p-}$nomial coefficients and not the patterns, respectively combinations, itself.

## Remark: implication example: ${\displaystyle [E \otimes P]}$

The example: ${\displaystyle [E \otimes P]}$ proves something different, which is the scalar mutliplication of any ${\displaystyle p-}$nomial triangle. It also proves the scalar multiplication theorem of the binomial theorem and it expands it to ${\displaystyle p-}$nomial triangles as well as to the complex numbers.

## Theorem: existence of unit ${\displaystyle -}$ and zero ${\displaystyle p-}$nomial triangles for all ${\displaystyle p}$

Let ${\displaystyle 1 < p \in \mathbb{N}}$. Then it exist a ${\displaystyle p-}$nomial unit triangle such that ${\displaystyle U^{p} \subset T_{\operatorname{com}}^{p}}$ and it exist a ${\displaystyle p-}$nomial zero triangle such that ${\displaystyle O^{p} \subset T_{\operatorname{com}}^{p}}$.

#### Proof

Let ${\displaystyle Q^{p} \subset T_{\operatorname{com}}^{p}}$ be an arbitrary ${\displaystyle p-}$nomial triangle with no zeros in it. After the theorem establishing the operations for ${\displaystyle p-}$nomial triangles, we can simply set then:

${\displaystyle Q^{p} \oslash Q^{p} = U^{p} \Longrightarrow U^{p} \subset T_{\operatorname{com}}^{p}}$


and:

${\displaystyle Q^{p} \ominus Q^{p} = O^{p} \Longrightarrow O^{p} \subset T_{\operatorname{com}}^{p}}$


## Theorem: all possible ${\displaystyle p-}$nomial Triangles

Let ${\displaystyle 1 < p \in \mathbb{N}}$ and let ${\displaystyle n}$, ${\displaystyle k_{i} \in \mathbb{Z}}$ for all ${\displaystyle i}$. Then all possible ${\displaystyle p-}$nomial triangles ${\displaystyle T_{\operatorname{com}}^{p}}$ can be compactified as follows:

${\displaystyle \begin{array}{ll} \displaystyle\binom{\pm n}{\pm k_{i}}_{\operatorname{com}_{nk_{i}}^{p}} & := \operatorname{com}_{nk}^{p} \in \mathbb{C} \\ \end{array}}$

whereby ${\displaystyle k_{i}}$ is a sequence with ${\displaystyle p}$ elements.

#### Proof

We can simply construct every possible ${\displaystyle p-}$nomial triangle by setting the ${\displaystyle p-}$nomial coefficients to:

${\displaystyle \begin{array}{llllll} \displaystyle\binom{0}{0}_{\operatorname{com}_{00}^{p}} & := 0 & \displaystyle\binom{\pm n}{0}_{\operatorname{com}_{n0}^{p}} & := \operatorname{com}_{n0}^{p} & \displaystyle\binom{\pm n}{\pm n}_{\operatorname{com}_{nn}^{p}} & := \operatorname{com}_{nn}^{p} \\ \end{array}}$

and:

${\displaystyle \begin{array}{llllll} \displaystyle\binom{\pm n}{\pm k_{i}}_{\operatorname{com}_{nk_{i}}^{p}} & := \operatorname{com}_{nk_{i}}^{p} \\ \end{array}}$

whereby it suffices:

${\displaystyle \displaystyle\sum\limits_{i = 1}^{p}~\vert k_{i} \vert = n}$

Together (but not necessarily) with either rule:

${\displaystyle \begin{array}{llll} \displaystyle\binom{n + 1}{k_{i} + 1}_{\operatorname{com}_{nk_{i}}^{p}} & = \displaystyle\binom{n}{k_{i}}_{\operatorname{com}_{nk_{i}}^{p}} & \pm & \displaystyle\binom{n}{k_{i} + 1}_{\operatorname{com}_{nk_{i}}^{p}} \\ & & & \\ \displaystyle\binom{-n - 1}{-k_{i} - 1}_{\operatorname{com}_{nk_{i}}^{p}} & = \displaystyle\binom{-n}{-k_{i}}_{\operatorname{com}_{nk_{i}}^{p}} & \pm & \displaystyle\binom{-n}{-k_{i} - 1}_{\operatorname{com}_{nk_{i}}^{p}} \\ \end{array}}$

and or with the operations:

${\displaystyle \begin{array}{lllllll} \bigoplus : T_{\operatorname{com}}^{p} & \times & T_{\operatorname{com}}^{p} \to T_{\operatorname{com}}^{p} & \wedge & \ominus : T_{\operatorname{com}}^{p} & \times & T_{\operatorname{com}}^{p} \to T_{\operatorname{com}}^{p}\\ & & & & \\ \bigotimes : \mathbb{R} & \times & T_{\operatorname{com}}^{p} \to T_{\operatorname{com}}^{p} & \wedge & \oslash : \mathbb{R} & \times & T_{\operatorname{com}}^{p} \to T_{\operatorname{com}}^{p}\\ \end{array}}$

every possible triangle can be constructed, since ${\displaystyle \operatorname{com}_{nk_{i}}^{p}}$ can be any sequence or combination and the operations produce even triangles that can not be constructed other than via a definition.