Combinatorial Geometric Series

Chinnaraji Annamalai

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Chinnaraji Annamalai. Combinatorial Geometric Series. 2022. hal-03723640

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# Combinatorial Geometric Series

Chinnaraji Annamalai

School of Management, Indian Institute of Technology, Kharagpur, India

Email: anna@iitkgp.ac.in

https://orcid.org/0000-0002-0992-2584

**Abstract:** This paper presents a geometric series with binomial coefficients or a combinatorial geometric series. The binomial coefficient of combinatorial geometric series is different from the traditional binomial coefficient.

**MSC Classification codes:** 05A10, 40A05 (65B10)

**Keywords:** computation, combinatorics, binomial coefficient

## 1. Introduction

Combinatorial geometric series [1-4] is produced by multiple summations of a geometric series as follows:

$$ \sum_{i_1=0}^{n} \sum_{i_2=i_1}^{n} \sum_{i_3=i_2}^{n} \cdots \sum_{i_r=i_{r-1}}^{n} x^{i_r} = \sum_{i=0}^{n} V_i^{r} x^i. \text{ Here, } \sum_{i=0}^{n} V_i^{r} x^i \text{ is the combinatorial geometric series.} $$

The binomial coefficient of combinatorial geometric series is given below:

$$ V_r^n = \frac{(r+1)(r+2)(r+3)\cdots(r+n)}{n!}, (n \ge 1, r \ge 0 \ \& \ n, r \in \mathbb{N}), $$

where $N = \{0, 1, 2, 3, \cdots\}$ is set of natural numbers including zero element

The traditional binomial coefficient denotes $\binom{n}{r} = \frac{n!}{r!(n-r)!}$, where $n, r \in \mathbb{N}$.

Let us compare the binomial coefficient $V_x^y$ with the traditional binomial coefficient:

$$ \text{Let } z = x + y. \text{ Then, } \binom{z}{x} = zC_x = \frac{z!}{x!y!}. \text{ Here, } V_x^y = V_y^x \Rightarrow zC_x = zC_y, (x, y, z \in \mathbb{N}). $$

$$ \text{For example, } V_3^5 = V_5^3 = (5+3)C_3 = (5+3)C_5 = 56. $$

$$ \text{Also, } V_n^0 = V_n^n = nC_0 = nC_n = \frac{n!}{n!0!} = 1 \text{ and } V_0^0 = 0C_0 = \frac{0!}{0!} = 1 (\because 0! = 1). $$

## 2. Binomial Expansion

The following binomial identity [1, 2] is created using the binomial coefficients of the combinatorial geometric series.

$$ V_0^r + V_1^r + V_2^r + \cdots + V_n^r = V_n^{r+1} \text{ and } V_n^0 + V_n^1 + V_n^2 + \cdots + V_n^r = V_{n+1}^r, (\because V_n^r = V_n^{r+1}). $$

From the binomial identity $V_0^r + V_1^r + V_2^r + \cdots + V_n^r = V_n^{r+1}$, we can derive the following binomial expansions:
$$ (1). \sum_{i=0}^{n} V_i^0 = \sum_{i=0}^{n} \frac{1}{1!} = \frac{(n+1)}{1!}. $$

$$ (2). \sum_{i=0}^{n} V_i^1 = \sum_{i=0}^{n} \frac{(i+1)}{1!} = 1 + 2 + 3 + \cdots + n + (n+1) = \frac{(n+1)(n+2)}{2!}. $$

$$ (3). \sum_{i=0}^{n} V_i^2 = \sum_{i=0}^{n} \frac{(i+1)(i+2)}{2!} = 1 + 3 + \cdots + \frac{(n+1)(n+2)}{2!} = \frac{(n+1)(n+2)(n+3)}{3!}. $$

$$ (4). \sum_{i=0}^{n} V_i^3 = \sum_{i=0}^{n} \frac{(i+1)(i+2)(i+3)}{3!} = \frac{(n+1)(n+2)(n+3)(n+4)}{4!}. $$

$$ (5). \sum_{i=0}^{n} V_i^4 = \sum_{i=0}^{n} \frac{(i+1)(i+2)(i+3)(i+4)}{4!} = \frac{(n+1)(n+2)(n+3)(n+4)(n+5)}{5!}. $$

Similarly, we can continue this process up to r times. The r-th binomial expansion is as follows:

$$ (r). \sum_{i=0}^{n} V_i^r = \sum_{i=0}^{n} \frac{(i+1)(i+2)(i+3)\cdots(i+r)}{r!} = \frac{(n+1)(n+2)\cdots(n+r)(n+r+1)}{(r+1)!}, $$

From the binomial identity $V_n^0 + V_n^1 + V_n^2 + \cdots + V_n^r = V_{n+1}^r$, we can derive the following binomial expansions.

$$ 1 + \frac{(n+1)}{1!} + \frac{(n+1)(n+2)}{2!} + \frac{(n+1)(n+2)(n+3)}{3!} + \cdots + \frac{(n+1)(n+2)\cdots(n+r)}{r!} = \frac{(n+2)(n+3)\cdots(n+r)(n+r+1)}{r!}. $$

### 3. Conclusion

In this article, a combinatorial geometric series or geometric series with binomial coefficients has been developed using the multiple summations of a geometric series. The idea is a sort of methodological advance which is useful for researchers working in science, economics, engineering, computation, and management.

### References

[1] Annamalai, C. (2020) Optimized Computing Technique for Combination in Combinatorics. HAL 02865835. https://hal.archives-ouvertes.fr/hal-02865835.

[2] Annamalai, C. (2020) Novel Computing Technique in Combinatorics. HAL 02862222. https://hal.archives-ouvertes.fr/hal-02862222.

[3] Annamalai, C. (2022) Annamalai's Binomial Identity and Theorem. SSRN 4097907. http://dx.doi.org/10.2139/ssrn.4097907.

[4] Annamalai, C. (2022) My New Idea for Optimized Combinatorial Techniques. SSRN 4078523. http://dx.doi.org/10.2139/ssrn.4078523.