Diverse Factorial Operators - wseas

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[12] Weisstein Eric W., Factorial Prime, MathWorld publisher (on the internet: http:/ / mathworld. wolfram. com/ FactorialPrime. html). Applied Mathematics inΒ ...
Applied Mathematics in Electrical and Computer Engineering

Diverse Factorial Operators 1

CLAUDE ZIAD BAYEH1, 2

Faculty of Engineering II, Lebanese University 2 EGRDI transaction on mathematics (2002) LEBANON Email: [email protected]

NIKOS E.MASTORAKIS

WSEAS (Research and Development Department) http://www.worldses.org/research/index.html Agiou Ioannou Theologou 17-2315773, Zografou, Athens,GREECE [email protected] Abstract: -The Diverse factorial operators is an original study developed by the first author in the mathematical domain, it is similar to the ordinary Factorial numbers (n!) but it is more sophisticated and more general than the former. The Diverse factorial operators will be encountered in many different areas of mathematics, notably in combinatorics, algebra and mathematical analysis. The main idea of introducing the Diverse factorial operation is to replace the operation (*) between numbers by another operation, such as (*, +, -, and /). For example, n!^(+) is equal to n+(n-1)+(n-2)+(n-3)…+1, the same thing can be applied for the other operators. Moreover, the Diverse factorial operators is not limited to introducing only different operators, but it is designed to change the sequence of the operational numbers, for example, n!^(i+) is equal to n+(n-1*i)+(n2*i)+(n-3*i)…(n-j*i), with j 0 π‘₯π‘₯ ⟹ 𝑖𝑖 βˆ™ 𝑛𝑛 < π‘₯π‘₯ ⟹ 𝑖𝑖 < with 𝑖𝑖 is the highest integer

The N-Multiplication factorial function is defined as following: π‘₯π‘₯!π‘›π‘›βˆ— = π‘₯π‘₯ βˆ™ (π‘₯π‘₯ βˆ’ 𝑛𝑛) βˆ™ (π‘₯π‘₯ βˆ’ 2𝑛𝑛) βˆ™ (π‘₯π‘₯ βˆ’ 3𝑛𝑛) βˆ™ (π‘₯π‘₯ βˆ’ 4𝑛𝑛) βˆ™ (π‘₯π‘₯ βˆ’ 5𝑛𝑛) βˆ™ … π‘šπ‘š with π‘šπ‘š β‰₯ 1

value for 𝑖𝑖
0 π‘₯π‘₯ ⟹ 𝑖𝑖 βˆ™ 𝑛𝑛 < π‘₯π‘₯ ⟹ 𝑖𝑖 < with 𝑖𝑖 is the highest integer value for 𝑖𝑖 < Therefore: π‘›π‘›βˆ—

π‘₯π‘₯!

𝑛𝑛

𝑖𝑖

With 𝑖𝑖 is the highest integer value for 𝑖𝑖
0 π‘₯π‘₯ ⟹ 𝑖𝑖 βˆ™ 𝑛𝑛 < π‘₯π‘₯ ⟹ 𝑖𝑖 < with 𝑖𝑖 is the highest integer π‘₯π‘₯

(π‘₯π‘₯βˆ’3) (π‘₯π‘₯βˆ’4) (π‘₯π‘₯βˆ’7)

βˆ™

Formally it is defined as: (π‘₯π‘₯ βˆ’ 3)!4βˆ— π‘₯π‘₯!4βˆ— βˆ™ π‘₯π‘₯!/ = (π‘₯π‘₯ βˆ’ 1)!4βˆ— (π‘₯π‘₯ βˆ’ 2)!4βˆ—

2.6 Definition 6:

value for 𝑖𝑖