Diverse Factorial Operators

Size: px

Start display at page:

Download "Diverse Factorial Operators"

Eric Mills
5 years ago
Views:

1 Diverse Factorial Operators CLAUDE ZIAD BAYEH 1, 1 Faculty of Engineering II, Lebanese University EGRDI transaction on mathematics (00) LEBANON claude_bayeh_cbegrdi@hotmail.com NIKOS E.MASTORAKIS WSEAS (Research and Development Department) Agiou Ioaou Theologou , Zografou, Athens,GREECE mastor@wseas.org 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-)+(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)+(n- *i)+(n-3*i) (n-j*i), with j<n/i. In Addition, there is a rising up factorial operators which is the inverse of the normal factorial operator, for example, n ^(+) is equal to n+(n+1)+(n+)+(n+3) +(n+m), more details about these new definitions are developed in this paper. Moreover the first author has developed many definitions that might be very useful to be applied in mathematics or any other scientific domain. Key-words:- Diverse factorial operation, double factorial operator, falling factorial operator, rising up factorial operator, mathematics. 1 Introduction In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = The value of 0! is 1, according to the convention for an empty product [1]. The factorial operation is encountered in many different areas of mathematics, notably in combinatorics, algebra and mathematical analysis. Its most basic occurrence is the fact that there are n! ways to arrange n distinct objects into a sequence (i.e., permutations of the set of objects). This fact was known at least as early as the 1th century, to Indian scholars []. The notation n! was introduced by Christian Kramp in 1808 [3]. The definition of the factorial function can also be extended to non-integer arguments, while retaining its most important properties; this involves more advanced mathematics, notably techniques from mathematical analysis. For this moment, many scientists and researchers introduced several forms of factorial operations, as the alternating factorial [10], exponential factorial [11], Factorial Prime [1], and many others [4], [5], [6], [7], [8], [9]. In this paper, the first author introduces and develops a new way of factorial operators which is called Diverse factorial Operator. The concept is similar to the normal factorial operator which is a particular case of the Diverse factorial operator. The difference is explained in details in the following sections. Briefly, there are two major Diverse factorial Operators, the first one is the Diverse falling Factorial Operator (detailed in the section ) and the second one is the Diverse rising Up Factorial Operators (detailed in the section 3). The main goal of introducing these definitions and formulae is to use them in mathematics and in all applied mathematical domains; such as physics, engineering and chemistry. Many researchers will follow this new study in order to find the application of these definitions and formulae in several domains. ISBN:

2 Diverse Falling Factorial Operators Let s consider that xx N and < xx with N.1 Definition 1: The Multiplication factorial function is defined as xx! = xx! = xx (xx 1) (xx ) (xx 3) 1 This is the traditional factorial operator. xx! = xx =1 (1) 5! = = 10 8! = = Definition : The N-Multiplication factorial function is defined as xx! = xx (xx ) (xx ) (xx 3) (xx 4) (xx 5) mm with mm 1 xx! = xx (xx ) (xx ) (xx 3) (xx 4) (xx 5) (xx ) with xx > 0 < xx < xx with is the highest integer value for < xx Therefore: xx! = (xx jj ) With is the highest integer value for < xx () 5! = 5 (5 ) (5 ) = 15 it is similar to the double factorial function for the odd numbers as 5!! = One can deduce that the double factorial function is a particular case of the N-Multiplication factorial function 0! 3 = 0 (0 3) (0 3) (0 3 3) (0 4 3) (0 5 3) (0 6 3) = = 4,188,800.3 Definition 3: The Plus factorial function is defined as xx! + = xx + (xx 1) + (xx ) + (xx 3) xx! + = xx(xx+1) This equation is similar to an existing summation as xx 1 xx! + = (xx ) = =0 xx(xx + 1) (3) 9! + = = 45 1! + = = 78.4 Definition 4: The N-Plus factorial function is defined as xx! + = xx + (xx ) + (xx ) + (xx 3) + (xx 4) + mm with mm 1 xx! + = xx + (xx ) + (xx ) + (xx 3) + (xx 4) + + (xx ) with xx > 0 < xx < xx with is the highest integer value for < xx xx! + = (xx jj ) = ( + 1) xx (4) 9! + = 9 + (9 ) + (9 ) + (9 3 ) + (9 4 ) = = 5 1! 3+ = 1 + (1 3) + (1 3) + (1 3 3) = = 30 The result can be easily determined using the equation (4) developed above. For example, calculate the result of 11! 3+ =?? using the equation (4). We have, xx = 11, = 3 and < 11 = = 37 because is the highest integer value for < xx. Therefore it is easy now to apply the formula, so 11! 3+ = ( + 1) xx = (37 + 1) = Definition 5: The Minus factorial function is defined as xx! = xx (xx 1) (xx ) (xx 3) 1 ISBN:

3 xx! = xx xx(xx 1) (5) 9! = = 7 1! = = 54.6 Definition 6: The N-Minus factorial function is defined as xx! = xx (xx ) (xx ) (xx 3) mm with mm 1 xx! = xx (xx ) (xx ) (xx 3) (xx 4) (xx ) with xx > 0 < xx < xx with is the highest integer value for < xx xx! = xx (xx jj ) = xx(1 ) + jj =1 ( + 1) (6) 9! = 9 (9 ) (9 ) (9 3 ) (9 4 ) = = 7 1! 3 = 1 (1 3) (1 3) (1 3 3) = = 6 The equation (6) can be used in order to obtain the For example calculate 59! =?? Therefore, xx = 59, =, < 59 = 19.5 = ! = xx(1 ) + (+1) = 59(1 19) + 19(19+1) = Definition 7: The Divide factorial function is defined as xx! / = xx/(xx 1)/(xx )/(xx 3)/ /1 = xx (xx 3) (xx 4) (xx 7) (xx 8) (xx 1) (xx ) (xx 5) (xx 6) (xx 9) Remark: do not confuse between the divisions of many numbers for example: 5/4/3//1 is the same as (5/4)*(/3)*(1/1) And 6/5/4/3//1 is the same as (6/5)*(3/4)*(/1) Do not put into your calculator the equation (5/4/3//1) because it will not give you the correct answer try to write (5/4)*(/3)*(1/1), this is the correct form. So xx! / = xx (xx 3) (xx 4) (xx 7) (xx 1) (xx ) (xx 5) (xx 6) xx! / xx! 4 (xx 3)!4 = (xx 1)! 4 (xx )! 4 (7) 9! / = 9/8/7/6/5/4/3//1 = = ! / = 1/11/10/9/8/7/6/5/4/3//1 = The equation (7) can be applied in order to obtain the same result..8 Definition 8: The N-Divide factorial function is defined as xx! / = xx/(xx )/(xx )/(xx 3)/ /mm with mm 1 xx! / = xx (xx 3) (xx 4) (xx 7) (xx 8) (xx ) (xx ) (xx 5) (xx 6) (xx 9) Remark: do not confuse between the divisions of many numbers for example: 5/4/3//1 is the same as (5/4)*(/3)*(1/1) And 6/5/4/3//1 is the same as (6/5)*(3/4)*(/1) Do not put into your calculator the equation (5/4/3//1) because it will not give you the correct answer try to write (5/4)*(/3)*(1/1), this is the correct form. So xx! / = xx (xx 3) (xx 4) (xx 7) (xx 8) (xx ) (xx ) (xx 5) (xx 6) (xx 9) xx! / xx! 4 (xx 3)!4 = (xx )! 4 (xx )! 4 (8) 9! / = ( = 9 7 (3) = ) 1! 3/ = 1 (1 3 3 ) = (3) = ISBN:

4 The equation (8) can be applied in order to obtain the same result..9 Definition 9: Let s consider that the operator (*, +, - or /) are indicated by (Op), for example xx! (OOOO) for the Falling Factorial operator. We define, the Alternate Falling Factorial Operator as the 1 AAAAAA xx! (OOOO ) = ( 1) xx+ xx! (OOOO ) =xx (9) 3 Diverse Rising Up Factorial Operators In mathematics, the Diverse Rising Up Factorial Operator, is an original study introduced by the first author and it is similar to the Diverse Falling Factorial Operator but the different is that the sequence numbers are rising up instead of falling down, for example the sequence will be xx (xx + 1) (xx + ) (xx + 3) (xx + mm) instead of xx (xx 1) (xx ) (xx 3) Definition 10: The Rising up Multiplication factorial function is defined as xx mm = xx (xx + 1) (xx + ) (xx + 3) (xx + mm) with mm 0 mm xx (xx + mm)! mm = (xx + ) = (xx 1)! =0 (10) 5 3 = 5 (5 + 1) (5 + ) (5 + 3) = = 8 (8 + 1) (8 + ) (8 + 3) (8 + 4) = Definition 11: The Rising up N-Multiplication factorial function is defined as xx mm = xx (xx + ) (xx + ) (xx + 3) (xx + 4) (xx + ) with mm xx mm = (xx + jj ) = (xx + )! (xx 1)! (11) with mm mm/ 8 6 = 8 (8 + ) (8 + ) (8 + 3 ) = = 0 (0 + 3) (0 + 3) = The equation (11) can be applied in order to obtain the same result. 3.3 Definition 1: The Rising up Plus factorial function is defined as xx mm + = xx + (xx + 1) + (xx + ) + (xx + 3) + + (xx + mm) with mm 0 mm xx + mm = (xx + jj) = (mm + 1)(xx + mm ) (1) = 9 + (9 + 1) + (9 + ) + (9 + 3) = = 1 + (1 + 1) + (1 + ) + (1 + 3) + (1 + 4) = 70 The equation (1) can be used in order to obtain the 3.4 Definition 13: The Rising up N-Plus factorial function is defined as xx mm + = xx + (xx + ) + (xx + ) + (xx + 3) + (xx + 4) + (xx + ) with mm mm/. takes the highest value for mm/. xx + mm = (xx + jj ) = ( + 1)(xx + ) With takes the highest value for mm/ = 9 + (9 + ) + (9 + ) = = 1 + (1 + ) + (1 + ) + (1 + 3 ) = 60 (13) The equation (13) can be used in order to obtain the ISBN:

5 3.5 Definition 14: The Rising up Minus factorial function is defined as xx mm = xx (xx + 1) (xx + ) (xx + 3) (xx + mm) mm xx mm = xx (xx + jj) jj =1 mm(mm + 1) = xx(1 mm) (14) 1 7 = = = = 46 The equation (14) can be used in order to obtain the 3.6 Definition 15: The Rising up N-Minus factorial function is defined as xx mm = xx (xx + ) (xx + ) (xx + 3) (xx + ) with mm mm/ xx mm = xx (xx + jj ) jj =1 ( + 1) = xx(1 ) With takes the highest value for mm/. 9 5 = 9 (9 + ) (9 + ) = = 1 + (1 + 3) + (1 + 3) + ( ) = 4 (15) The equation (15) can be used in order to obtain the 3.7 Definition 16: The Rising up Divide factorial function is defined as / xx mm = xx/(xx + 1)/(xx + )/(xx + 3)/ /(xx + mm) = xx (xx+3) (xx+4) (xx+7) (xx+8) (xx+1) (xx+) (xx+5) (xx+6) (xx+9) / (xx + 4jj) xx mm = (xx + (4jj + 1)) (xx + (4jj + 3)) (xx + (4jj + )) with (4 + pp) mm (mm pp) with pp is the maximum value that verify (4 + pp) mm / = 9/(9 + 1)/(9 + )/(9 + 3)/(9 + 4) = / = 1/(1 + 1)/(1 + ) = Definition 17: The Rising up N-Divide factorial function is defined as xx mm / = xx/(xx + )/(xx + )/(xx + 3)/ /(xx + ) with mm mm/ / (xx + 4jj ) xx mm = (xx + (4jj + 1) ) (xx + (4jj + 3) ) (xx + (4jj + ) ) with (4 + pp) mm (mm pp) with pp is the maximum value that verify (4 + pp) mm 9 6 / = 9/(9 + )/(9 + )/(9 + 3 ) = / = 1/(1 + 3)/(1 + 3) = Definition 18: Let s consider that the operator (*, +, - or /) are (OOOO ) indicated by (Op), for example xx mm for the Rising Up Factorial operator. We define, the Alternate Rising Up Factorial Operator as the 1 AAAAAA xx mm (OOOO ) = ( 1) xx+ xx mm (OOOO ) =xx 4 (18) 4 Conclusion The Diverse factorial operators is an original study introduced by the first author in the mathematical domain; the basic idea is derived from the Factorial Operator (!) but in this paper, the first author has developed new useful definitions that can be helpful in the mathematical domains. The main idea is how to manipulate the Operators between digits and how to vary digits between the operators. Many researchers will follow this study in order to find the application of these definitions and formulae in several domains. References: [1] Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics, Addison- ISBN:

6 Wesley, Reading MA. ISBN , (1988), p. 111 [] N. L. Biggs, The roots of combinatorics, Historia Math. 6, (1979), pp [3] Higgins Peter, Number Story: From Counting to Cryptography, New York: Copernicus, ISBN , (008), p. 1 [4] Peter Borwein, On the Complexity of Calculating Factorials, Journal of Algorithms 6, (1985), pp [5] Hadamard M. J. (in French), Sur L Expression Du Produit 1 3 (n 1) Par Une Fonction Entière, OEuvres de Jacques Hadamard, Centre National de la Recherche Scientifiques, Paris, (1968). [6] Ramanujan Srinivasa, The lost notebook and other unpublished papers, Springer Berlin, ISBN X, (1988), p [7] Olver Peter J., Classical Invariant Theory, Cambridge University Press, ISBN , (1999). [8] Knuth Donald E., Two notes on notation, American Mathematical Monthly 99 (5), (199), pp [9] Steffensen J. F., Interpolation (nd ed.), Dover Publications, ISBN (A reprint of the 1950 edition by Chelsea Publishing Co.),(1950), p. 8. [10] Weisstein Eric W., Alternating Factorial, MathWorld publisher (on the internet: / mathworld. wolfram. com/ AlternatingFactorial. Html). [11] Jonathan Sondow, Exponential Factorial, MathWorld publisher (on the internet: / mathworld. wolfram. com/ ExponentialFactorial. html). [1] Weisstein Eric W., Factorial Prime, MathWorld publisher (on the internet: / mathworld. wolfram. com/ FactorialPrime. html). ISBN:

Introduction to the General operation of Matrices

Introduction to the General operation of Matrices CLAUDE ZIAD BAYEH 1, 1 Faculty of Engineering II, Lebanese University EGRDI transaction on mathematics (004) LEBANON Email: claude_bayeh_cbegrdi@hotmail.com