A Comparative Study of Positive and Negative Factorials

Transcription

1 A Comparative Study of Positive ad Negative Factorials A. M. Ibrahim, A. E. Ezugwu, M. Isa Departmet of Mathematics, Ahmadu Bello Uiversity, Zaria Abstract. This paper preset a comparative study of the various types of positive factorial fuctios, amog which iclude the covetioal factorial, double factorial, quadruple factorial, superfactorial ad hyperfactorial. Subsequetly, a extesio of the cocepts of positive! to egative umbers! is itroduced. Based o this extesio, a formulatio of specific geeralizatio cases for differet forms of egative factorials are aalyzed ad preseted. Keywords: Factorial, egative factorial, covetioal factorials, factorial fuctios. 1. Itroductio I factorial fuctios emphasis is geerally placed o its applicatio as i combiatorics, calculus, umber theory ad probability theory. Study of factorial fuctio helps us to uderstad more about it applicatio i factorial Desigs to optimize aimal experimets ad reduce aimal use ad Ecoomic factorial aalysis. Differet types of factorials like: Double factorial!!, quadruple factorial (2)!, superfactorial Sf() ad hyperfactorial H() were! derived ad exteded to factorials of o-positive itegers (i.e. egative factorials) uder the study of factorial fuctios. Multiple Scietists worked o this subject, but the pricipal ivetors are [1] who gives the asymptotic formula after some work i collaboratio with (De Moivre), [2], fially [3] ad [4] itroduces the actual otatio!. Of course other Scietists such as Taylor also worked a lot with this otatio. The otatio! was itroduced by Frech mathematicia Christia Kramp ( ) i 1808 i Elemets d'arithmétique uiverselle. The term factorial was first coied (i Frech as factorielle) by Frech mathematicia Louis Fracois Atoie Arbogast ( ) ad Kramp decided to use the term factorial so as to circumvet pritig difficulties icurred by the previous used symbol. 2. Related Work The factorial of a atural umber is the product of the positive itegers less tha or equal to. This is writte as! ad proouced factorial. The factorial fuctio i is formally defied by! = k=1 k for all 0, or recursively defied by! = { 1 if = 0, ( 1)! if > 0. 18

2 That is 0! = 1, because the product of o umber at all is 1 (i.e. there is exactly oe permutatio of zero objects) [6]. For example: By defiitio of factorial;! = ( 1)( 2)( 3) 1 1! = 1 2! = 2(2 1)! = 2.1 = 2 3! = 3(3 1)(3 2)! = = 6 4! = 4(4 1)(4 2)(4 3)! = = 24 5! = 5(5 1)(5 2)(5 3)(5 4)! = = 120 Double factorials A fuctio related to the factorial is the product of all odd (eve) values up to some odd (eve) positive iteger. It is ofte called double factorial (eve though it ivolves about half the factors of the ordiary factorial, ad its value is closer to the square root of the factorial), ad deoted by!! [6] k For odd positive iteger = 2k 1, k 1, it is; (2k 1)!!= i=1 (2k 1). For eve positive iteger = 2k, k 2, it is; (2k)!! = It is recursively defied by,. ( 2)( 4) > 0 odd!! = {. ( 2)( 4) > 0 eve 1 = 1,0 Like i the ordiary factorial, 0!! = 1!! = 1 k i=1 2!! = 2(2 2)!! = 2(0)!! = 2.1 = 2 3!! = 3(3 2)!! = 3(1)!! = 3.1 = 3 (2i) = k! 4!! = 4(4 2)(4 4)!! = 4(2)(0)!! = = 8 5!! = 5(5 2)(5 4)!! = 5(3)(1)!! = = 15 I geeral, a commo related otatio is to use multiple exclamatio poits to deote a multifactorial, as the product of itegers i steps of two ()!!, three ()!!!, or more. The double factorial is the most commoly used variat, but oe ca similarly defie the triple factorial (!!!) ad so o. Oe ca defie the k th factorial, detoated by! (k), recursively for o-egative itegers as; 2 k!! = { 1, (( k)! (k) ), if 0 < k, if k Or (k)! (k) = k! Here are the computed values for! ad!! (startig from = 0 to 5) 19

3 !!! Graph of! ad!!!!! 5, 120 4, 24 0, 1 1, 1 2, 2 3, 63 4, 8 5, Figure 1: Graph of factorial (!) ad double factorial (!!) for = 0 to 5 From the above figure, we oticed that the graph of! is the same as the graph of!! for = 0, 1 ad 2 this is because;0! = 0!! = 1, 1! = 1!! = 1 ad 2! = 2!! = 2 ad they diverged from = 3, 4 ad 5 as a result of; 3! = 6 3 = 3!!, 4! = 24 8 = 4!! ad 5! = = 5!! ad so o. Relatios betwee double factorials ad covetioal factorials There are may idetities relatig double factorials to covetioal factorials. Sice we ca express (2 + 1)!! 2! = [(2 + 1)(2 1) 1][2][2( 1)][2( 2)] 2.1 It follows that = [(2 + 1)(2 1) 1][2][2(2 2)(2 4) 2.1 = (2 + 1)(2)(2 1)(2 2)(2 3)(2 4) 2.1 = (2 + 1)!. (2 + 1)!! = (2+1)! 2! for = 0, 1, 2,..(1) 20

4 Also, (2)!! = (2)(2 2)(2 4) 2 = [2()][2( 1)][2( 1)] 2 i.e. (2)!! = 2! for eve...(2) (2 1)!! = (2)! for 2! odd..(3) It follows that, for eve;! ( 1)( 2) (2) =!! ( 2)( 4) (2) = ( 1)( 3) (2) = ( 1)!!. For odd;! ( 1)( 2) (1) =!! ( 2)( 3) (1) Therefore, for ay ; = ( 1)( 3) (1) = ( 1)!!.! = ( 1)!!!!! =!! ( 1)!!... (4) Quadruple factorials The so-called quadruple factorial, however, is ot the multiple factorial! (4) ; it is a much larger umber give by (2)! [6]! For example, the quadruple factorials = 0, 1, 2, 3, 4, 5 ad 6 are; (2 0)! 0! (2 1)! 1! (2 2)! 2! (2 3)! 3! (2 4)! 4! = 0! 0! =1 = 2! 1! = 2 1 =2 = 4! 2! = 24 2! =12 = 6! 3! = =120 = 8! = =1680 4! 24 21

5 (2 5)! 5! (2 6)! 6! = 10! = = = 12! 6! = 479,001, =665,280 Graph of (2)!/! (2)!/! 5, , 1 1, 2 2, 12 3, 120 4, Figure 2: Graph of quadruple factorial for = 0 to 5 The figure above is colliear o x axis from = 0, 1, 2 ad 3 ad its makes a shaped curve alog the yaxis at = 3 which projects upward from = 4. Superfactorials I [7], the superfactorial is defied as the product of the first factorials. Superfactorial is defied by; Sf() = k! = k k+1 = ( 1) 2. 1 k=1 k=1 The sequece of superfactorials starts (from = 0) For example, the superfactorialsof =0, 1, 2, 3, 4 ad 5 are; Sf(0) = 1 Sf(1) = 1 Sf(2) = = 2 Sf(3) = = 12 Sf(4) = = 288 Sf(5) = =

6 Sf(6) = = Equivaletly, the superfactorial is give by the formula; Sf() = 0 i<j (j i) which is the determiat of a Vadermode matrix. Hyperfactorials Occasioally the hyperfactorial of cosidered, it is writte as H() ad defied by, H() = k=1 k! = ( 1) 1. For example the hyperfactorials for = 1, 2, 3, 4 ad 5 are; H(1) = 1 H(2) = = 4 H(3) = = 108 H(4) = = 27,648 H(5) = = 86,400,000 Here are the computed values for S()ad H() for = 1 to 5 Sf() H() Graph of Sf() ad H() Sf() H() 5, , 1 1, 1 2, 24 3, , , Figure 2: Graph of superfactorial Sf() ad hyperfactorial H() for = 0 to 5 23

7 From the figure above, we see that for = 0, 1, 2 ad 3 the graph of H() is the same as the graph of Sf() which are colliear o x-axis ad from = 3 the graph of H() diverges by makig a curve alog the egative y axis ad projects vertically alog the y-axis from = 4 while the graph of Sf() maitai its symmetric o x-axis for all values of. Relatioship betwee Factorial Powers I [4], a proof o relatioship betwee positive ad egative factorial power was discussed. We itroduce this cocept to support the poit that there exist correspodig relatioships betwee egative ad positive factorials. The proof o the cocept of egative factorials is provided i sectio 3. Positive factorial power is defied as: k = k(+1) (k ), (5) we ote that k = k(k 1) (k + 1)...(6) by substitutig (+1) for ito (6), we have k (+1) = k(k 1) (k + 1)(k )..(7) Therefore, k (+1) = k (k ) Similarly, we defie egative factorial power as: 1 k ( ) = (k+1) ().....(8) We claim that by mathematical iductio, that if (10) is true the so is, whe we substitute + 1 for 1 k ( +1) = (k+ 1) ( 1).. (9) We recall that, (k + ) () = (k + )(k + 1) (k + 1..(10) ad (k + 1) ( 1) = (k + 1) (k + 1)...(11) Hece by substitutig these i (8) ad (9) k ( ) = 1 (k+)(k+ 1) (k+1).... (12) 24

8 k ( +1) = 1 (k+ 1) (k+1). (13) From (12) ad (13) we ote: This is true by defiitio (5). 3. Cocept of Negative Factorial k ( ) = k +1 (k + ) The classical case of the iteger form of the factorial fuctio! cosists of the product of ad all itegers less tha, dow to 1. This defied the factorial of positive itegers o the right side o real lie as:! = ( 1)( 2) 1.. (14) We ca exted the above defiitio to the left side of the real lie as show below; Figure 3: Factorial of egative ad positive itegers o the left ad right side o real lie Kurba s Classical Case of Negative Factorials Fuctio The Kurba s classical case of egative iteger form of the factorial fuctio! cosists of the product of ad all itegers greater tha up to -1. This defied the factorial of egative itegers o the left had side of the real lie recursively defied by; ( )! = ( + 1)( + 2) ( 1)... (15) For example: By defiitio of egative factorial;! = ( + 1)( + 2) ( 1) 1! = 1 2! = 2( 2 + 1)! = 2. 1 = 2 3! = 3( 3 + 1)( 3 + 2)! = = 6 4! = 4( 4 + 1)( 4 + 2)( 4 + 3)! = = 24 5! = 5( 5 + 1)( 5 + 2)( 5 + 3)( 5 + 4)! = = 120 Therefore, the egative factorial fuctio is formally defied by! = k=1 ( 1) k k for all 1, or recursively defied by ( + 1)! if = 1,! = { 1 if < (16) Negative Double Factorials 25

9 A fuctio related to the egative factorial is the product of all odd (eve) egative values up to some odd (eve) egative iteger. It is ofte called egative double factorial (eve though it ivolves about half the factors of the ordiary egative factorial.) which is deoted by!!. Recursively defied as; ( + 2)( + 4) ( 6)( 4)( 2)for eve!! = { ( + 2)( + 4) ( 5)( 3)( 2)for odd 1 for = 1 Simplified as follows:!! = ( + 2)!! if = 1, { 1 if < 1....(17) Like i the egative factorial, 0! = 0!! = 1 ad 1! = 1!! = 1 2!! = 2( 2 + 2)!! = 2.0!! = 2.1 = 2 3!! = 3( 3 + 2)!! = 3. 1!! = 3. 1 = 3 4!! = 4( 4 + 2)( 4 + 4)!! = !! = = 8 5!! = 5( 5 + 2)( 5 + 4)!! = !! = = 15 Here are the computed values for! ad!! (startig from 1 to 5)!!!

10 Graph of (-)! ad (-)!! (-)! (-)!! -4, 24-4, 8-3, 3 2 0, 1-3, -6-2, -2-1, , , -120 Figure 4: Graph of egative factorial! ad egative double factorial!! from 0 to 5 From the above figure, we oticed that the graph of! is the same as the graph of!! oly from 0 ad 1 this is because; 0! = 0!! = 1 ad 1! = 1!! = 1 which are symmetric o the egative x axis ad they started oscillatig ad itersectig by makig curves alog positive ad egative y axis respectively immediately after 1 as a result of; 2! = 2 2 = 2!!, 3! = 6 3 = 3!!, 4! = 24 8 = 4!! ad 5! = = 5!! ad so o. Relatios betwee egative double factorials to egative factorials There are may idetities relatig egative double factorials to egative factorials. Sice we ca express ( 2 1)!! 2.! = [( 2 1)( 2 + 1) 1][ 2][2( + 1)][2( + 2)] 2. 1 = [( 2 1)( 2 + 1) 1][ 2][ 2( 2 + 2)( 2 + 4) 2. 1 = ( 2 1)( 2)( 2 + 1)( 2 + 2)( 2 + 3)( 2 + 4) 2. 1 It follows that ( 2 1)!! = ( 2 1)! 2! = ( 2 1)!. for = 0, 1, 2,..(18) Also, ( 2)!! = ( 2)( 2 + 2)( 2 + 4) 2 = [2( )][2( + 1)][2( + 1)] 2 i.e. ( 2)!! = 2! for eve (19) ( 2 + 1)!! = ( 2)! 2! 27

11 for odd..(20) It follows that, for eve;! ( + 1)( + 2) ( 2) =!! ( + 2)( + 4) ( 2) = ( 1)( 3) (2) = ( + 1)!!. For odd;! = ( +1)( +2) ( 1)!! ( +2)( +3) ( 1) = ( + 1)( + 3) ( 1) = ( + 1)!!. Therefore, for ay ;!!! = ( + 1)!!! =!! ( + 1)!!.....(21) For example; 1. 1! = 1!! ( 1 + 1)!! = 1!! 0!! = 1 1 = ! = 2!! ( 2 + 1)!! = 2!! 1!! = 2 1 = ! = 3!! ( 3 + 1)!! = 3!! 2!! = 3 2 = ! = 4!! ( 4 + 1)!! = 4!! 3!! = 8 3 = ! = 5!! ( 5 + 1)!! = 5!! 4!! = 15 8 = 120 (Usig the fact that 0! = 0!! = 1.) Negative quadruple factorial for egative itegers ca be defied as ( 2)! ordiary quadruple factorial. For example, the egative quadruple factorials are; (2 0)! 0! (2 ( 1))! 1! (2 ( 2))! ( 2)! (2 ( 3))! ( 3)! (2 ( 4))! ( 4)! (2 ( 5))! ( 5)! = 0! 0! =1 = ( 2)! ( 1)! = 2 1 = 2 = ( 4)! ( 2)! = 24 2 = 12 = ( 6)! ( 3)! = = 120 = ( 8)! = = 1680 ( 4)! 24 = ( 10)! ( 5)! = = 30240! like i the 28

12 Graph of (-2)!/(-)! (-2)!/(-)! -4, , , 12-1, -2 0, , Figure 5: Graph of egative quadruple factorial from 0 to 5 The figure above is colliear o the egative x axis from 0, 1, 2 ad 3 ad its makes a shaped curve alog the positive yaxis from 3 which declied to the egative yaxis from 4. Negative superfactorial for egative iteger ca be defied as; Sf( ) = ( k)! = ( k) k+1 = ( 1). ( 2) 1. ( 3) 2 ( + 1) 2. ( ) 1 k=1 k=1 For example, the egative superfactorials from 0 to 5 are give below; Sf(0) = 1 Sf( 1) = 1 Sf( 2) = ( 1) 2. ( 2) 1 = 2 Sf( 3) = ( 1) 3. ( 2) 2. ( 3) 1 = 12 Sf( 4) = ( 1) 4. ( 2) 3. ( 3) 2. ( 4) 1 = 288 Sf( 5) = ( 1) 5. ( 2) 4. ( 3) 3. ( 4) 2. ( 5) 1 = Negative hyperfactorial for egative itegers H( ) is give by; H( ) = ( k)! = k=1 ( 1) 1. ( 2) 2. ( 3) 3 ( + 1) 1. ( ) For example the hyperfactorials for k= 1, 2, 3, 4 ad 5 are; H( 1) = 1 H( 2) = ( 1) 1. ( 2) 2 = 4 H( 3) = ( 1) 1. ( 2) 2. ( 3) 3 = 108 H( 4) = ( 1) 1. ( 2) 2. ( 3) 3. ( 4) 4 = 27,648 29

13 H( 5) = ( 1) 1. ( 2) 2. ( 3) 3. ( 4) 4. ( 5) 5 = 86,400,000 Graph of Sf(-) ad H(-) Sf(-) H(-) -5, , , , , 1-1 0, , Figure 6: Graph of egative superfactorial ad egative hyperfactorial from 0 to 5 From the figure above, we see that for = 0, 1, 2 ad 3 the graph of H( ) is the same as the graph of Sf( ) which are both colliear o egativex-axis ad at = 3 the graph of H( ) shoots up a curve alog the positive y-axis which declied to the egative y axis from = 4 while the graph of Sf( ) maitai its colliearity o egative x-axis for all values of egative. 4. Coclusio ad Discussio of Result Factorials of positive itegers i.e.! is the product of ad all itegers less tha, dow to 1while factorials of egative itegers i.e.! is the product of ad all egative itegers greater tha up to -1.Therefore factorials of both positive ad egative itegers exists. We oticed that both factorials of positive ad egative itegers have the same values with same (differet) sigs as i the case of egative factorial! ad egative quadruple factorial ( 2)! due to whether factorials of egative itegers are odd or eve. If the egative! iteger is eve the it has the same value with that of positive itegers ad if the egative iteger is odd the it has differet sig to that of positive iteger. While egative superfactorial Sf( ) has the same value with that of positive superfactorial Sf() if the egative iteger is odd. We fially observed that egative double factorial!! ad egative hyperfactorial H( ) follows the same patter i which they have the same value but differet sig to that of their positive itegers for the first ad secod egative odd ad eve itegers ad the same value, same sig for the third ad fourth odd ad eve egative itegers cotiuously. Therefore,! ad ( 2)!/! has the same properties with! ad ( 2)!/! for eve,sf( ) has the same properties with Sf() for odd ad!! ad H( ) has the same properties with!! ad H() for the third ad fourth odd ad eve egative itegers respectively. 30

14 Refereces [1]. J. Stirlig. Methodus Differetialis. Lodo, [2]. C. Kramp, Elemes ď arithmetique Uiversele (Cologe, 1808) [3]. K. A Stroud, Egieerig Mathematics, with additios by Dexter J. Booth.-5th ed. pp [4]. Ke Ward s Mathematics Pages, Polyomial Factorials Negative, ialnegative.html [accessed, 2012] [5]. Peter Brow. O the Complex of Calculatig Factorials. Joural of Algorithm 6, (1985) [6]. Factorial Wikipedia, the free ecyclopedia, ] [7]. J. Borwei, R. Corless (1996). The Ecyclopedia of Iteger Sequeces ( N. J. A. Sloae ad Simo Plouffe. SIAM Review 38 (2): dio:101137/

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