Sequences of Definite Integrals, Factorials and Double Factorials

Transcription

1 47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology Havaad Haleumi Str. POB 6 Jerusalem 96 Israel daa@jct.ac.il Abstract We study sequeces of defiite itegrals. Some of them provide closed forms ivolvig factorials ad/or double factorials. Other examples are associated with either sequeces or pairs of sequeces of ratioal umbers, for which summatios are foud. Itroductio. The study of sequeces of either defiite or improper itegrals has coectios with various fields, such as combiatorics (see [6], []), ifiite series (see [4]), ad others. I this paper, we study sequeces of itegrals, depedig mostly o oe parameter, sometimes o two parameters. The method geerally used is a telescopic method (see [5, p. 579]); whe a recurrece relatio based o multiplicatio by a homographic fuctio exists, the implemetatio of the method is easy. Otherwise it ca be very hard, sometimes impossible. I Sectio, improper itegrals depedig either o oe or two parameters (that are o-egative itegers) are cosidered, where the itegrad ivolves a logarithm. I these examples, various situatios are described (differet roles ad iflueces of the parameters, existece or o-existece of a closed expressio for the geeral term of the sequece, etc.). I Sectio, three sequeces of itegrals are studied; the mai oe is the sum of a sequece of ratioal umbers ad a sequece of ratioal multiples of. The formulas for related itegrals are derived.

2 Logarithmic itegrals with a parameter. For every atural umber, we defie the improper itegral As x (l x) dx. () lim xp (l x) q = x + holds for ay two positive itegers p ad q, we ca work with I as if it is a defiite itegral, i.e., by writig ordiary expressios for the itegrals ad ot writig limits for λ arbitrarily close to of x (l λ x) dx. We take u(x) = (l x) ad v(x) = x / i order to perform a itegratio by parts; we have thus, By telescopig, we have I = [ ] x (l x) x (l x) dx, I. () ( ) ( ) I = = ( )! I. A straightforward computatio provides I = /, ad we have proved the followig propositio: Propositio. For ay atural umber, x (l x) dx = ( )! +. Note that the sequece whose geeral term is ( ) + I provides a itegral represetatio of factorials. Now cosider the sequece of itegrals defied by x (l x) +/ dx. () The atural logarithm is a egative fuctio over the iterval (, ), thus the value of this itegral, if it exists, is a pure imagiary complex umber. Actually, the square root fuctio is a multi-valued fuctio with two braches. Each brach is aalytic, thus a itegral of the form x (l x) +/ dx where ε, ε

3 is path-idepedet (i fact, as x is a positive real umber, l x < ad the ivolved path ca be a segmet o the y axis), whe computed alog a path which does ot itersect the stadard brach cut (see []). Moreover, for ay o-egative iteger, we have lim x (l x)+/ =. x + Therefore the give itegral I is well-defied (we use a method as i [, p. 6]). I a maer similar to the method used above, we obtai the followig recurrece relatio: ( + ) I = + I. (4) 4 We eed ow to compute the first itegral of the sequece: I = x (l x) / dx = i π. 8 We recall the defiitio of the double factorial of a odd umber (see Sloae s sequece A47 ad [9]): N, ( )!! = 5 ( ), ad for a eve umber Therefore the followig formula holds: Propositio. For ay atural umber : N, ()!! = 4 6 (). x (l x) +/ dx = ( )!! + i π. Now let p ad q be o-egative itegers. We defie I p,q = x p (l x) q dx. (5) We take u(x) = (l x) q ad v(x) = x p+ /(p + ) i order to perform a itegratio by parts ad get [ ] I p,q = p + xp+ (l x) q q p + x xp+ (l x) q dx, }{{} i.e., = I p,q = q p + I p,q. (6)

4 Now we have I p,q = q p + I p,q ( = q ) ( q ) I p,q p + p + ( = q ) ( q ) ( q ) p + p + p + =... I p,q As I p, = /(p + ), we have fially Propositio. For ay pair (p, q) of atural umbers, x p (l x) q dx = ( )q q! (p + ) q+. This example of a parametric itegral depedig o two parameters that are o-egative itegers, together with aother itegral described i [], shows the great differece betwee the iflueces of the parameters: the whole computatio is cocetrated o oe parameter oly, ad the other oe is passive. Nevertheless, the fial result is depeds o both parameters. Remark. Equatio (.) is equivalet to q! = ( ) q (p + ) q+ x p (l x) q dx. This itegral form for a factorial is surprisig, as it cotais a parameter without ifluece. I the examples studied above, the reaso for the computatio of a closed form to be so easy lies i the fact that, whe performig the itegratio by parts, the itegrated part of the result is equal to. This provides a recurrece relatio for the sequece (I ) of the form f() I, (7) where f is a homographic fuctio of with iteger coefficiets. Other examples of this kid have bee studied i [4,, ]. Whe this situatio does ot occur, computatios ca be more complicated, as the ext example shows. For every atural umber, we defie the itegral We have I = e e x dx = x (l x) dx. (8) [ ] e x = (e ). 4

5 Choosig as above u(x) = (l x) ad v(x) = x /, a itegratio by parts yields [ ] e x (l x) e x (l x) dx ad leads to the followig recurrece relatio: e I. (9) The presece of a o-zero itegrated term makes the work harder tha i previous examples. We have e ( e ) I = ( e ) ( ) + I 4 = ( e ) ( ( ) + 4 e ) I = ( e ) ( ) ( )( ) + I 4 8 =... = ( e ( )! ( ) 4 = ( e ( ) + + ( ) +! +. ) + ( )! I ( )( ) + + ( ) )! + ( )! Recall that for ay two o-egative itegers ad k such that k, A k =! ( k)! (A k is the umber of arragemets without repetitio of elemets by k). Hece, the followig holds: Propositio.4 e x (l x) dx = e k= ( ) k Ak k + ( )+! +. 5

6 Three related parametric ratioal itegrals For a positive iteger, we defie the itegrals J = K =. First itegral: complete computatios dx, () (x + x + ) x dx, () (x + x + ) x dx. () (x + x + ) As i the previous examples, we wish to fid a recurrece relatio for the sequece (I ), the a closed form for the geeral term, if possible. We perform a itegratio by parts; let whece It follows that u(x) = u (x) = (x + x + ) ad v(x) = x, (x + ) (x + x + ) + ad v (x) =. [ x (x + x + ) ] + = + x(x + ) dx. (x + x + ) + }{{} =T x(x + ) dx (x + x + ) + I order to compute K, we decompose the itegrad ito partial fractios: x(x + ) x R, (x + x + ) = x + (x + x + ) + (x + x + ) + + (x + x + ) = x + (x + x + ) + (x + x + ) + + (x + x + ). Thus, T = = [ ( ] (x + x + ) I + + I ) I + + I. By re-arragig the terms, we obtai the followig relatio of recurrece: I + = ( ) ( ) + I. () 6

7 We first compute the itegral I : I = x + x + dx = ( x + ) + 4 dx = 4 [ 4 ( ) ] x + dx. + Usig the substitutio t = (/ )(x + /), we obtai: I = / t + dt = ( arcta arcta ) = π 9. (4) From Equatios () ad (4) follows that I is give by a relatio of the form a + b π (5) where a ad b are ratioal umbers. We study separately the sequeces (a ) ad (b ). Cosider b first: ( ) b + = b = ( ) = ( ) = =... ( ) = b b 5 b ( )( )( 5)... ( )( )... b. Isertig suitable factors ito the umerator, ad dividig out by the same factors, we obtai a closed factorial form for b + : b + = ( ) ()! (!) b = ()! (!) b = ()! + (!). (6) Note that ()! (!) = ( ). The iterested reader will fid cocrete occureces of these umbers (special paths i graphs, etc.) i Sloae s ecylopedia, sequece A984. Aother represetatio ca be give for the sequece (b ), usig the double factorial (see Sloae s ecyclopedia, A47 ad [9]). We have N, b = ( )!! + ( )!. (7) 7

8 A closed form for a is harder to derive. From Equatio (), we derive the followig recurrece relatio for a : a + = ( ) ( ) + a. (8) By Equatio (4), a =, whece the sequece (a ) is well defied by the above relatio. Let s ow use a telescopic process: a + = ( ) ( ) + a = ( ) [ ( ) ] ( ) + ( ) ( ) + ( ) a = ( ) + ( ) ( ) ( ) ( ) ( )( ) + a ( ) = ( ) + ( ) ( ) ( ) ( ) [ ( ) ] ( )( ) + ( ) ( ) ( 5) + ( ) a =.... Iteratios are eeded util a is reached, because a = a =. Fially, the followig formula is derived: a + = ( ) + k ( k ( )( )... ( k + ) ). ( )( )... ( k + ) k= The ratioal fractio o the right ca be tured ito a closed factorial formula. Shiftig the idex + to, we obtai a = ( ) ( ) + ( + k ) k= (( ))! ( k )! ( ) ( k )!. (9) ( )! A formula ivolvig double factorials looks a little more compact (see Sloae s sequece A47 ad [9]): a = ( ) ( ) + I coclusio, we have k ( + k ) k= Propositio. For ay o-egative iteger, dx (x + x + ) = a + b π, 8 ( )!! ( k )! ( k )!! ( )!. ()

9 with a = ad b = ( ) ( ) + ( )!! + ( )!. k ( + k ) k= ( )!! ( k )! ( k )!! ( )!. Extesios From the results above, the two related parametric itegrals J ad K ca be computed: We have i.e., I + J = J = = = ( ) + x (x + x + ) dx [ (x + x + ) ( ) +, ] ( ) I. () A expressio of J as a fuctio of is obtaied by substitutio. A closed form for K is obtaied by substitutio, accordig to the followig remark: I + J + K = 4 Ackowledgemets + x + x (x + x + ) dx = (x + x + ) dx = I. () The author wishes to thak the referee for his remarks ad appreciatios, ad the editor for his care, icludig log-distace phoe calls. Refereces [] Th. Daa-Picard, Explicit closed forms for parametric itegrals, Iterat. J. Math. Educ. i Sciece ad Techology 5 (4), [] Th. Daa-Picard, Parametric itegrals ad Catala umbers, Iterat. J. Math. Educ. i Sciece ad Techology 6 (5), [] Th. Daa-Picard, Sequeces of defiite itegrals, preprit, 4. 9

10 [4] P. Glaister, Factorial sums, Iterat. J. Math. Educ. i Sciece ad Techology 4 (), [5] H. Johsto ad J. Mathews, Calculus, Addiso-Wesley,. [6] B. Sury, T. Wag ad F.-Z. Zhao, Idetities ivolvig reciprocal of biomial coefficiets J. Iteger Sequeces 7 (4), Article [7] N. J. A. Sloae, The O-Lie Ecyclopedia of Iteger Sequeces. [8] I. M. Viogradov, Elemets of Number Theory, 5th rev. ed., Dover, New York, 954. [9] E. W. Weisstei, Double Factorial, MathWorld. [] A. David Wusch, Complex Variables with Applicatios, d editio, Addiso-Wesley, Readig, Massachussetts, 994. Mathematics Subject Classificatio: Primary 5A; Secodary 6A. Keywords: parametric itegrals, double factorials, combiatorics. (Cocered with sequeces A984 ad A47.) Received April 9 5; revised versio received September 9 5. Published i Joural of Iteger Sequeces, September 9 5. Retur to Joural of Iteger Sequeces home page.