It. J. Cotemp. Math. Scieces, Vol. 3, 008, o. 3, 5-56 O the Iverse of a Certai Matrix Ivolvig Biomial Coefficiets Yoshiari Iaba Kitakuwada Seior High School Keihokushimoyuge, Ukyo-ku, Kyoto, 60-0534, Japa iava@kyoto-be.e.jp Abstract I this short essay we aim to give the explicit form for the iverse of a certai upper triagular matrix which cotais biomial coefficiets as its elemets. This matrix has played a importat role i the article by Burrows & Talbot [], i which a certai extesio for a approximatio formula of r= rk has bee give. We shall obtai our result by makig use of some idetities cocerig Beroulli umbers. Mathematics Subject Classificatio:B68, 5A09, Y55 Keywords: iverse matrix, Beroulli umbers, sums of powers. Itroductio I the paper Burrows & Talbot [], a approximatio formula of the sums of powers of itegers, r k ( + )k+ k + r= has bee refied ad exteded to the case where k is ay real umber. Especially i the case where k is positive real umber, for the polyomials i, g(k) = k + {( + )k+ ( )k+ }, (k ), g( ) = log( +) ad for the power sums S(k) = r= rk (k>0), the relatio i a matrix form ( ) k... 3 4 5 4 6 7 ( 6 k... 3 4 5 ) 4... 3 g(k) S(k) g(k ) 0 S(k ) ( g(k 4) = 0 0 k 4 S(k 4) g(k 6) 0 0 0... S(k 6).......
5 Y. Iaba has bee show. Here, we shall compute the explicit form of the iverse of this upper triagular matrix whose (i, j)-elemet is ( ) k i (0 (j i) (j i)+ (j i) i, j ). Ad we shall see the fact that the iverse of it also cotais biomial coefficiets i (j i)).. Some idetities cocerig Beroulli umbers First, we shall see two elemetary idetities cocerig Beroulli umbers. They are show by usig the well kow Euler Maclauri sum formula. Lemma. Let B be the -th Beroulli umber, the we have followig elemetary idetities. ( ) m + m + () B k =, m =,, 3,...<, k ( ) m + () k m + B k =. m =0,,, 3,...< k Proof. For the Euler Maclauri sum formula, f(i) = f(x)dx + 0 (f() f(0)) + B k (k)! (f (k ) () f (k ) (0)). k= i= Let us start with settig f(x) =x m (m =,, 3,...< ). I this case, sice f(x) is fiite times differetiable, we do ot eed cosider the remaider term. m i m = m+ m + + m + B k (m)! (k)! (m k + )! m k+. i= Settig =, we have whece = Therefore we obtai (), k= m + + m + m = m + = = k= B k (m)! (k)! (m k + )!, B k (m)! (k)! (m k + )!. B k (m + )! (k)! (m k + )! ( ) m + B k. k
Iverse of a certai matrix 53 Next let us advace to the proof of (). Startig with f(x) =(x ) m (m =,, 3,...< ) ad processig with same above brig us the fact, (i ) m = ( )m+ + + ( )m (m +) B k (m)! + (k)! (m k + )! k (( ) m k+ +). i= Settig =, we have = k= m m + + k= B k (m)! (k)! (m k + )! k, whece = B k (m)! (k)! (m k + )! k. Thus ( ) m + k B k = k m +. This idetity is also true for the case m =0. This lemma tells us the followig oe immediately, ad we ca state a corollary. ( ) m + B k ) = 0, k Corollary. k (k)!((m k) + )! B k =0. m =0,,, 3,...< Next we see a idetity which is cocered with the Beroulli umber, ad by which we ca express our result more simply. Lemma. Let B (x) be the -th Beroulli polyomial, the we fid the followig idetity. B = B ( )
54 Y. Iaba Proof. Let us show it by usig the well kow (for istace, see []) geeratig fuctios. B (x) z! = zexz e z, B z! = z e z, we have z B! = (B B )z! z = B! (z/) B! = z e z z/ e z/ = zez/ e z. 3. Explicit iverse of the matrix We ow proceed to our mai cosequece. Let A be the matrix whose (i, j)- elemet is A ij = ( ) k i (j i) (j i)+ (j i), where k is a costat such that k =,, 3,... <. Ad the biomial coefficiet ( m) is take to be zero for the case <m, m<0. Sice this matrix is upper triagular matrix, to compute the iverse of this oe is ot so difficult i terms of the elemetary study of the matrix theory. But to deduce the explicit expressio for the iverse of this matrix is ot so easy. Propositio. Let A ad B are ( +) ( + ) upper triagular matrices whose (i, j)-elemets are as follows. ( ) k i A ij = (j i) (j i)+, ( ) (j i) k i B ij = B (j i) ( (j i) ). The we have A ilb lj = δ ij, where δ ij is Kroecker delta. I other words, the matrix B is the iverse of the matrix A.
Iverse of a certai matrix 55 Proof. First it is clear that A ilb lj = A ii B ii = for i = j. Next we compute the matrix elemets of AB for i j. ( ) k i A il B lj = (l i) (l i)+ (l i) ( ) k i ((j i))! = (j i) ((l i))!((j l))! ( ) k i ((j i))! = (j i) (j i) Thus we should just show the followig idetity, (j l) ((j l))!((l i) + )! B (j l) =0. By settig j l = t, j i = s, The LHS is foud that, s t (t)!((s t) + )! B t t=0 ( ) k l (j l) B (j l) (j l) (j l) (j l) (j i) (l i)+ B (j l) (j l) ((j l))!((l i) + )! B (j l). ad it is clearly equal to zero by corollary above. Note that we ca see the same biomial coefficiets i (j i)) both i the matrix A ad i the iverse B. Now let us see a example. Example For the matrix, ( ) k we get A 5 = 3 4 5 4 6 7 6 8 9 8 0 ( 0 k 3 4 5 4 6 7 6 8 9 ( 8 k 4 4 4 3 4 5 4 6 7 ( 6 k 6 6 3 4 5 ) 4 3 0 0 0 A 5 = 0 0 0 ( 0 0 0 0 k 8 0 0 0 0 0 ( ) k ) 7 3 4 5 4 6, ) 3 7 ( ) k 555 6 8 5 8 0 33 0 7 ( ) k 3 7 5 4 6 6 8 5 ) 8 4 ) 7 ( ) k 4 3 3 4 5 4 6 ) 6 6 ) 7 3 4 5 ) 4 3 0 ( ) k 3 4 0 0 4 0 0 0 6 0 0 0 0 0 0 0 0 0 8.
56 Y. Iaba To coclude this essay, we apped statig a formula of the sums of powers of itegers S(k) = r= rk with the help of cosequeces of [] ad of our aboves. Let k be a atural umber, we ca state the followig formula. S(k) = k/ i=0 ( ) k B i ( )g(k i) i where, g(k) = {( + k+ )k+ ( )k+ }, ad x meas the highest iteger less tha or equal to x. Example Puttig k = 9. The we have 9/ ( ) 9 S(9) = B i ( )g(9 i) i i=0 = 3 0 + 4 7 0 6 + 3 4 8 + 9 + 0 0. This formula is usually ot available because the computig process is too troublesome. However, this is ot meaigless because of oly a reaso that we have ot see ever. 4. Refereces [] B.L.Burrows ad R.F.Talbot, Sums of powers of itegers, America Mathematical Mothly 9(984), 394-403. [] R.Graham, D.Kuth, ad O.Patashik, Cocrete Mathematics, Addiso- Wesley, 989. Received: March 8, 008