FORMULAS FOR P + 3^ + + n p

Transcription

1 FORMULAS FOR + 2 P + 3^ + + n p G* F C. de Bruyn and J. M. de Villiers Department of Mathematics, University of Stellenbosch, Stellenbosch, South Africa (Submitted December 992). INTRODUCTION Let S (ri) = + 2 P + 3 P n p, withw and/? positive integers. In [3] R. A. Khan, using the binomial theorem and a definite integral, gave a proof of the general recurrence formula for S^ri), S 2 (n), S 3 (ri),... in terms of powers of n. A matrix formula for S (ri), obtained by solving a difference equation by matrix methods, is given in [2]. In this note the recurrence formulas in terms of powers of n and of n + are given in symbolic form, only using the binomial theorem. These formulas are both easily remembered and applied. These recurrence formulas are then used to establish, employing Cramer's rule, explicit expressions for S p (ri) in determinant form. Finally, the usual formulas for S (ri) as polynomials of degree p + \ mn and n + l, with coefficients in terms of the Bernoulli numbers, are derived from these determinants. It is noted that it is possible to do this without prior knowledge of the Bernoulli numbers. 2, A Recurrence Formula 2. FORMULAS IN TERMS OF POWERS OF n Let n GN, with TV the set of positive integers. For k GN, let S k () =, and take S (n) = n. Then n k =S k (n)-s k (n-l) r=l n f k r=iv/=(a ' = S k (n)m k \-\) k - i S i (n) (2.-) /=(A ' 994] 27

2 FORMULAS FOR + 2 P + 3 P + + n p The equation S k -(S-l) k =n k, in which the binomial power is expanded and S (i -,,2,, provides a mnemonic for (2..). For example, for k = 2, formula (2..3) yields and thus giving the well-known result Next, for & = 3, it similarly follows that S 2 -(S 2-2S + l) = n\ 2S (n)-n = n 2, (2..3) k) are then replaced by S^ri),,(/!) = - ( /! +!). (2..4) 3S 2 (n)-3s l (n) + n = n\ so that substitution of (2..4) leads to the formula 2.2 S p (n) as a Determinant,(/!) = - (» + l)(2/, + l). 6 Let / > # and let k = l,2,...,p,p + l in (2..2). Then, solving for S p (n) in the resulting (p +) x (p +) lower triangular linear system by means of Cramer's rule, the determinant representation SM = (p+iy. - c-ir (-i) p (?) (-l)'(f) (-)^) (5) (-l)^(f) (-\y + \r?) :/-,) (S) H» 2» 3 n p n" + (2.2.) p\!!! ni «i (2.2.2) (-) p-\ (-> / > + ( - i ) p ( - ) ' (p+iy. ("> (/>-!)! (p-2)! (p-)! (/?+!)! 272 [JUNE-JULY

3 FORMULAS FOR + 2 ^ +3 P + +U P is obtained. The step from (2.2.) to (2.2.2) follows by first multiplying the / th row of the determinant by /z! for i = l,2,...,p + l, and then multiplying the j i h column of the resulting determinant by (j -)! for j = 2,3,..., /?. 2.3 ^ ( / i ) as a Polynomial By expanding the determinant (2.2.2) with respect to the last column, with a p+l = -Jx and, for r = \,2,...,p, ^(») = I f l^'" r. p I r= (2.3.) 4 p+l-r i-iy P \ (p + l-r)\ (-D r (~l) r " (r+l)!! (-D r (r-)! ("l) r (~l) r (r-2)! c-d r (r-)! X! p\ {p + \-r)\ -is i : (-i) r + 2 (-ir 3 (r+l)!! _L (-D r+ (r-)! c-ir 2 ( - l ) r (r-2)! c-ir (r-)! (2.3.2) Now multiply the 2 n, 4?... columns of the determinant in (2.3.2) by - and then multiply the 2 nd, 4 th,... rows of the resulting determinant by -. Then a p+ip\ (p + l-r)\ Hs i (r+d!! (r-)!! (r-2)! (r-)!! (2.3.3) Next, recall (see, e.g., [4], p. 323) that the Bernoulli numbers Bj, j-,2,..., by the determinants A - o Bj = (-l) J ji! can be represented (2.3.4) i j _ i 994] 273

4 FORMULAS FOR + 2 P +3 P + +n p Hence, by (2.3.3) and (2.3.4), Thus, by (2.3.), Since B 2r+l = a p+l _ r = -(-\y[p; ]B r, r = l,2,...,p. p + l ^(") = -^ri(- ) r K + W + - r - (2-3.5) p + r=q V / (ren), s M - f «+ I -. + I^Bl «.- + I(^4 ^ +... with the last term either containing nor n 2. This is the form in which S p (n) was given by Jacques Bernoulli in []. 3. A Recurrence Formula 3, FORMULAS IN TERMS OF POWERS OF n + Let nen. For ICGN, let S;(w) = + 2*+3* + -»+w* =E / '* and take S 6 (n) = n +. Then, arguing as in the steps leading to (2..) and (2..2), The equation (n + l) k =S k (n + l)-s k (n) = t(f)sm)s k (») (3..) /= k-\ (3..2) wy<* /=o v J (5 + )*-^* =(w + l) fc, (3..3) in which the binomial power is expanded and S (i =,,2,..., k) are then replaced by Sj(n), provides a mnemonic for (3..). Note in particular that (3..3) can be obtained from (2..3) by merely increasing the values of S, S-, and n by one. For example, let k = 2 in (3..3). Then + IS = (n +) 2 and thus n (H) = (w +) 2, again yielding (2..4). 3.2 S p (n) as a Determinant Let pgn and let # = l,2,...,/?,/? + l in (3..2). It follows as in section 2.2, with (-l) r replaced by, that 274 [JUNE-JULY

5 FORMULAS FOR + 2 ^ +3 P +--+n p or, alternatively, S p {n)-- S p (n) = --p\ G> + )!! (o e?)! (?) c")! o..» o U 7 + {n + lf {n + lf (A) ( n + l Y ( i) (H+ir ^ o («+ ) 3 U (3.2.) (3.2.2) P } - (p+iy. (p-iy. I pi (/>-2)! (p-)! (n+l) P! />! (»+l) P + (p+)! Note in particular that the determinants in (3.2.) and (3.2.2) can be obtained from their counterparts in (2.2.) and (2.2.2) by merely replacing n by «+ l in the last column, and replacing all negative entries by their absolute values. 3.3 S p {n) as a Polynomial Proceeding as in section 2.3, (3.2.2) can now be employed to establish the formula with c p+ = -^ and, for r =,2 '"p+l-r Hence, by (2.3.4) and (3.3.2), Thus, by (3.3.), S p (n) = f j c p+l _ r (" + V P+ - r, (3.3.) P, (-!//>! (p + l - r )! (r+)! i_(w! (r-)! (r-2)! J. (r-)! 5 r = l,2, (3.3.2) S p (n) = - } ~ { P r )B r (n + iy +l ~ r. (3.3.3) This standard form of S (ri) is usually established with the aid of the generating function xe* I (e x -) of the Bernoulli polynomials. 994] 275

6 FORMULAS FOR + 2 ^ +3 P + +n p The question arises if this method could have led to formulas (2.3.5) and (3.3.3) with B =, B x = - y, B 2 = \, B 3 =, B 4 = --^-,... without prior knowledge of the Bernoulli numbers. This indeed is the case. First note that, by (2.3.) and (2.3.3), and (3.3.) and (3.3.2), and w^ifrv^with b = and, for r =,2,..., /?, SM = ^tim P r l ) b r np+l - r > P~T~ l r= Ar=(-l) r r (r+)!! v (r-)! (r-2)! _L (r-)! y = i_!!! * ol (r-)! (r-2)! (r+)! (r-)! Now observe that the last determinant differs from that of S r (n), as obtained by setting/? = r in (3.2.2), only with respect to the last column the entries of b r, from top to bottom, are,,..., while those of S r (n) are ^, ^ ^,..., ^ g ~. It follows [cf (3..2)] that Z>,A? * 2,... satisfy the recurrence formula *o = l, XffU=(r = 2,3,4,...), which generates the numbers, - y, ^,, - ^,..., i.e., the Bernoulli numbers. REFERENCES. J.Bernoulli. Arsconjectandi D. Kalman. "Sums of Powers by Matrix Methods." The Fibonacci Quarterly 28. (99): R. A. Khan. "A Simple Derivation of a Formula for Z" k= k r." The Fibonacci Quarterly 9.2 (98): L. A. Lyusternik & A. R. Yanpol'skii, eds. Mathematical Analysis Functions, Limits, Series, Continued Fractions. New York: Pergamon Press, 965. AMS Classification Numbers: 5A, 5A9 276 [JUNE-JULY