92.10 Evaluating againAuthor(s): ROBIN CHAPMANSource: The Mathematical Gazette, Vol. 92, No. 523 (March 2008), pp. 92-94Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/27821726 .

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92 THE MATHEMATICAL GAZETTE

= |(1)(1)?4

+ 32 33 j(l)(15)--(6)(l)

n3 + |(D(39)-j(3)(15) +j(ll)(l)

(1)(8)-|(1)(39) + j(2)(15)-^(6)(l)

27 4 9 3 9 2 = ?/? + -ft-ft 4 2 4

ft.

The author would like to thank the referee for his valuable suggestions.

References

1. S. Barnard & J. Child, Higher Algebra, MacMillan (1936) pp. 114-115.

2. S. Simons, An approach to evaluating S? = i (a + nd)m, Math. Gaz. 88 (November 2004) pp. 519-522.

3. M. D. Hirschhorn, Evaluating Sn = i {a + ndf, Math. Gaz. 90 (March 2006) pp. 114-116.

4. N. Gauthier, Explicit formula for power sums of an arithmetic

sequence, Math. Gaz. 91 (March 2007) pp. 97-103.

5. Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics (2nd edn.), (Addison-Wesley) 1994.

HUNG-PING TSAO 7757 Highland Drive, Novato, CA 94949, USA

N

92.10 Evaluating ? (a + again ? = i

In [1] Hirschhorn proves a formula for the sum n

Sp =

X (a + n= 1

involving Bernoulli numbers. When a = 0 and d = 1 the sum reduces to

the celebrated sum X? = i nP. In the book Concrete Mathematics [2], section 2.5 gives an illuminating comparison of various methods of evaluating the latter sum. Here we adapt one standard method, using exponential generating functions, for computing Zn = i^ (described in [2, ?7.6]) to calculate the sums Sp.

A sequence ao, ai9 a2, ... of numbers has the ordinary generating

function

p = 0

and the exponential generating function -

tp

2>V" p = 0 P'

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NOTES 93

Usually T is regarded as an indeterminate and these series as formal power series, but sometimes T is taken to be a real or complex variable chosen small enough to make the series converge.

We define the exponential generating function of the Sp: TP

We calculate:

TP

)P' n=\ p = Oi

N = 11

n=1p=0

(a + ndfTp = X exP((a + nd)T).

This last sum is a finite geometric progression with first term exp((q + d)T) and common ratio exp(oT). We can now use the standard formula for

summing a geometric progression to simplify the sum. The reader might check that if we had used the ordinary generating function, the

corresponding computation would have led to a sum lacking a

straightforward closed form. We continue, making further manipulations to cast the denominator in a more 'symmetric' form:

exp((q + (N + l)d)T) - exp((q + d)T)

edT - 1

exp((q + (N + ?)d)T) -

exp((q + \d)T)

!f(T) =

We can now write (T) as a product of two series. The first is

exp((a + (N + \)d)t) - exp((a + \d)f)

- ? f)'

- (. Cl

To obtain the second we write

1_ ~ 2 edTI2

_ e-dT/2

{edm + e-dmf edTI2 + e-dTI2

edm + e~dm

-dm edT/2 + e-dT/2

edTI2 _

e-dT/2

l r L dT ^dTi = - coth-coth

? . 2L 4 2 J

By a formula cited in [1] rp rjn

CO rp2m - coth- =

? #2??r. 2 2 m = 0 (2 )!

(1)

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94

It follows that

so that

and

THE MATHEMATICAL GAZETTE

2 2 m4-0 (2m)!

-coth? = ??2m2 Z 4 m = 0

X2m - 1

1 - 2m^2m

- l?_

(2m)!

-2m l)d

m = 0

T-2/n- 1 ,2m- _

(2m)! ' (2)

Taking the coefficient of Tp in the product of (1) and (2) gives

Sp= y B2m{2l-2m-\)d2m-l\l . d\p+l~2m I d\p+l P' 0<m<[p/2}

and we conclude that

(2m)l(p+l-2m)\ l(?ff-K) -2m"

0<m<Lp/2j ( 2m B2m{2l-2m-\)d ,2m-1

a + Nd + - 2

p+l-2m np+\-2m

in agreement with [1].

References 1. M. D. Hirschhorn, Evaluating rn=i(a + ndf, Math. Gaz. 90 (March

2006) pp. 114-116.

2. Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete

Mathematics (2nd edn.), (Addison-Wesley) 1994.

ROBIN CHAPMAN

Department of Mathematics, University of Bristol, Royal Fort Annexe, Bristol BS8 1TW

e-mail: marjc@bris.ac.uk

92.11 A remarkable formula In [1] we find a reference to the following beautiful formula, apparently

discovered by Bernoulli in 1697: 00

1 11

The amazing thing about this is that neither author had ever seen it before; and presumably the same can be said of the authors of the reference below. It is surely known and, as is shown below, its proof is such that this result

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