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easy to obtain an expanded expression for this, it is very easy to calculate its successive values in a manner analogous to that used in Ch. II. Art. 13.

Let C = numerical value of the coefficient of out in the expansion of aclm). Then since &*+1) = (a - n) c("), we obtain Crn+1 = C'm_, + n Cr"....

(22), and we can thus calculate the values of On+1 from those of C"; and we know that the values of Care 1, 0, 0, ... 11. Let us denote by C* the numerical value of the coeffi

1 cient of in the expansion of aclm) in negative powers of x,

ak

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(-1)^-1

1 Then ock-n) = !

(-1)-1

1
1
(where A now refers to p alone)
(-1)-1

(1
pp

. &c.
n-1

22 23
(-1)), AM-10

A9-10. An-102
+

&c
1 aca X 3

4*-10--1 :: C"=(-1)**

(23).

n-1 also evident from the following consideration : Dion) D(n) or 1

log 2 putting x=log 2 dzn

n

an

{

an

n

n

%=1

=

an

log (1 + 2) = coefficient of zn in the [n dza

2 = 0 expansion of log (1 + 2)" by Maclaurin's theorem. Thus this expansion may be written 1

*+1 +1 k+2 +2
{log (1+2)}" =

**+1+cm . &c.
K+1

K+2

к

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A formula analogous to (22) can also be obtained by means of Art. 13, Ch. II. This gives for numerical values

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and thus from the values of Cx-1 those of Ck can be obtained. The values of C, are of course 1, 0, 0, ...

12. Analogous series are those of the coefficients when ach and x* are expanded in factorials. By (5) page 11, we have

A'0"
Oc" = 0o + A0”.x +

&c. ....
1.2

2 (2)

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* It will be seen that, as in the analogous case we could expand {log (1 + 2)}" in terms of 0m, we can expand (6 - 1)" in terms of -(x+1)

In fact

K'

(6 - 1)* = *C

(x+1) z*+1
*+2 n+1

- (x+1)

gle+2 +C

*x+3 (n + 1) (n + 2)

+&C. ......... (26)

- (x+1) where we have given C its numerical value, disregarding its sign.

Ben-z=integer +(-15" (

+

11) –integer +1576.

13. There is another class of properties of Bernoulli's numbers that has received some attention; these relate to their connection with the Theory of Numbers. Staudt's theorem will serve to illustrate the nature of these properties. It is that

1 +Σ

2m + 1 where m is a divisor of n such that 2n +1 is a prime number. Thus, taking n=8, we have (since the divisors of 8 are 1, 2, 4, 8)

1 1 1 B15=integer +

12

5' 17 It will be found on reference to page 91 to be 7717. Staudt's paper will be found in Crelle (xxi. 374), but a simpler demonstration of the above property has been given by Schläfli (Quarterly Journal, vi. 75). On this subject see papers by Kummer (Crelle, XL. XLI. LVI.). Staudt's theorem has also been given by Clausen.

14. To Raabe is due the invention of what he names the BernoulliFunction, i.e. a function F (20) given by

F(2)=1"+ 2n + +(2C – 1)" when a is an integer, and which is given generally by AF(x)=x”. He has also given the name Euler-Function to the analogous one that gives the sum of

in – 2n + 3n - &c. + (2.C – 1)" when x is integral. See Brioschi (Tortolini, Series II. 1. 260), in which there is a review of Raabe's paper (Crelle, XLII. 348) with copious references, and Kinkelin (Crelle, LVII. 122). See also a note by Cayley (Quarterly Journal, II. 198).

15. The most important papers on the subject of this Chapter are a series by Blissard (Quarterly Journal, Vols. IV.—Ix.) under various titles. The demonstrations shew very strikingly the great power obtainable by the use of symbolical methods, which are here developed and applied to a much greater extent than in other papers on the subject. They include a most complete investigation into all the classes of numbers of which we have spoken in this Chapter; the results are too copious for any attempt to give them here, but Ex. 15 and 16 have been borrowed from them. The notation in the original differs from that here adopted. Ben there denotes what is usually denoted by B2n-1. See also two papers on Anom and its congeners by Horner (Quarterly Journal, iv.).

16. Attempts have been made to connect more closely Bernoulli’s and Euler's Numbers, which we know already to have markedly similar properties. Scherk (Crelle, IV. 299) points out that, since tan

+
4+2

=sec x + tan x, the expansion of this function in powers of a will have its coefficients depending alternately on each set of numbers {see (7) and (11), of this Chapter. This idea has been taken up by others. Schlömilch (Crelle, xxxII. 360) has written a paper upon it. It enables us to represent both series by one expression, but there is no great advantage in doing so, as the expression referred to is very complioated. · Another method is by finding the coefficient of an in the ex

T

1 pansion of from which both series of numbers can be deduced by

aed -1' taking a=+1 (Genocchi, Tortolini, Series I. Vol. 111. 395).

17. Schlömilch has connected Bernoulli's numbers and factorial coeffi. cients with the coefficients in the expansions of such quantities as Dnf (log »), Dn &c. (Grunert, VIII. IX. XVI. XVIII.) Most of his analysis could be

() rendered simpler by the use of symbolical methods. This is usually the case in papers on this part of the subject, and the plan mentioned in the last Chapter has therefore been adhered to, of giving characteristic examples out of the various papers with references, instead of referring to them in the text. We must mention, in conclusion, that the numbers of Bernoulli as far as B31 have been calculated by Rothe, and will be found in Crelle (xx. 11).

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EXERCISES.

1. Prove that

2n+1 02n+1)

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(402x+1 A2024+1

A
Byn_1 = (-1)"+1

1%

+ &c. +

(2n + 1)? 2. Prove that if n be an odd integer 1

n (n-1) (n − 2) 2

3

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B. +

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3. Obtain the formula of page 107, for determining successively Bernoulli's numbers, by differentiating the identity

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6. Apply Herschel's Theorem to find an expression for a Bernoulli's number.

7. Demonstrate the following relation between the even Bernoulli's numbers:

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9. Prove that the coefficient of Azn in the expansion of A

22n (2n – 1)
is to
0

2n

B2n-1

10. Express log sin x and log tan u in a series proceeding by powers of x by means of Bernoulli's numbers.

[Catalan, Comptes Rendus, Liv.]

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