Computation of Tangent, Euler, and Bernoulli Numbers* · 2018-11-16 · Computation of Tangent, Euler, and Bernoulli Numbers* By Donald E. Knuth and Thomas J. Buckholtz Abstract

Computation of Tangent, Euler, andBernoulli Numbers*

By Donald E. Knuth and Thomas J. Buckholtz

Abstract. Some elementary methods are described which may be used to cal-

culate tangent numbers, Euler numbers, and Bernoulli numbers much more easily

and rapidly on electronic computers than the traditional recurrence relations which

have been used for over a century. These methods have been used to prepare an

accompanying table which extends the existing tables of these numbers. Some the-

orems about the periodicity of the tangent numbers, which were suggested by the

tables, are also proved.

1. Introduction. The tangent numbers Tn, Euler numbers En, and Bernoulli

numbers Bn, are defined to be the coefficients in the following power series:

(1) tan z = To/0! + T&/V. + T2z2/2\ + • • • = Z Tnzn/n\ ,

(2) sec z = E0/0\ + Eiz/V. + E2z2/2\ +•••«= 5Z Enzn/n\ ,

(3) z/(ez - 1) = tfo/O! + Biz/1\ + ^2272! + ••• = £ Bnzn/n\ .

Much of the older mathematical literature uses a slightly different notation for

these numbers, to take account of the zero coefficients. Thus we find many papers

where tan z is written Tiz + T2z3/3\ + T-¡zb/5\ + - - -, sec z is written

Eo + Eiz2/2\ + E2z4/Al + ■ ■ -, and z/ (e* - 1) is written 1 - z/2 + Biz2/2l - B^/Al

+ Bsz6/6l ••'. Some other authors have used essentially the notation defined

above but with different signs; in particular our E2n is often accompanied by the

sign ( — 1)".

In Section 2 we present simple methods for computing T„, En, and Bn which are

readily adapted to electronic computers, and in Section 3 more details of the com-

puter program are explained. A table of Tn and En for n ^ 120, and Bn for n ^ 250,

is appended to this paper, thereby extending the hitherto published values of Tn

for n á 60 [6], En for n á 100 [2, 3], and Bn for n ^ 220 [7, 4].Using the methods of this paper it is not difficult to extend the tables much

further, and the authors have submitted a copy of the values of Tn On gj 835),

En (n ^ 808), Bn (n ^ 836) to the Unpublished Mathematical Tables repository

of this journal.

Section 4 shows how the formulas of Section 2 lead to some simple proofs of

arithmetical properties of these numbers.

2. Formulas for Computation. The traditional method of calculating Tn and En

is to use recurrence relations, such as the following: Let cos z = 2~2nô Cnzn/n;

Received February 6, 1967.

* Supported in part by NSF Grant GP 3909.

663

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664 D. E. KNUTH AND T. J. BUCKHOLTZ

then the coefficient of zn/n\ in (tan z) (cos z) is

T,(nk)TkCn-k

and in (sec z) (cos z) it is

Z{nk)EkCn-k.

Hence, making use of the fact that T2n = E2n+i = 0, we have the recurrence re-

lations

(4) (2» + >>, - (2»3+ l)r. + ■■■ + (-irfc + ;)r«, _ 1,

(« (^-(ïH+'-' + '-'K^K-0- ">o-The disadvantage of these formulas is that the binomial coefficients as well as the

numbers Tn, En become very large when n is large, so a time-consuming multiplica-

tion of multiple-precision numbers is implied. As Lehmer [4] has observed, we may

simplify the calculations if we remember the values of

(2\+1K (2;>'so that when n increases by 1 we need only multiply

Í2n + 1\ k

by

(2n + 2) (2n + 3)

(2n + 2 - k) (2n + 3 - k)

to get the next value ; but the method to be described here is even simpler and has

other advantages.

The tangent numbers may be evaluated by noting that D(t&nnz) is

n tan71-1 z (1 + tan2 z) ; hence the nth derivative of tan z is a polynomial in tan z.

We have Z>n(tan z) = P„(tan z), where the polynomials Pn(x) are defined by

(6) Pi(x) = X , Pn+lix) = (1 + X2)Pn'ix) .

Thus if we write

7>"(tan z) = Tno + Tni tan z + Tn2 tan2 z + ■ ■ -

the coefficients Tnk satisfy the recurrence equation

(7) Tok = oik ; Tn+i,k = (k — l)Tn,k-i + (k + l)Tn,k+i ■

Since Tn = Z)"(tan z)\z=o = Tno, and since Tnk is zero except for at most On + 3)/2

values of k, formula (7) shows that the calculation of all T„+i,k from the values of

Tn,k essentially requires only On + 2)/2 multiplications of a small number k by a

n


TANGENT, EULER AND BERNOULLI NUMBERS 665

iarge number Tn<k and n/2 additions of large numbers. Since we are interested only

In T„o for odd values of n, we might try to use the relation

Tn+2,k = (k - 2)(k - l)Tn,k-2 + 2k2Tn,k + (k + l)(k + 2)Tn.k+2

but a count of the operations involved shows this provides little if any improve-

ment over (7), and so the simpler form (7) is preferable.

Similarly, we have D(sec z tan" z) = sec z (n tan"-1 z + (n + l)tann+1 z), hence

if we write

(8) 7)"(sec z) = (sec z) (En0 + Eni tan z + En2 tan2 z + • • • )

we have the recurrence

(9) Eok = Ôok ; En+itk = kEn¡k_i + (k + l)En,k+i.

Since En = En0, this relation yields an efficient method for calculating the Euler

numbers. A somewhat similar recurrence relation was used by Joffe [3] to calculate

Euler numbers; his method requires essentially the same amount of computation,

but as explained in the next section there is a way to modify (9) to obtain a con-

siderable advantage.

The identities tan (7r/4 + z/2) = tan z + sec z and Z)n(tan (v/A + z/2)) =

2-nP„(tan (7r/4 + z/2)) imply that the sums of the numbers Tnk have a very simple

form:

(10) 2-"P„(l) = 2~n E Tnk = if" n eVf? 'kô [Tn , wodd .

This relation can be used to advantage when both E„ and Tn are being calculated.

The definition of tan z implies

sin z (eiz - e~iz) 1 ( 2iz . \ l( 2iz Aiz . \tan z = -= -:-— = -1 IT--iz I = -1 -T-.--r--iz )

cosz i(e" + e—) 2 V»2 + l / z \e2™ _ 1 eilz - 1 J

= -(-iz+Tl ((2iz)n - (Aiz)n)Bn/n\) ;Z \ n£0 /

and by equating coefficients we obtain the well-known identity

(11) Bn = -i~nnTn-i/2n(2n - 1) , n > 1 .

Hence, the Bernoulli numbers may be obtained from the tangent numbers by a

calculation which (on a binary computer) is especially simple.

The celebrated von Staudt-Clausen theorem [8, 1] states that

(12) B2n = C2n- Z Ip prime '.(p—1) \2n y

where C2n is an integer. The table appended to this paper expresses Bn in this form,

and, as shown below, the calculation of (11) may be carried out without any

multiple-precision division.

3. Details of the Computation. By the recurrence (7) we may discard the value

of Tn,k once T„+i,k+i has been calculated, so only about n of the values T„fk need



to be retained in the computer memory at any one time. A further technique can

be employed when the memory size has been exceeded; for example, suppose we

start with the computation of Tnk for n %. A:

k = 0 fc = 1 k = 2 fc = 3 k = 4 k = 5

n = 0 0 1

n = 1 1 0 1

n = 2 0 2 0 2

n = 3 2 0 8 0 6n = A 0 16 0 40 0 24

and suppose that very little memory space is available, so that we cannot com-

pletely evaluate all of the entries for n = 5 ; we might obtain

n = 5 16 0 136 0 240 0 *

where "*" denotes an unknown value. The calculation may still proceed, keeping

track of unknown values :

n = 6 0 272 0 1232 0 *ii = 7 272 0 3968 0 *n = 8 0 7936 0 *n = 9 7936 0 * etc.

In this way we may compute the values of about twice as many tangent numbers

as were produced before overflow occurred, avoiding much of the calculation of

the Tn,k.

Since the numbers Tn become very large (Tss¡, has 1866 digits, and Tn is asymp-

totically 2n+2n\/vn+l when n is odd), care needs to be taken for storage allocation of

the numbers Tn.k if we are to make efficient use of memory space. The program we

prepared makes use of two rather small areas of memory (say A and B) each of

which is capable of holding any one of the numbers T„,k, plus a large number of

consecutive locations used for all the remaining values. By sweeping cyclically

through this large memory area, it is possible to store and retrieve the values in a

simple manner.

For the sake of illustration let us suppose the word size of our computer is very

small, so that only one decimal digit may be stored per word ; and suppose there

are just 14 words of memory used for the table of Tn¡k. After the calculation of the

values for n = A, the memory might have the following configuration :

l~6~i- [116 1, | 4 | 0 | , | 2 | 4 | . | , | 8 | , |

(13) î ÎP Q

Here P and Q represent variables in the program that point to the current places

of interest in the memory; P points to the number that will be accessed next, and

Q points to the place where the next value is to be written. Only locations from P

to Q contain information that will be used subsequently by the program. The sym-

bols "." and "," represent special negative codes in the table which delimit the

numbers in an obvious fashion. As we begin the calculation for n = 5, we set area

A to zero and a variable k to 1. The basic cycle is then :



(a) Set area B to k times the next value indicated by P, and move P to the

right.

(b) Store the value of A + B into the locations indicated by Q, and move Q

to the right.

(c) Transfer the contents of B to area A.

(d) Increase fc by 2.

In the case of (13) we would change the memory configuration to

I 6 | . | 1 | 6 | , | 4 | 0 | , | 2 | 4 | . | 1 | 6 | , |

(14) î îQ P

k = 3 A = 16 B = 16

Notice that the value 16 has been stored, the pointer Q has moved to the right and

(treating the memory as a circular store) then to the far left. The next two itera-

tions of steps (a)-(d) give

| 1 | 3 | 6 [ , | 2 | 4 | 0 | , | 2 | 4 | . | 1 | 6 | , |

(15) T îQ P

k = 7 A = 120 I B = 120

Now since the terminating "." was sensed, the program attempts to store the

value from area A ; but since this would make pointer Q pass P, the "memory over-

flow" condition is sensed, and the memory configuration becomes

|~1~T3 | 6 | , | 2 | 4 | 0 | , |* | 2 | 0 | 1 | 6 | , |

(16) î îQ P

where "*" is another internal code symbol. The computation for n = 6 is similar

but it uses a different initialization since n is even; after n = 6 has been processed

we would have

J2[3|2|,|*|4¡0|,|*|2|7|2|,|1|

(17) T îQ P

and so on.

The above discussion has been slightly simplified for purposes of exposition. In

the actual program, it is preferable to keep the numbers stored with least signifi-

cant digit first, so that for example (16) would really be

16|3|1|,|0|4|2|,|*|2|1|6[1|,|

(18) î îQ P

in order to simplify the multiple-precision operations. A few other changes in the

sequence of operations were made in order to use memory a little more efficiently

(for example the value Tno need never be retained).

A similar method may be used for En. This arrangement of the computation

gives a substantial advantage over Joffe's method [3] because of the "*", and it



also has advantages over (10) for the same reason.

It remains to consider the calculation of the Bernoulli number B2n from T2n-i.

Consider formula (12) ; if p is an odd prime, 2P-1 = 1 (modulo p), hence if (p — l)\2w,

then 22" — 1 is divisible by p. So we first compute the integer

(19) n= (-iy x2nT2n-i + Z .p prime \(j>—1) \2n

(2n)(22n)(22"

V

A

by referring to an auxiliary table of primes that may be calculated at the beginning

of the program. Then it is merely a question of computing

C2n = N/22n(22n - 1) = N/2in + N/2«n + N/2Sn +(20)

The calculation of N/2k is of course merely a "shift right" operation in a binary

computer, so all the terms of the infinite series on the right side of (20) are readily

computed. This series converges very rapidly, and we know C2n is an integer, so we

need only carry out the calculation indicated in (20) until it converges one word-

size (35 bits) to the right of the decimal point. It is simple to check at the same

time that C2n is indeed very close to an integer, in order to verify the computations.

4. Periodicity of the Sequences. Examination of the tables produced by the com-

puter program shows that the unit's digits of the nonzero tangent numbers repeat

endlessly in the pattern 2, 6, 2, 6, 2, 6, starting with Tz ; furthermore the two least

significant digits ultimately form a repeating period of length 10: 16, 72, 36, 92,

56, 12, 76, 32, 96, 52, 16, 72, ... . The three least significant digits have a period

of length 50, and for four digits the period-length is 250. These empirical observa-

tions suggest that theoretical investigation of period-length might prove fruitful.

Theorem 1. Let p be an odd prime, and let X be the period-length of the sequence

(T„ mod p). Then

(21)

and

X = p-1,2ip - 1)

p = 1 (mod 4)

p = 3 (mod 4)

(22) Tn+x = Tn (mod p) for all n à 0 .

Proof. It is clear from the recurrence relation (7) that the sequence (Tn mod p)

is determined by the recurrence equation

(23) yn+i = Ayn

where the vector yn and the matrix A are defined by

(24)

0 21 0 3

2 0 4

3 •

0

0 p-1

p - 2 0 .

!Jn

Tn,l

I n,1

L J n,p—1 J



For Tn¡k can contribute nothing to any subsequent value of T„ when fc 2: p.

We will show below that the minimum polynomial equation satisfied by A is

(25) Ap-1 - (-l)(^-1)/27 = 0 (modulo p)

hence (22) is valid for the value of X given by (21). It remains to show that X is

the true period-length of the sequence, not merely a multiple of the period.

Accordingly, suppose Tn+y = Tn(mod p) for some positive X' ^ X and all large

n. In view of (22) this congruence must hold for all n ^ 0. Let y = y y — yo', then

p(Any) = 0 for all n ^ 0 where p denotes the projection onto the first component

of the vector Any. But this implies n\an = 0 (mod p) for all components an of y,

hence y = 0, i.e., y0 = y\' = A^'yo. It follows that yn = Ax'yn for all n ^ 0, and

since the vectors y0, • ■ -, yP~2 are obviously linearly independent we must have

Ax = / (modulo p). Therefore, X' is à X, and the proof is complete.

It remains to verify (25), which seems to be a nontrivial identity. Clearly, the

minimum polynomial of A must be of degree p — 1, since y0, • • •, yP-2 are linearly

independent; therefore, it suffices to calculate the characteristic polynomial of A.

Let

x —On — 1)

-n x —(n — 2)

(26) Dn = det

in- 1)

x-2

then Dn — xDn-i — On — l)nD„-2 so we have

Di = x ,

D2 = x2 - 1-2,

D3 = x3 - (1-2 + 2-3).c,

Da = x" - (1-2 + 2-3 + 3-4)a:2+ 1-2-3-4,

Db = xb - (1-2 + 2-3 + 3-4 + 4-5).r3+ (1-2-3-4 + 1-2-4-5 + 2-3-4-5).r ,

and in general

7-s, n n—2 i n—4 n—6 ,

Dn = X — SnlX + S„2X — SnZX + ■ • • ,(27)

where

(28) Snk = 2 ai(ai + l)ß2(a2 + 1) • • -ak(ak + 1)

is summed over all values 1 ^ oti <C a2 <<C • • • <3C ak < n. (Here u <3C v, for integers

u, v, denotes v ^ u + 2.) Thus, snk is the sum of all products of fc of the pairs

1 -2, 2 -3, • • -, (n — 1) -n with no "overlapping" pairs allowed in the same term.

To evaluate sip-i)k mod p, it is convenient to allow also the pairs (p — l)-p

and p • 1, since these contribute nothing to the sum. Thus for example,



s62 = 1-2-3-4+ 1-2-4-5 + 1-2-5-6 + 1-2-6-7 + 2-3-4-5 + 2-3-5-6

+ 2-3-6-7+ 2-3-7-1 + 3-4-5-6 + 3-4-6-7 + 3-4-7-1

+ 4-5-6-7 + 4-5-7-1 + 5-6-7-1

(modulo 7). Let us say two terms ai(ai + 1) • • ■ ak(ak + 1) and a/(ai' +1) • • •

ak'(ak + 1) are "equivalent" if, for some r and t and for all j, a, = a'(y+r)mod p + t;

thus, in the above example the terms 1-2-4-5, 2-3-5-6, 3-4-6-7, 4-5-7-1,

5-6-1-2, 6-7-2-3, 7-1-3-4 are mutually equivalent. It is impossible for a term to

be equivalent to itself when 0 < t < p, since this would imply ai + • • • + ak

= öi+ • • • + o,k + kt, and t = 0. Therefore, each equivalence class has precisely

p terms in it. When fc < Op — l)/2 the sum over an equivalence class has the form

Z iai + t)iai + t+!)■■■ iak + t)(ak + t + 1)0Si<p

where the summand is a polynomial of degree ^ p — 2 in t. Any such summation

may be expressed modulo pasa sum of terms of the form

co&0) = cC+\) = °' since o^^"1'

0. It follows that

,(j>-l)/2.

SO skp

(29) Dp-i = x*-L + (-ly^'Xp - 1)! (modulo p)

and an application of Wilson's theorem completes the proof of (25).

Theorem 2. Let p be an odd prime, and let X be the period-length of the sequence

(En mod p). Then

\p — 1 , p = 1 Cmod 4)

l2(p - 1) , p = 3(mod4)

and

(31) En+X = En (mod p) for all n ^ 1 .

Proof. Make the following changes in the proof of Theorem 1 :

(30) X =

(32) A =

p-10 J

Un

En,0

En,i

L-En,js_i Jp-1

Then the minimum polynomial equation satisfied by A is

(33) Ap - (-l)(p-1)/2A = 0 (modulo p) .

The proof is a straightforward modification of the proof of Theorem 1.

The congruences (22) and (31) were obtained long ago by Kummer (see for ex-

ample [5, p. 270]), but it was not shown that the true period-length could not be a

proper divisor of the number X given by (21), (30). More general congruences given



by Kummer make it possible to establish further results about the period-length :

Theorem 3. Let p be an odd prime, and let X be given by (30). Then

(34) Tn+Xvh-i = Tn (modulo pk) , n ^ fc ,

(35) En+Xpk-i = En Omodulo ph) , n ^ fc .

Proof. Assume to ^ fc and define the sequence (um) by the rule

(36) um = i-l)^1)m/2Tn+(p-i)m , mÔ.

Kummer's congruence for the tangent numbers may be written

(37) Akum =■ 0 (modulo pk) , m ^ 0 , k ^ 1 ,

where Akum denotes

Um+k — I . jUm+k-l + l o )Um+k-2 — • • • + (— l)kUm .

We will prove that (37) implies

(38) Um+pr-i = Um (modulo pr) , m ^ 0 , r ^ 1 ,

and this will establish (34). Eq. (35) follows in the same way if we let

t*m — V XJ *-'n+(p—l)m •

Assume Eq. (37) is valid for some sequence of real numbers (not necessarily

integers) uo, U\, • • • ; thus, Akum is an integer multiple of pk when fc ^ 1, but not

necessarily when fc = 0. We will prove that the sequence um/p, um+p/p, um+2p/p, • • •,

for fixed m also satisfies Eq. (37), and this suffices to prove (38) by induction on r.

Let E be the operator Eum = um+i. Eq. (37) may be written (E — l)kum = 0

(modulo pk), and our goal as stated in the preceding paragraph is to show that

(Ep — l)k(um/p) = 0 (modulo pk), i.e. (Ep — l)kum = 0 (modulo pk+1). Let

f(E) = Ep-2 + 2EP-* + ■■■ + (p - 2)E + (p - 1); then E" - 1 =(E - l)(p + f(E)(E - 1)), hence

(Ep - 1)V» = Z ( * )pj(E - l)2k~jf(E)k-jum

and each term in the sum on the right is an integer multiple of p2k. Hence, we have

proved in fact that (Ep — l)kum = 0 (modulo p2k), which is more than enough to

complete the proof of the theorem.

Note that Eqs. (34), (35) do not necessarily give the true period-length of the

sequence mod pk when fc > 1; although (34) is "best possible" when p = 5 and

fc = 2, 3, 4, the tangent numbers have the same period-length modulo 9 as they

do modulo 3.

The tangent number T2n+i is divisible by 2", so the period length of T„ mod 2r

is 1 for all r. Eq. (35) is valid for X = 2 when p = 2, since Kummer's congruence

(37) holds for um = En+2m. In particular, we may combine the results proved above

to show that for any modulus m the sequences Tn mod m, En mod m are periodic,

and the period-length divides 2<p(m).



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Mathematics Department

California Institute of Technology

Pasadena, California 91109

1. Thomas Clausen, "Theorem," Asir. Nachrichten, v. 17, 1840, cols. 351-352.

2. S. A. Joffe, "Calculation of the first thirty-two Eulerian numbers from central differences

of zero," Quart. J. Math., v. 47, 1916, pp. 103-126.3. S. A. Joffe, "Calculation of eighteen more, fifty in all, Eulerian numbers from central

differences of zero," Quart. J. Math., v. 48, 1917-1920, pp. 193-271.4. D. H. Lehmee, "An extension of the table of Bernoulli numbers," Duke Math. J., v. 2,

1936, pp. 460-464.5. Niels Nielsen, Traité Élémentaire des Nombres de Bernoulli, Paris, 1923.

6. J. Peters & J. Stein, Zehnstellige Logarithmentafel, Berlin, 1922.

7. S. Z. Serebrennikoff, "Tables des premiers quatre vingt dix nombres de Bernoulli,"

Mém. Acad. St. Petersbourg 8, v. 16, 1905, no. 10, pp. 1-8.

8. K. G. C. von Staudt, "Beweis eines Lehrsatzes die Bernoullischen Zahlen betreffend,"

J.für Math., v. 21, 1840, pp. 372-374.


Documents

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