Summation formulae for finite cotangent sums

Applied Mathematics and Computation 215 (2009) 1135–1140

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Summation formulae for finite cotangent sums Djurdje Cvijovic´ Atomic Physics Laboratory, Vincˇa Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Serbia

a r t i c l e

i n f o

Keywords: Trigonometric sums Finite summation Cotangent sums Alternate cotangent sums Contour integration Cauchy residue theorem Higher order Bernoulli polynomials Bernoulli polynomials Bernoulli numbers

a b s t r a c t Recently, a half-dozen remarkably general families of the finite trigonometric sums were summed in closed-form by choosing a particularly convenient integration contour and making use of the calculus of residues. In this sequel, we show that this procedure can be further extended and we find the summation formulae, in terms of the higher order Bernoulli polynomials and the ordinary Bernoulli polynomials, for four general families of the finite cotangent sums. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction Recently, a half-dozen remarkably general families of the finite trigonometric sums were summed in closed-form by choosing a particularly convenient integration contour and making use of the calculus of residues [1–4]. In this sequel, we show that the following families of finite alternating cotangent sums

S2nþ1 ðq; rÞ :¼

    q1 X 2rpp pp cot2nþ1 ð1Þp sin q q p¼1

ð1:1Þ

(n 2 N0 :¼ N [ f0g; q is an even positive integer greater than 2; r = 1, . . . ,q  1),

C 2n ðq; rÞ :¼

    q1 X 2rpp pp cot2n ð1Þp cos q q p¼1

ð1:2Þ

(n 2 N; q is an even positive integer greater than 2; r = 1,. . .,q  1) and

C 2n ðqÞ :¼

  q1 X pp ð1Þp cot2n q p¼1

ð1:3Þ

(n 2 N; q is an even positive integer greater than 2), as well as the family of sums given in (2.14) below, can be considered in the same way and we find their closed-form summation formulae in terms of the higher order Bernoulli polynomials and the ordinary Bernoulli polynomials and numbers. 2. Statement of main results Observe that, throughout the text, we set an empty sum to be zero. We use the floor function bxc, also called the greatest integer function or integer value, which gives the largest integer less than or equal to x. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.06.053

D. Cvijovic´ / Applied Mathematics and Computation 215 (2009) 1135–1140

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In what follows, we denote by BðmÞ n ðxÞ the Bernoulli polynomial of order m and degree n defined by means of the following generating functions (see, for details, [5, p. 53, et seq.] and [6, Section 1.6])

 et

t 1

m

etx ¼

1 X

BðmÞ n ðxÞ

n¼0

tn ; n!

ðjtj < 2p; m 2 N0 Þ:

ð2:1Þ

For m ¼ 1 we have

Bn ðxÞ :¼ Bnð1Þ ðxÞ;

ðn 2 N0 Þ;

ð2:2Þ

where Bn ðxÞ is the relatively more familiar (ordinary) Bernoulli polynomial (see, for instance, [5, p. 35, et seq.]). The (ordinary) Bernoulli number Bn is given by

Bn :¼ Bn ð0Þ ðn 2 N0 Þ:

ð2:3Þ

Our results are as follows. Theorem 1. Let BðmÞ n ðxÞ be the Bernoulli polynomial of order m and degree n defined by (2.1) and let Bn ðxÞ be the Bernoulli polynomial defined as in (2.2). Let di;j be the Kronecker delta and suppose that fxg ¼ x  bxc where bxc is the greatest integer function. Then, the sums S2nþ1 ðq; rÞ in (1.1) are given by

S2nþ1 ðq; rÞ ¼ S2nþ1 ðq; rÞ;

ð2:4Þ

where

S2nþ1 ðq; rÞ ¼ ð1Þn1 qdq;2r þ

  n 2nþ1 X  2n þ 1  2n þ 1  ð1Þn1 X r 1 ð2nþ1Þ B2n2a ðbÞq2aþ1 B2aþ1 þ  q 2 ð2n þ 1Þ! a¼0 b¼0 2a þ 1 b

ð2:5Þ

(n 2 N0 ; q is an even positive integer greater than 2; r = 1, . . . ,q  1), while the sums C 2n ðq; rÞ in (1.2) are given by

C 2n ðq; rÞ ¼ C2n ðq; rÞ;

ð2:6Þ

where

C2n ðq; rÞ ¼ ð1Þn qdq;2r þ

    n X 2n  2n 2n ð1Þn1 X r 1 ð2nÞ B2n2a ðbÞq2a B2a þ q 2 ð2nÞ! a¼0 b¼0 2a b

ð2:7Þ

(n 2 N; q is an even positive integer greater than 2; r = 1, . . . ,q  1). Theorem 2. Let BðmÞ n ðxÞ be the Bernoulli polynomial of order m and degree n defined by (2.1) and let Bn ðxÞ be the Bernoulli polynomial defined as in (2.2). Then, the sums

S2nþ1 ðq; rÞ :¼

q1 X

sin

p¼1

    2rpp pp cot2nþ1 q q

ð2:8Þ

ðn 2 N0 :¼ N [ f0g; q 2 N n f1g; r ¼ 1; . . . ; q  1Þ are given by

S2nþ1 ðq; rÞ ¼ S2nþ1 ðq; rÞ;

ð2:9Þ

where

S2nþ1 ðq; rÞ ¼

  n 2nþ1 X  2n þ 1  2n þ 1  ð1Þn1 X r ð2nþ1Þ B B2aþ1 ðbÞq2aþ1  q 2n2a ð2n þ 1Þ! a¼0 b¼0 2a þ 1 b

ð2:10Þ

(n 2 N0 ; q 2 N n f1g; r = 1,. . . ,q  1), while the sums

C 2n ðq; rÞ :¼

q1 X p¼1

cos

    2rpp pp cot2n ðn 2 N; q 2 N n f1g; r ¼ 1; . . . ; q  1Þ q q

ð2:11Þ

are given by

C 2n ðq; rÞ ¼ C2n ðq; rÞ; where

ð2:12Þ

D. Cvijovic´ / Applied Mathematics and Computation 215 (2009) 1135–1140

C2n ðq; rÞ ¼

    n X 2n  2n 2n ð1Þn1 X r ð2nÞ ðbÞq2a B B2a q 2n2a ð2nÞ! a¼0 b¼0 2a b

1137

ð2:13Þ

ðn 2 N; q 2 N n f1g; r ¼ 1; . . . ; q  1Þ: Remark 1. Two families of the finite sums which are evaluated in closed-form in Theorem 1, to the best of our knowledge, were never studied before (see [8–13]). In the case of Proposition 3 below, it should be noted that the sums C 2n ðqÞ and C 2n ðqÞ were first summed by Chu and Marini [8, p. 137 and p. 155]. Moreover, C 2n ðqÞ were differently evaluated by Cvijovic´ and Klinowski [7, p. 156] and by Berndt and Yeap [9, p. 364]. The results given in Theorem 2 were previously obtained by Cvijovic´ et al. [4, p. 201, Eq. (2.1) and pp. 207 and 208, Eqs. (3.2) and (3.3)], by the method which is also applied in this study, and we reproduce them without the proofs for the sake of completeness and comparison (see Theorem 1). Observe that the sums S2nþ1 ðq; rÞ and C 2n ðq; rÞ were introduced first by Williams and Zhang [14] who closed-form evaluated them using expressions equivalent to those in (2.10) and (2.13). Berndt and Yeap [9, pp. 376–378, Theorem 4.1] studied S2nþ1 ðq; rÞ and C 2n ðq; rÞ by making use of the contour integration and explicitly obtained another equivalent closed-form formula for S2nþ1 ðq; rÞ. Proposition 3. Let us suppose the assumptions of Theorem 2. Then, the sums

C 2n ðqÞ :¼

q1 X

cot2n

p¼1

  pp ; q

ðn 2 N; q 2 N n f1; 2gÞ

ð2:14Þ

are given by

C 2n ðqÞ ¼ C2n ðqÞ;

ð2:15Þ

where

C2n ðqÞ ¼ ð1Þn q þ

  n X 2n  2n 2n ð1Þn1 X ð2nÞ B2a B2n2a ðbÞq2a ð2nÞ! a¼0 b¼0 2a b

ð2:16Þ

ðn 2 N; q 2 N n f1; 2gÞ; while the sums C 2n ðqÞ in (1.3) are given by

C 2n ðqÞ ¼ C2n ðqÞ; where

C2n ðqÞ ¼

    n X 2n  2n 2n ð1Þn1 X 1 ð2nÞ B B2a ðbÞq2a 2 2n2a ð2nÞ! a¼0 b¼0 2a b

ð2:17Þ

ð2:18Þ

ðn 2 N; q is an even positive integer greater than 2Þ: Corollary 1. In terms of C2n ðqÞ defined as in (2.16) we have

 pp 1 ðn 2 N; q 2 N n f1gÞ; ¼ C2n ð2qÞ 2 2q p¼1   q X pp 1 ¼ C2n ð2q þ 1Þ cot2n ðn 2 N; q 2 N n f1gÞ: 2 2q þ 1 p¼1 q1 X

cot2n



ð2:19Þ ð2:20Þ

Example 1. It is not difficult to verify that, in general, C 2n ðqÞ in (2.14) is a polynomial of degree 2n in q. These first few polynomials generated by the formula in (2.16) are

1 2 2 q qþ ; 3 3 1 4 4 2 26 C 4 ðqÞ ¼ q  q þq ; 45 9 45 2 6 2 4 23 2 502 C 6 ðqÞ ¼ q  q þ q qþ ; 945 45 45 945 8 16 6 44 4 176 2 7102 q8  q þ q  q þq ; C 8 ðqÞ ¼ 4725 2835 675 315 14; 175 2 2 86 6 718 4 563 2 44834 C 10 ðqÞ ¼ q10  q8 þ q  q þ q qþ ; 93; 555 2835 8505 8505 945 93; 555 1382 8 106 8 544 6 21; 757 4 6508 2 295; 272; 982 q12  q10 þ q  q þ q  q þqþ C 12 ðqÞ ¼ 638; 512; 875 93; 555 70; 875 35; 721 212; 625 10; 395 638; 512; 875 C 2 ðqÞ ¼

and they are the same as those listed in literature (see, for instance, [7, pp. 155 and 156] and [8, p. 137]).

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3. Proof of the results Our derivation of the above-given summation formulae would make use of contour integration and calculus of residues and is based, in part, upon the following three power-series expansions. Note that the residue of a meromorphic function U at a pole z0 is denoted by ResðU; z0 Þ. Suppose that k is a real and m is a nonnegative integer. If jzj < p, then:

  1  z m X 1 ð2zÞ2n ðmÞ cosð2kzÞ ¼ ð1Þn B2n kþ m ; sin z 2 ð2nÞ! n¼0   1  z m X 1 ð2zÞ2nþ1 ðmÞ sinð2kzÞ ¼ ð1Þn B2nþ1 kþ m ; sin z 2 ð2n þ 1Þ! n¼0

ð3:1Þ ð3:2Þ

and

ðzcotzÞm ¼

 1 m  X m ðmÞ 1 X ð2zÞ2n n ð1Þ B2n ðkÞ ; m ð2nÞ! 2 n¼0 k k¼0

ð3:3Þ

where fxg ¼ x  bxc. Observe that the derivation of the expansions in (3.1), (3.2) can be, for instance, found in [1, Lemma 1] while for (3.3) see [4, p. 202, Eq. 2.9]. It should be noted that the same contours, the contours C and C defined as follows, are required for all needed integrations. Let C denote the indented square with vertices at p  ıp and p  ı p and two semicircular indentations, both of radius Rq , 0 < Rq < p=q ðq 2 N n f1gÞ, one to the left of z ¼ p and one to the right of z ¼ p. Let C denote the circle

  p 0 < Rq < ; q 2 N n f1g : q

 p  z   ¼ Rq ; 2

It is assumed that the contours C and C are positively-oriented. Proof of Theorem 1. To prove (2.4) and (2.5) consider the contour integral of Us ðzÞ along the contour C, where Us ðzÞ is given by

sinð2rzÞ cot2nþ1 z sinðqzÞ ðn 2 N0 ; q is an even positive integer greater than 2; r ¼ 1; . . . ; q  1Þ:

Us ðzÞ ¼ q

ð3:4Þ

Suppose, for the moment, that r – 2q. Firstly, it can be shown that

I C

Us ðzÞdz ¼ 4pıS2nþ1 ðq; rÞ þ

I C

Us ðzÞdz;

ð3:5Þ

S2nþ1 ðq; rÞ being the sums in (1.1) and where the contours C and C are traversed in the positive (counter-clockwise) direction. Then, since

I C

Us ðzÞdz ¼ 

I C

Us ðzÞdz;

ð3:6Þ

we have

S2nþ1 ðq; rÞ ¼ 

1 2pı

I C

Us ðzÞdz ¼ ResðUs ðzÞ; 0Þ:

ð3:7Þ

We omit here the proofs of (3.5), (3.6) and (3.7) because they are the same as those already given in detail several times before [1–4]. Secondly, we compute the required residues in (3.7) by extracting the coefficient of z1 in the Laurent-series expansion of Us ðzÞ. By applying the series (3.2) (with m ¼ 1) and (3.3) (with m ¼ 2n þ 1), upon making use of the Cauchy product of two power-series, we obtain the Laurent expansion of Us ðzÞ

Us ðzÞ ¼ z2n2

  X1 qz r ð1Þk ck z2k2n1 sin 2 qz ðzcotzÞ2nþ1 ¼ k¼0 sinðqzÞ q

with the coefficients ck

D. Cvijovic´ / Applied Mathematics and Computation 215 (2009) 1135–1140

ck ¼

1139

  k 2nþ1 X  2k þ 1  2n þ 1  ð1Þk1 X r 1 ð2nþ1Þ B2aþ1 þ B2n2a ðbÞ q2aþ1 : q 2 ð2k þ 1Þ! a¼0 b¼0 2a þ 1 b

This expression, when k ¼ n, yields the coefficient of z1 , thus we obtain the residue in (3.7) and finally arrive at the proposed summation formula for S2nþ1 ðq; rÞ in (2.5) valid when r– 2q. For the case r ¼ 2q, (3.4) and (3.5) are still valid but not and (3.6) since, now, these integrals are equal. Hence, S2nþ1 ð2r; rÞ ¼ S2nþ1 ð2r; rÞ ¼ 0. However, we choose to use the same formula for all r; r ¼ 1; . . . ; q  1; with an additional term in the case of r ¼ 2q (i.e when dq;2r ¼ 1). This term cancels the other one and it could be found easily upon noting that

Resðqcot2nþ1 ðzÞ; 0Þ ¼ ð1Þn q

ðn 2 N0 Þ:

To prove (2.6) and (2.7) consider the contour integral of U

Uc ¼ q

ð3:8Þ  c ðzÞ

along the contour C, where U

 c ðzÞ

is given by

cosð2rzÞ cot2n z sinðqzÞ

ð3:9Þ

(n 2 N; q is an even positive integer greater than 2; r ¼ 1; . . . ; q  1). The proof proceeds along exactly the same lines as the above one. The series (3.1) (with m ¼ 1) and (3.3) (with m ¼ 2n) are used for the Laurent expansion of Uc ðzÞ, but now, for the case r ¼ 2q, we have one more singularity with residue ð1Þn q (see (3.8)) which should be added at the right-hand side of the formula (2.7). h Proof of Proposition 3. In order to prove (2.15) and (2.16) we use the function

Wc ¼ q

cos½ð2r  qÞz cot2n z ðn 2 N; q 2 N n f1g; r ¼ 1; . . . ; q  1Þ sinðqzÞ

ð3:10Þ

and consider its contour integral along the contour C. Observe that Wc is used in a closed-form evaluation of the sums C 2n ðq; rÞ (see [1, Proof of Theorem 1]) which are given in the above Theorem 2 and the result of this summation is the formula in (2.13). In the case when r ¼ 0 to this formula ð1Þn q should be added. Similarly, to prove (2.17) and (2.18), we use (3.9) and subtract ð1Þn q from the formula (2.7). h 4. Concluding remarks We remark that by making use of such elementary series identities as (for example) 2q1 X p¼1

  X   2q1   q1 X pp pp pp cot cot cot ¼ þ 2q 2q 2q p¼1 p¼qþ1

we could derive not only the sought formulae in Corollary 1 (see (2.19) and (2.20)), but also many other summations, such as q1 X p¼1

tan2n

  X   q pp ð2p  1Þp 1 tan2n ¼ ¼ C2n ð2qÞ ðn 2 N; q 2 N n f1gÞ: 2 2q 2q p¼1

In order to demonstrate an application of our results, we list several of the sums S2nþ1 ðq; rÞ defined by (1.1) and evaluated in (2.5)

  r 1  qdq;2r ; S1 ðq; rÞ ¼ 2qB1 þ q 2     4 r 1 r 1 S3 ðq; rÞ ¼ q3 B3 þ þ 2qB1 þ þ qdq;2r ; 3 q 2 q 2       4 r 1 20 3 r 1 r 1 S5 ðq; rÞ ¼  q5 B5 þ  q B3 þ  2qB1 þ  qdq;2r ; 15 q 2 9 q 2 q 2         8 7 r 1 28 5 r 1 392 3 r 1 r 1 q B5 þ þ q B5 þ þ q B3 þ þ 2qB1 þ þ qdq;2r ; S7 ðq; rÞ ¼ þ 315 q 2 45 q 2 135 q 2 q 2         4 r 1 8 7 r 1 76 5 r 1 3272 3 r 1    S9 ðq; rÞ ¼  q9 B5 þ q B5 þ q B5 þ q B3 þ 2835 q 2 105 q 2 75 q 2 945 q 2   r 1  2qB1 þ  qdq;2r ; q 2 and several of the sums C 2n ðq; rÞ (see (1.2) and (2.7))

D. Cvijovic´ / Applied Mathematics and Computation 215 (2009) 1135–1140

1140

 r 1 2 þ  qdq;2r ; þ q 2 3     2 r 1 8 r 1 26 C 4 ðq; rÞ ¼  q4 B4 þ  q2 B2 þ  þ qdq;2r ; 3 q 2 3 q 2 45       4 6 r 1 4 r 1 46 r 1 502 C 6 ðq; rÞ ¼ q B6 þ þ q4 B4 þ þ q2 B2 þ þ  qdq;2r ; 45 q 2 3 q 2 15 q 2 945         2 8 r 1 32 6 r 1 88 r 1 352 2 r 1 7102 C 8 ðq; rÞ ¼  q B8 þ  q B6 þ  q4 B 4 þ  q B2 þ  þ qdq;2r ; 315 q 2 135 q 2 45 q 2 105 q 2 14; 175         4 r 1 4 8 r 1 172 6 r 1 1346 4 r 1 þ þ þ C 10 ðq; rÞ ¼ q10 B10 þ q B8 þ q B6 þ q B4 þ 14; 175 q 2 189 q 2 405 q 2 567 q 2   1126 2 r 1 44; 834 þ þ q B2 þ  qdq;2r : 315 q 2 93; 555

C 2 ðq; rÞ ¼ 2q2 B2



Next, for convenience, we compute several C 2n ðqÞ

1 2 C 2 ðqÞ ¼  q2 þ ; 6 3 7 4 2 2 26  C 4 ðqÞ ¼  q þ q  ; 360 9 45 31 7 4 23 2 502 C 6 ðqÞ ¼  q6 þ q  q þ ; 15120 180 90 945 127 31 6 77 4 88 2 7102 C 8 ðqÞ ¼  q8 þ q  q þ q  ; 604; 800 5670 1350 315 14175 73 127 1333 6 359 4 563 2 44; 834 q10 þ q8  q þ q  q þ ; C 10 ðqÞ ¼  3; 421; 440 181440 136; 080 4860 1890 93; 555 1; 414; 477 73 6731 527 6 21; 757 4 q12 þ q10  q8 þ q  q C 12 ðqÞ ¼  653; 837; 184; 000 855; 360 4; 536; 000 35; 721 243; 000 3254 2 295; 272; 982 þ q  : 10; 395 638; 512; 875 We conclude by remarking that four remarkably general families of the finite cotangent sums are evaluated in closed-form by choosing a particularly convenient integration contour and making use of the calculus of residues. The obtained summation formulae involve the higher order Bernoulli polynomials and the ordinary Bernoulli polynomials and numbers. Acknowledgements The author thankfully acknowledges the valuable comment of the anonymous referee of this journal. Present investigation was supported by Ministry of science and technological development of the Republic of Serbia under Research Projects 142025 and 144004. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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