Applied Mathematics and Computation 218 (2012) 6744–6747

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Another discrete Fourier transform pairs associatedwith the Lipschitz–Lerch zeta function

Djurdje CvijovicAtomic Physics Laboratory, Vinca Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Serbia

a r t i c l e i n f o

Keywords:Discrete Fourier transformAlternating zeta functionsHurwitz–Lerch zeta functionLipschitz–Lerch zeta functionLerch zeta functionHurwitz zeta functionLegendre chi functionBernoulli polynomials

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2011.12.041

E-mail address: djurdje@vinca.rs

a b s t r a c t

It is demonstrated that the alternating Lipschitz–Lerch zeta function and the alternatingHurwitz zeta function constitute a discrete Fourier transform pair. This discrete transformpair makes it possible to deduce, as special cases and consequences, many (mainly new)transformation relations involving the values at rational arguments of alternating variantsof various zeta functions, such as the Lerch and Hurwitz zeta functions and Legendre chifunction.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction and definitions

In a recent note [1] it was shown that numerous (known or new) seemingly disparate results involving various specialfunctions, such as the Lipschitz–Lerch zeta function, Lerch zeta function, Hurwitz zeta function and Legendre chi function,could be established in a more general context provided by a discrete Fourier transform (DFT). In this sequel to [1], weaim at deriving the analogous transformation formula which would be valid in the case of the alternating counterparts ofthese functions.

The alternating Hurwitz–Lerch zeta function defined by (cf. [2, p. 121 et seq.])

U�ðz; s; aÞ :¼X1n¼0

ð�1Þn zn

ðnþ aÞsða 2 C n Z�0 ; Z�0 :¼ f0;�1;�2;�3; . . .gÞ ðs 2 C when jzj < 1; RðsÞ > 0 when jzj ¼ 1Þ

ð1:1Þ

contains, as its special cases, not only the alternating Lipschitz–Lerch zeta function (cf. [2, p. 122, Eq. 2.5(11)]; see also [3, p.761]):

/�ðs; a; nÞ :¼X1n¼0

ð�1Þn e2npın

ðnþ aÞs¼ U�ðe2pın; s; aÞ ðı :¼

ffiffiffiffiffiffiffi�1p

Þ ðRðsÞ > 0; a 2 C n Z�0 ; n 2 RÞ ð1:2Þ

and the alternating Hurwitz zeta function ([3, p. 761]; cf. [2, p. 88]):

f�ðs; aÞ :¼X1n¼0

ð�1Þn 1ðnþ aÞs

¼ U�ð1; s; aÞ; ð1:3Þ

but also other functions such as the alternating Lerch zeta function (cf. [2, p. 122, Eq. 2.5(11)]):

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D. Cvijovic / Applied Mathematics and Computation 218 (2012) 6744–6747 6745

‘�s ðnÞ :¼X1n¼1

ð�1Þn�1 e2npın

ns¼ e2pınU�ðe2pın; s;1Þ ðn 2 R; RðsÞ > 0Þ ð1:4Þ

and the alternating Legendre chi function vs(z) (cf. [4, p. 1056]):

v�s ðzÞ :¼X1n¼0

ð�1Þn z2nþ1

ð2nþ 1Þs¼ 1

2s zU� z2; s;12

� �ðjzj51; RðsÞ > 0Þ: ð1:5Þ

Finally, the Bernoulli and Euler polynomials, Bn(x) and En(x), are defined by (see, for details, [2, p. 61 et seq.]):

t ext

et�1¼X1n¼0

BnðxÞtn

n!ðjtj<2pÞ and

2 t ext

etþ1¼X1n¼0

EnðxÞtn

n!ðjtj<pÞ ðn2N0 :¼N[f0g; N :¼f1;2;3; . . .gÞ: ð1:6Þ

2. The main results and their proofs

To begin, we observe that, in what follows, it is assumed that p, r and q are positive integers and we set an empty sum tobe zero. For more details regarding the DFT, the reader is referred to such standard text on the subject as the book by Weaver[5]. In order to simplify the presentation, we shall use the following definition:

Zqðs; xqÞ :¼X1n¼0

ð�1Þqn

ðnþ xqÞs¼

f�ðs; xqÞ ðq is oddÞ;fðs; xqÞ ðq is evenÞ;

�ð2:1Þ

xq :¼ x(q) being any real sequence, xq R Z�0 ; f�ðs; aÞ is the above-defined function, while f(s,a) is the Hurwitz zeta functiongiven by [2, p. 88, Eq. 2.2(1)]

fðs; aÞ :¼X1k¼0

1ðkþ aÞs

ða R Z�0 ; RðsÞ > 1Þ: ð2:2Þ

Our main results in this section are stated and proved as follows. The formula given by (2.3) is essentially the same as thatderived in a different way by Chang and Ha [3, p. 761, Eqs. (2.5) and (2.6)], while (2.4)–(2.6) are presumably new. Note thatthe two more new DFT pairs are given in Section 3.

Theorem. Suppose that s and a are complex numbers, s – 1 when q is even and a R Z�0 . Then Zq(s,xq) and /⁄(s,a,n) form thefollowing DFT pair:

ð�1ÞpZq s;pþ a

q

� �¼ 1

q

Xq�1

r¼0

qs/� s; a;rq

� �e�

2pırpq ðp ¼ 0; . . . ; q� 1Þ ð2:3Þ

and

/� s; a;rq

� �¼ 1

qs

Xq�1

p¼0

ð�1ÞpZq s;pþ a

q

� �e

2pıprq ðr ¼ 0; � � � ; q� 1Þ: ð2:4Þ

Proof. Assume that RðsÞ > 1. We omit the proof of (2.3) which proceeds in the same way as the proof of (2.1) in [1] and isbased on the use of Simpson’s series multisection formula (see, for instance, [6, p. 131]) and Abel’s theorem (see [7, p. 148]).To prove (2.4) it suffices to recall the elementary series identity (see, for instance, [2, p. 337] and [8, p. 79, Eq. (2.2)])

X1n¼0

an ¼Xq�1

p¼0

X1k¼0

aqkþp;

and the sought formula follows since we have

/� s; a;rq

� �¼Xq�1

p¼0

ð�1ÞpX1k¼0

ð�1Þkq e2pıðkqþpÞr=q

ðkqþ pþ aÞs¼ 1

qs

Xq�1

p¼0

ð�1Þpe2pıpr

q

f� s; pþaq

� �q is odd;

f s; pþaq

� �q is even:

8><>:

We next show that the relations in (2.3) and (2.4) form a discrete Fourier transform pair. Indeed, upon substituting from (2.3)into (2.4) and using the following orthogonality relation

Xq�1

m¼0

e�2pı

q mne2pı

q mn0 ¼ qdn;n0 ðn; n0 ¼ 0; � � � ; q� 1Þ;

6746 D. Cvijovic / Applied Mathematics and Computation 218 (2012) 6744–6747

we obtain

/� s; a;rq

� �¼ 1

q

Xq�1

p¼0

Xq�1

r0¼0

/� s; a;r0

q

� �e�

2pıpr0q � e

2pıprq ¼ /� s; a;

rq

� �:

Thus the proposed transform relations are established for RðsÞ > 1.Lastly, it is obvious that the above formulas may be extended by applying the principle of analytic continuation on s as far

as possible. It is well-known that f(s,a) is a meromorphic function in s 2 C, with a single simple pole as s = 1. On the otherhand, f⁄(s,a) is an entire function in s 2 C. Similarly, /⁄(s,a,n) is also an entire function in s 2 C. In view of (2.1), we shouldconsider separately the cases when q is even and when q is odd. If q is odd, then the formulas (2.3) and (2.4) are valid for anycomplex s. Otherwise, they are valid for any complex s (s – 1). h

Corollary. Suppose that n is a positive integer and that a 2 C n Z�0 . Then, in terms of Bn(x), En(x) and /⁄(s,a,n), the following DFTpair holds true:

12 En�1

pþaq

� �q is odd

� 1n Bn

pþaq

� �q is even

9>=>; � ð�1Þp ¼ 1

qn

Xq�1

r¼0

e�2pırp

q /� 1� n; a;rq

� �ðp ¼ 0; � � � ; q� 1Þ ð2:5Þ

and

/� 1� n; a;rq

� �¼ 1

q1�n

Xq�1

p¼0

e2pıpr

q ð�1Þp12 En�1

pþaq

� �q is odd

� 1n Bn

pþaq

� �q is even

8><>: ðr ¼ 0; � � � ; q� 1Þ: ð2:6Þ

Proof. The above Corollary follows from our Theorem in conjunction with the following familiar relationships [2, p. 85, Eq.(17)]:

fð1� n; aÞ ¼ �BnðaÞn

ðn 2 NÞ

and [3, p. 761, Eq. (2.3)]

f�ð1� n; aÞ ¼ 12

En�1ðaÞ ðn 2 NÞ;

where Bn(x) and En(x) are the Bernoulli and Euler polynomials defined by (1.6). h

3. Concluding remarks

We remark that a great deal of interesting and novel results could be obtained as special cases of Theorem and Corollary,but we choose only to deduce and record here, in the following Examples, respectively, the cases when a = 1 and a ¼ 1

2 inCorollary. All that is needed is to use the easily derivable relations /�ðs;1; nÞ ¼ e�2pın‘�s ðnÞ and /� s; 1

2 ; n�

¼ 2se�pınv�s ðepınÞ.

Example 1. Assume that n 2 N. Then, we have

12 En�1

pq

� �q is odd

� 1n Bn

pq

� �q is even

9>=>; � ð�1Þp�1 ¼ 1

qn

Xq�1

r¼0

e�2pırp

q ‘�1�nrq

� �ðp ¼ 1; � � � ; qÞ ð3:1Þ

and

‘�1�nrq

� �¼ 1

q1�n

Xq

p¼1

e2pıpr

q ð�1Þp�112 En�1

pq

� �q is odd

� 1n Bn

pq

� �q is even

8><>: ðr ¼ 0; � � � ; q� 1Þ; ð3:2Þ

where ‘�s ðnÞ is the alternating Lerch zeta function defined as in (1.4).

Example 2. If n 2 N, then

12 En�1

2p�12q

� �q is odd

� 1n Bn

2p�12q

� �q is even

9>=>; � ð�1Þp�1 ¼ 1

qn

Xq�1

r¼0

e�2pırp

q v�1�n epırq

� �ðp ¼ 1; � � � ; qÞ ð3:3Þ

D. Cvijovic / Applied Mathematics and Computation 218 (2012) 6744–6747 6747

and

v�1�n epırq

� �¼ 1

ð2qÞ1�n

Xq

p¼1

epıð2p�1Þr

q ð�1Þp�112 En�1

2p�12q

� �q is odd

� 1n Bn

2p�12q

� �q is even

8><>: ðr ¼ 0; � � � ; q� 1Þ; ð3:4Þ

where v�s ðzÞ is the alternating Legendre chi function (1.5).Observe that there exists the connection /�ðs; a; nÞ ¼ /ðs; a; nþ 1

2Þ between the Lipschitz–Lerch zeta function andalternating Lipschitz–Lerch zeta function, /(s,a,n) and /⁄(s,a,n). However, the previously obtained DFT pair involving /(s,a,n) [1, Theorem] and the DFT pair given by the above Theorem and involving /⁄(s,a,n) cannot be deduced from each otherand they should be considered as two independent (and complementary) pairs associated with /(s,a,n). Together, theyprovide a very general context for a unified treatment of numerous (known or new) seemingly disparate results involvingvarious special functions, such as the Lipschitz–Lerch zeta function and alternating Hurwitz–Lerch zeta function and theirspecial cases.

In conclusion, note that recently derived closed-form formulas for the values of all derivatives of tangent and secantfunctions at rational multiple of p ([9, Theorem 2]; see also [10]) are, in fact, special cases of the here presented results.Indeed, they respectively follow from (3.2) and (3.4): it is enough to notice that (for more details, see [9, Eqs. (2.17) and(2.18)])

‘�1�nðnÞ ¼ı

2ð2pıÞn�1

dn�1

dnn�1 tanðpnÞ n 2 Rn ð2kþ 1Þ12jk 2 Z

� ; n 2 N n f1g

� �

and

v�1�nðepınÞ ¼ ı

2ðpıÞn�1

dn�1

dnn�1 secðpnÞ n 2 Rn ð2kþ 1Þ12jk 2 Z

� ; n 2 N

� �:

Acknowledgement

The present investigation was supported by the Ministry of Science of the Republic of Serbia under Research Projects 45005and 172015.

References

[1] D. Cvijovic, H.M. Srivastava, Some discrete Fourier transform pairs associated with the Lipschitz–Lerch zeta function, Appl. Math. Lett. 22 (2009) 1081–1084.

[2] H.M. Srivastava, J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.[3] C.-H. Chang, C.-W. Ha, A multiplication theorem for the Lerch zeta function and explicit representations of the Bernoulli and Euler polynomials, J. Math.

Anal. Appl. 315 (2006) 758–767.[4] D. Cvijovic, Integral representations of the Legendre chi function, J. Math. Anal. Appl. 332 (2007) 1056–1062.[5] J.H. Weaver, Theory of Discrete and Continuous Fourier Analysis, John Wiley and Sons, New York, 1989.[6] J. Riordan, Combinatorial Identities, John Wiley and Sons, New York, 1968.[7] T.J.I’A. Bromwich, An Introduction to the Theory of Infinite Series, AMS Chelsea Publishing Company, Providence, Rhode Island, 2005.[8] H.M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc. 129 (2000) 77–84.[9] D. Cvijovic, Closed-form formulae for the derivatives of trigonometric functions at rational multiples of p, Appl. Math. Lett. 22 (2009) 906–909.

[10] D. Cvijovic, Values of the derivatives of the cotangent at rational multiples of p, Appl. Math. Lett. 22 (2009) 217–220.