## Abstract

Maximon has recently given an excellent summary of the properties of the Euler dilogarithm function and the frequently used generalizations of the dilogarithm, the most important among them being the polylogarithm function Li_{s}(*z*). The polylogarithm function appears in several fields of mathematics and in many physical problems. We, by making use of elementary arguments, deduce several new integral representations of the polylogarithm Li_{s}(*z*) for any complex *z* for which |*z*|<1. Two are valid for all complex *s*, whenever Re *s*>1. The other two involve the Bernoulli polynomials and are valid in the important special case where the parameter *s* is a positive integer. Our earlier established results on the integral representations for the Riemann zeta function *ζ*(2*n*+1), *n*∈*N*, follow directly as corollaries of these representations.

## 1. Introduction

Recently, Maximon (2003) has given an excellent summary of the defining equations and properties of the Euler dilogarithm function and the frequently used generalizations of the dilogarithm, the most important among them being the polylogarithm function Li_{s}(*z*). These include integral representations, series expansions, linear and quadratic transformations, functional relations and numerical values for special arguments.

Motivated by this paper we have begun a systematic study of new integral representations for Li_{s}(*z*) since it appears that only half a dozen of them can be found in the literature. Among known representations, the following are best-known (Prudnikov *et al*. 1990, p. 762):and (Prudnikov *et al*. 1986, p. 494)

Here, by making use of fairly elementary arguments, we deduce several new integral representations of the polylogarithm function for any complex *z* for which |*z*|<1. Two are valid for all complex *s*, whenever Re *s*>1 (see theorem 3.1). The other two given in corollary 3.1 are valid in the important special case where the parameter *s* is a positive integer. Our earlier published results (Cvijović & Klinowski 2002) on the integral representations for the Riemann zeta function *ζ*(2*n*+1), *n*∈*N*, follow directly as corollaries of these representations (see corollary 3.2).

## 2. The polylogarithm function

The polylogarithm function (also known as Jonquiére's function), Li_{s}(*z*), is defined for any complex *s* and *z* as the analytic continuation of the Dirichlet series(2.1)which is absolutely convergent for all *s* and *z* inside the unit disc in the complex *z*-plane. Sometimes, Li_{s}(*z*) is referred to as the polylogarithm of order *s* and argument *z*; however, most frequently it is simply called the polylogarithm.

The function Li_{s}(*z*), for fixed *s*, has no poles or essential singularities anywhere in the finite complex *z*-plane and, for fixed *z*, has no poles or essential singularities anywhere in the finite complex *s*-plane. It has only one essential singular point at *s*=∞. For fixed *s*, the function has two branch points: *z*=1 and ∞. The principal branch of the polylogarithm is chosen to be that for which Li_{s}(*z*) is real for *z* real (when *s* is real), 0≤*z*≤1, and is continuous except on the positive real axis, where a cut is made from *z*=1 to ∞ such that the cut puts the real axis on the lower half-plane of *z*. Thus, for fixed *s*, the function Li_{s}(*z*) is a single-valued function in the *z*-plane cut along the interval (1, ∞) where it is continuous from below.

In the important case where the parameter *s* is an integer, Li_{s}(*z*) will be denoted by Li_{n}(*z*) (or Li_{−n}(*z*) when the parameter is negative). Li_{0}(*z*) and all Li_{−n}(*z*), *n*=1, 2, …, are rational functions in *z*. Li_{n}(*z*), *n*=1, 2, …, is the polylogarithm of order *n* (i.e. the *n*th-order polylogarithm). The special case *n*=1 is the ordinary logarithm Li_{1}(*z*)=−log(1−*z*), while the cases *n*=2, 3, 4, …, are classical polylogarithms known, respectively, as dilogarithm, trilogarithm, quadrilogarithm, etc.

For more details and an extensive list of references in which polylogarithms appear in physical and mathematical problems, we refer the reader to Maximon (2003). The polylogarithm Li_{n}(*z*) of order *n*=1, 2, 3, … is thoroughly covered in Lewin's standard text (1981), while many formulae involving Li_{s}(*z*) can be found in Erdélyi *et al*. (1953, pp. 30–31) and Prudnikov *et al*. (1990). Berndt's treatise (1985) can serve as an excellent introductory text on the polylogarithm (and numerous related functions) and as an encyclopaedic source.

## 3. Statement of the results

In preparation for the statement of the results, we need to make several definitions.

The Riemann zeta function and the Hurwitz zeta function, *ζ*(*s*) and *ζ*(*s*, *a*), are, respectively, defined by means of the series (Abramowitz & Stegun 1972, p. 807; eqns (23.2.1), (23.2.19) and (23.2.20))(3.1)and (Erdélyi *et al*. 1953, pp. 24–27)(3.2)Both are analytic over the whole complex plane, except at *s*=1, where they have a simple pole.

Next, for any real *x* and any complex *s* with Re *s*>1, we define(3.3a)and(3.3b)and note that these functions are related to the polylogarithm Li_{s}(*z*),

We also use the Bernoulli polynomials of degree *n* in *x*, *B*_{n}(*x*), defined for each non-negative integer *n* by means of their generating function (Abramowitz & Stegun 1972, p. 804, eqn (23.1.1))(3.4a)and given explicitly in terms of the Bernoulli numbers *B*_{n}=*B*_{n}(0) by (Prudnikov *et al*. 1990, p. 765)(3.4b)

Our results are as follows.

*Assume that s and z are complex numbers, let* Li_{s}(*z*) *be the polylogarithm function and let S*_{s}(*x*) *and C*_{s}(*x*) *be defined as in equations* (3.3*a*) *and* (3.3*b*), *respectively*. *If* Re *s*>1 *and* |*z*|<1, *then*(3.5a)(3.5b)(3.5c)

*δ*=1 or

*δ*=1/2.

Observe that our integral representations in theorem 3.1 essentially involve either the Hurwitz zeta function or the generalized Clausen function. Indeed, between the functions *S*_{s}(*x*) and *C*_{s}(*x*) and the Hurwitz zeta function defined as in (3.2), there exists the relationship (Prudnikov *et al*. 1986, p. 726, Entry 5.4.2.1)where *Γ*(*s*) is the familiar gamma function. In the cases when this relationship does not hold, *S*_{s}(*x*) and *C*_{s}(*x*) are used to define the generalized Clausen function (Lewin 1981, p. 281, eqn (3))What is most important, however, is that *S*_{2n−1}(*x*) and *C*_{2n}(*x*) (*n*∈*N*) are expressible in terms of the Bernoulli polynomials and this makes possible our corollaries.

*Assume that n is a positive integer*, Li_{n}(*z*) *be the polylogarithm function and B*_{n}(*x*) *be the Bernoulli polynomials of degree n in x given by equations* (3.4*a*) *and* (3.4*b*). *Then, provided that* |*z*|<1, *we have*(3.6a)(3.6b)(3.6c)

*Assume that n is a positive integer, let ζ*(*s*) *be the Riemann zeta function defined as in* (3.1) *and* *B*_{n}(*x*) *be the Bernoulli polynomials of degree n in x given by equations* (3.4*a*) *and* (3.4*b*). *We then have*(3.7a)(3.7b)

The integral for *ζ*(2*n*+1) in (3.7*a*) when *δ*=1 is well known (Abramowitz & Stegun 1972, p. 807, eqn (23.2.17)), while the other integrals in corollary 3.2 were recently deduced (Cvijović & Klinowski 2002, theorem 1). However, we have failed to find the integral representations given in theorem 3.1 and corollary 3.1 in the literature.

The integral representations given in (3.5*a*)–(3.5*c*) and (3.6*a*)–(3.6*c*) hold for |*z*|<1. However, our results may be extended to any complex *z*, |*z*|>1, by means of the inversion formula (Erdélyi *et al*. 1953, pp. 30–31; Prudnikov *et al*. 1990, pp. 762–763)where *ζ*(*s*,*a*) is the Hurwitz zeta function (3.2) or, by means of the particularly simple inversion formula,which is valid for the *n*th-order polylogarithm.

We remark that the existence of the integrals in (3.7*a*) and (3.7*b*) is assured since the integrands on [0, *α*], 0<*α*<1, have only removable singularities. This can be demonstrated easily by making use of some basic properties of *B*_{n}(*x*). For instance, knowing that the odd-indexed Bernoulli numbers *B*_{n}, apart from *B*_{1}=−1/2, are zero (Abramowitz & Stegun 1972, p. 805, eqn (23.1.19)), we havesince (Abramowitz & Stegun 1972, p. 805, eqn (23.1.21)).

## 4. Proof of the results

In what follows, we shall need the following lemma 4.1 and we provide two proofs for it. The second proof was suggested by one of the anonymous referees and it makes use of well-known Chebyshev polynomials and Fourier series.

*Assume that n is a positive integer and that δ*=1 *and* 1/2. *Then*, *we have*(4.1a)(4.1b)

It is clear that the integrals in (4.1*a*) and (4.1*b*) can be rewritten as follows:(4.2a)(4.2b)

We first establish the case *δ*=1 in (4.2*a*) and (4.2*b*). For any positive integer *n* and arbitrary complex *z*, consider the integrals *I*_{1} and *I*_{2} with parameters given by(4.3a)and(4.3b)In order to derive the integrals *I*_{1} and *I*_{2}, we make use of contour integration and calculus of residues. By settingwhere , we arrive at(4.4a)and(4.4b)

Combining (4.3*a*), (4.3*b*) and (4.4*a*), (4.4*b*) and equating the real and imaginary parts on both sides gives the integrals in (4.2*a*) and (4.2*b*) where *δ*=1. In this way, we evaluate the integrals in (4.1*a*) and (4.1*b*) where *δ*=1. Observe that in both integrals in (4.4*a*) and (4.4*b*), the contour is the unit circle and is traversed in the positive (counterclockwise) direction and the only singularities of the integrands that lie inside the contour are at *τ*=*z*.

Next, it can be easily shown that the case *δ*=1/2 in (4.2*a*) and (4.2*b*) reduces to the above considered case *δ*=1. Indeed, in view of the following well-known property (Prudnikov *et al*. 1986, p. 272 and 273, eqns (2.1.2.20) and (2.1.2.21)) it suffices to show that both integrands in (4.2*a*) and (4.2*b*) are such that *f*(2*π*−*t*)=*f*(*t*):This completes the proof of our lemma 4.1.

Consider the Chebyshev polynomials of the first and second kind, *T*_{m}(*x*) and *U*_{m}(*x*), defined by (Abramowitz & Stegun 1972, p. 776, eqns (22.3.15) and (22.3.16))and recall that their generating functions are given by (Abramowitz & Stegun 1972, p. 783, eqns (22.9.9) and (22.9.10))Now, by applying the orthogonality relation of the trigonometric functions we, for instance, have the sought formulae in (4.1*b*)The integrals given in (4.1*a*) follow in a similar manner.

The proof of theorem rests on lemma 4.1. From (4.1*a*), we obtain the formulaDividing the formula by *k*^{s} and summing over *k*, we haveso that(4.5)

Now, the required integral formula in (3.5*a*) directly follows from the last expression in light of the definitions of Li_{s}(*z*) in (2.1) and *S*_{s}(*x*) in (3.3*a*). It should be noted that inverting the order of summation and integration on the right-hand side of (4.5) is justified by absolute convergence of the series involved.

Starting from (4.1*b*), the formula in (3.5*b*) is derived in precisely the same way. In order to prove (3.5*c*) note thatthus we havesinceThis proves the theorem 3.1.

We need the Fourier expansions for the Bernoulli polynomials *B*_{n}(*x*) (Abramowitz & Stegun 1972, p. 805, eqns (23.1.17)),where 0≤*x*≤1 for *n*=2, 3, …, 0<*x*<1 for *n*=1 and (Abramowitz & Stegun 1972, p. 805, eqn (23.1.18))where 0≤*x*≤1 for *n*=1, 2, …. These Fourier expansions can be rewritten as follows:andin terms of the functions *S*_{s}(*x*) and *C*_{s}(*x*) defined in (3.3*a*) and (3.3*b*).

Finally, the integral formulae for *n*th polylogarithms proposed in (3.6*a*)–(3.6*c*) are obtained from the above expansions in conjunction with theorem 3.1.

First, note that for *z*=1 and −1, the polylogarithm reduces to the Riemann zeta function(4.6a)and(4.6b)Second, it is not difficult to show that(4.7a)and(4.7b)

Finally, we deduce the integral formulae in (3.7*a*) and (3.7*b*) starting from (3.6*a*) and by employing the expressions in (4.6*a*), (4.6*b*) and (4.7*a*), (4.7*b*).

## Acknowledgments

The author would like to thank the two anonymous referees of this journal for their extremely valuable comments on an earlier draft of this article. In particular, it should be gratefully acknowledged that the substantially shorter proof of lemma 4.1 is suggested by one of the referees. This work was financially supported by Ministry of Science and Environmental Protection of the Republic of Serbia under research project number 142025.

## Footnotes

- Received August 3, 2006.
- Accepted November 16, 2006.

- © 2006 The Royal Society