## Abstract

The purpose of this paper is to establish a connection between the polylogarithm function and a continuous probability distribution. We provide closed-form expressions in terms of the polylogarithm function for all the moments of a continuous random variable related to the Bass diffusion model, which was introduced by Bass and is widely used in marketing science. In addition, a new integral representation of the polylogarithm of order *n* is achieved from a probabilistic formulation.

## 1. Introduction

The polylogarithm function has attracted a great deal of attention over the last two centuries or so and has a long history connected with some of the great mathematicians of the past. The simplest expression is called dilogarithm and was initially studied in its integral representation by Leibniz in a series of letters to Bernoulli in 1696 (cf. Leibniz 1855) and later by Euler (1768) and Landen (1760, 1780). The next simplest form is called trilogarithm and is due to Landen (1780). Abel (1881) and Ramanujan (cf. Berndt 1985) also contributed to the study of the properties of the polylogarithm function in the nineteenth and twentieth centuries, respectively, among other eminent mathematicians. For complex numbers *s* and *z*, the modern polylogarithm function Li_{s}(*z*), also known as Jonquière's function, is a generalization of Euler's and Landen's functions (see §2 for a precise definition).

Among the well-known integral representations of Li_{s}(*z*), the following stands out (Lewin 1981, p. 236, eqn (7.188); Prudnikov *et al*. 1990, p. 762):(1.1)whenever this integral converges (see §2 for details). Statistical mechanics is the field where the integral representation (1.1) arises in a natural way closely related to the Fermi–Dirac and Bose–Einstein functions. Indeed, since the pioneering paper of Lee (1995), the polylogarithms have unified the statistical mechanics of ideal gases (cf. also Lee & Kim 2002 and the references therein). More recently, Ciccariello (2004) has noted that the unified description given by Lee (1995) can be obtained from Lerch's function, which generalizes the polylogarithm function (for details on Lerch's function see Erdélyi (1953), §1.11, pp. 27–32).

In Prudnikov *et al*. (1986, p. 494, eqn (14)), we find another integral representation of the polylogarithm function that reads as follows:(1.2)which can be directly derived from (1.1) by a change of variable. Motivated by these integral expressions involving Li_{s}(*z*), we have begun to explore the polylogarithm function and its applications in probability theory.

Kulasekera & Tonkyn (1992) introduced a discrete probability model, which is useful for analysis and modelling of survival processes, and showed that all its moments can be written in closed form as polylogarithms. However, as far as we know, in the probability and statistical literature, we have not found any continuous random variable whose moments can be expressed as functions of polylogarithms.

In this paper, we give closed-form expressions for all the moments of a continuous probability distribution in terms of the polylogarithm function. The random variable involved is related to the Bass diffusion model, which was developed by Bass (1969) and is often used in marketing research. As a consequence, we obtain a new integral representation of the polylogarithm function Li_{s}(*z*) and provide a probabilistic proof of the integral equality (1.2) in the important case where *s* is a non-negative integer. The remainder of this paper is organized as follows. We briefly present the polylogarithm function and the Bass diffusion model in §§2 and 3, respectively. The main results are stated in §4 and, for the sake of clarity, their proofs are postponed to §5.

## 2. The polylogarithm function

For complex numbers *s* and *z*, let Li_{s}(*z*) be the polylogarithm function defined by the Dirichlet series(2.1)which is absolutely convergent for all complex *s* and *z* inside the unit circle in the complex *z*-plane. It is known that for fixed *s* the definition above can be extended by analytic continuation as a single-valued function to the whole complex *z*-plane, with the exception of the points on the cut along the real axis from 1 to +∞, for example by using the integral representation (1.1). It should be noted that the point *z*=1 is a singularity if Re(*s*)≤1 and that for Re(*s*)>1 the polylogarithm function Li_{s}(1) reduces to the Riemann zeta function *ζ*(*s*) (cf. for instance, Lee 1997 and Paulsen 2002 for further details). Moreover, the series in (2.1) can be extended for all *s* and *z* by means of the contour integral given in Lewin (1981, p. 236, eqn (7.189)).

In the case where the parameter *s* is a non-negative integer, the polylogarithm function is denoted by Li_{n}(*z*) and is called polylogarithm of order *n*, *n*=1, 2, …. The name of the function comes from the fact that Li_{n}(*z*) can be recursively defined as follows:(2.2)where *n*=1, 2, 3, …. The case *n*=1 corresponds to the natural logarithm Li_{1}(*z*)=−log(1−*z*), while the cases *n*=2, 3 are the polylogarithms known as dilogarithm and trilogarithm, respectively.

An exhaustive review of the polylogarithm function and its properties is given in Lewin (1981, 1991). Formulae involving Li_{s}(*z*) can be found in Prudnikov *et al*. (1986, 1990). Maximon (2003) provides an excellent overview of the basic properties of the polylogarithm function, historical notes, as well as an extensive list of references with different applications in mathematics and physics. To conclude this section, let us mention that Cvijović (2007) has recently published a paper dealing with new integral representations of Li_{s}(*z*).

## 3. The Bass diffusion model

The modelling and forecasting of the diffusion of innovations have been a topic of increasing research interest in marketing and other disciplines as the recent survey of Meade & Islam (2006) points out. In the late Sixties, Bass (1969) developed a growth model for the timing of first purchase (adoption) of new products (innovations) by consumers. Since this pioneering paper, the Bass diffusion model and its more advanced versions have been successfully used for forecasting the adoption of an innovation in marketing science (cf. for instance, Mahajan *et al*. 1993, 2000).

The Bass diffusion model assumes a finite population of potential buyers who are homogeneous in their propensity to innovate. The likelihood that a potential buyer buys a new product at time *x*, given that he/she has not yet bought, is a linear function of the proportion of buyers at time *x*. More specifically, each individual time to adoption of an innovation is a random variable *X* with cumulative distribution function *F*(*x*)≔*P*(*X*≤*x*) and probability density function *f*(*x*)≔d*F*(*x*)/d*x*, satisfying the following Riccati differential equation with constant coefficients(3.1)where *p* and *q* are the parameters that determine the shape of the diffusion process. The parameter *p* is interpreted as a coefficient of innovation and *q* as a coefficient of imitation, and the conditions 0<*p*≤1 and 0≤*q*≤1 are usually assumed for practical purposes. The solution to the differential equation (3.1) provides the following expression for the probability density function of *X* (see for more details Bass 1969):(3.2)

A reparametrization of the density function (3.2) yields the expression(3.3)with the corresponding parameters *α*=*p*+*q* and *β*=*q*/*p*. For our immediate purpose, it may be noted that the function *f*(*x*) given in (3.3) is a well-defined probability density function when the parameters take on values *α*>0 and *β*>−1, that is, . Moreover, observe that the special case *β*=0 in (3.3) leads to the exponential distribution with mean 1/*α*.

## 4. Main results

In this section, we provide explicit expressions in terms of the polylogarithm of order *n* for all the moments of the random variable *X* described in §3. Here, and throughout the rest of this paper, *X* will denote the random variable with probability density function given in (3.3) with parameters *α*>0 and *β*>−1.

To start with, recall that the *n*th (non-central) moment of the random variable *X* is defined by(4.1)Our goal is to obtain a closed-form expression for *E*[*X*^{n}] as a function of Li_{n}(*z*) and, at the same time, from a probabilistic viewpoint to achieve an integral representation of Li_{n}(*z*) without taking advantage of the integral equality (1.2). To this end, we shall compute such moments through the characteristic function of *X*, which is expressed as a series expansion, instead of computing *E*[*X*^{n}] directly by means of definition (4.1).

For any real number *t*, let *ϕ*_{X}(*t*) be the characteristic function of *X*, that is, *ϕ*_{X}(*t*)≔*E*[e^{itX}], where i denotes the imaginary unit. With the preceding notations, we state the following:

*For any α*>0 *and* −1<*β*<1*,* *we have**where* .

In the light of the above theorem, we are in a position to establish a relation between the *n*th moment of *X* and the polylogarithm of order *n*.

*Let n be a non-negative integer. Then, we have for any α*>0

*X*can be evaluated with high accuracy in a straightforward manner.

Finally, in view of corollary 4.2, a new integral representation of the polylogarithm of order *n* is stated in the following result (see remark 5.1 at the end of this paper):

*Let n be a non-negative integer. For any z*<1*,* *we have*

## 5. The proofs

In the proof of theorem 4.1, we shall use the Gauss hypergeometric function defined by the series expansionwhere *a*, *b*, *c* are complex values with *c*≠0, −1, −2, …, (.)_{k} denotes the Pochhammer symbol and *s* is a complex variable inside the unit disc (cf. Abramowitz & Stegun 1972, p. 556, eqn (15.1.1)).

Let *t* be a real number, *α*>0, *β*>−1 and . From the definition of the characteristic function of *X* and considering (3.3), we have(5.1)where the last equality follows from the change of variable *u*=e^{−αx}.

On the other hand, taking into account the relation between the Gauss hypergeometric function and the Euler integral (see Abramowitz & Stegun 1972, p. 558; eqn (15.3.1)), we get(5.2)

From equation (5.1), together with (5.2) and the recurrence relation *Γ*(*s*+1)=*sΓ*(*s*) (Re(*s*)>0), we obtain(5.3)Moreover, it can also be checked whether the coefficients of (−*β*)^{k} in the series expansion of _{2}*F*_{1}[2, 1−i*t*/*α*; 2−i*t*/*α*; −*β*] satisfy the equalityso that the hypergeometric function involved in formula (5.3) can be rewritten as follows:(5.4)Finally, by substituting equation (5.4) into (5.3), the proof of theorem 4.1 is complete. ▪

In order to compute the *n*th moment of *X*, in the proof of corollary 4.2 we shall use the fact that if *E*[*X*^{n}] exists then , *n*=1, 2, …, where denotes the *n*th derivative of the characteristic function of *X* and (cf. for instance, Dudewicz & Mishra 1988, pp. 273–274).

Let *t* be a real number, *α*>0 and −1<*β*<1. From the series representation of *ϕ*_{X}(*t*) stated in theorem 4.1, the *n*th derivative of *ϕ*_{X}(*t*) is given by(5.5)Setting *t*=0 in equation (5.5) and considering that , we get(5.6)where the last equality holds only for *β*∈(−1,0)∪(0,1).

Taking into account equation (5.6), together with definition (2.1) and analytic continuation, we arrive at(5.7)Since Li_{0}(−*β*)=−*β*/(1+*β*) by definition (2.2), it is clear that equation (5.7) is also true for *n*=0.

On the other hand, as it was said in §3, the case *β*=0 in (3.3) corresponds to the exponential distribution with mean and, from definition (4.1), it can be checked whether *E*[*X*^{n}]=*Γ*(*n*+1)/*α*^{n}, *n*=0, 1, 2, …. This completes the proof of corollary 4.2. ▪

Let *n* be a non-negative integer, *α*>0 and *β*>−1. By definition (4.1) and (3.3), the *n*th moment of *X* can be directly computed by(5.8)The change of variable *u*=e^{−αx} in (5.8) yields(5.9)Accordingly, for any *β*∈(−1, 0)∪(0, ∞), by virtue of corollary 4.2 and equality (5.9) the integral representation of Li_{n}(*z*) claimed in corollary 4.3 is achieved, where we have set *z*=−*β* for notational convenience. Note that the statement of this corollary holds trivially for *z*=0, that is, Li_{n}(0)=0, thus completing the proof. ▪

As it has been mentioned in §1, in Prudnikov *et al*. (1986, p. 494, eqn (14)) we find the following integral representation of Li_{s}(*z*):(5.10)where the last equality was derived by Lee (1995) from (1.1). An integration by parts in (5.10) leads to the expressionThereby, from a probabilistic perspective, we have obtained in corollary 4.3 an integral representation of Li_{n}(*z*) closely related to the integral representations of Li_{s}(*z*) given in (5.10) in the special case where *s* is a non-negative integer.

## Acknowledgments

The author wishes to express his thanks to the anonymous referees for their careful reading of the manuscript and valuable suggestions, which led to an improvement of the paper. Many thanks also to Profs José Luis Arregui and Juan Ignacio Pardo for their helpful comments.

## Footnotes

- Received May 12, 2008.
- Accepted June 26, 2008.

- © 2008 The Royal Society