Abstract
Many of the properties of Nielsen generalized polylogarithm , for example, the special value and the transformation formulas, play important roles in the computation of higher order radiative corrections in quantum electrodynamics. In this paper, some transformation formulas of , and are obtained. In particular, the last three transformation formulas are new results so far in the literature. By use of these transformation formulas presented, new fast algorithms for Nielsen generalized polylogarithm can be designed. For , a new recurrence formula is also given. The identities and the calculation of and are also investigated.
1. Introduction
The Nielsen generalized polylogarithm, introduced by the Danish mathematician Niels Nielsen, reads as follows: in which is a complex and and are positive integers. It is a generalization of the polylogarithm function . In fact, . In particular, the values of for , and are usually used in the theory of the generalized polylogarithms. According to Nielsen's notation as follows, the following relationship exists: where is Riemann zeta function and is the Dirichlet eta function.
The Nielsen generalized polylogarithm plays an important role in quantum electrodynamics (see [1, 2]). The properties of it can also find applications in group theory and geometry [3]. For the Nielsen generalized polylogarithm , Kölbig has done much in-depth and systematic study in [4, 5]. For example, the transformations and are given as follows: where (see (5.3) and (5.12) in [5]). In [5], the author also noted that by repeated use of (4)–(6) it is possible to find formulae for which only contains generalized polylogarithms apart from logarithms and known constants. These formulae soon become very complicated and a more systematic investigation needs to be done later.
In this paper, we use different methods from that in [5] to give the properties of the transformations , and for the Nielsen generalized polylogarithm . For , we also give a new recurrence formula. Finally, we consider the identities and the calculation of and high accuracy and fast calculation of .
2. The Properties of the Transformations , and
For the Nielsen generalized polylogarithm, in [5], by use of the following results (see [6, (2.15), (2.16)]), where and some identities, the transformation properties (4)–(6) were obtained.
In this section, we use different methods to give the properties of the transformations , and . First we introduce the following function: It is easy to get the following relation: where and are the Beta function and the incomplete Beta function, respectively.
Lemma 1. When , for , there are the following properties:
(1) or :
(2) :
where
(3) or :
(4) :
(5) :
Proof. (1) By (10) we have
So (12) holds.
(2) For we have
where the second integral path does not go through the real axes. By using the variable substitution, we have
Substituting (20) into (19) we get that
Letting , we obtain that
By (22) and (21) we get that
Then by (11) one can see that (13) holds. By (13) and (12) we deduce that (15) holds.
Fully in accordance with the discussion of (2), we can also get that (15)–(17) hold.
By means of the application of Hadamard integral (the Hadamard finite part integral or neutrix calculus; see [6–8]) and integration by parts we have
By Lemma 1, (24), and (25), we will consider the transformation properties of on . First we prove (4) in another approach.
Theorem 2. When the integers , (4) for the transformation holds for all complexes .
Proof. By (11) and (12), we have In (26), letting and using (25), we have where can be extended to the whole complex plane. Then by (27) and , we deduce that (4) holds.
Theorem 3. When the integers , for the transformation , there is where .
Proof. By (11) and (13) and noting , we have In (29), letting and using (24), we get that (28) holds.
Theorem 4. When the integers , for the transformation , there is where and .
Proof. By (11) and (15) we have In (31), letting and by (25), we have Noting , one can see that (30) holds.
Theorem 5. When the integers , for the transformation and , there is where , and where .
Proof. By (11), (16), and (17) we obtain that In (35), letting and using (25), we can deduce that (33) and (34) hold.
3. The Recursive Formula on
For in Theorems 2–5, we can give the following recursive formula.
Theorem 6. For the positive integers , there is
Proof. By calculating -order partial derivatives on for , we have
where is the digamma function defined as
By calculating -order partial derivatives on in (38), we have
Using , we have
By the property of the Beta function, it holds that
Combining (10), (11), and (38)–(42), we obtain that (36) holds.
Remark 7. In [5], there is the following result: where , and where is Euler's constant. From the perspective of recurrence relation, one can see that (36) is simpler and clearer than (43).
4. On the Identities of and and Their Calculations
In (28), letting and using , we have where Using (3) and (36), from (45) and (46), we have Here (48) is a new formula.
In (34) and (4) letting we have By (49) we get Noting the bracket part independent of in (50), it follows that Taking into account the real and imaginary parts, we have In (52), letting , we get that Thus we have established two recursive formulas for .
In (30), letting , we have In (33), letting , we have Equation (55) is just consistent with (9.11) in [6]. By (54) and (55), we have Similarly, noting bracket part independent of in (56), we have which is equivalent to (9.8) in [5].
5. High Accuracy and Fast Calculation for the Nielsen Generalized Polylogarithm
In this section, we apply the transformation formulas established previously to design algorithms of high accuracy and fast calculation for the Nielsen generalized polylogarithm . In Mathematica, there is an internal command, PolyLog[n, p, z], designed to calculate the Nielsen generalized polylogarithm . When , and the integers in the Nielsen generalized polylogarithm , using (1) and where satisfying the following recursive formula, we have
Concerning the computational speed and accuracy, we make a comparison between PolyLog and (59), (4), (28), (30), (33), and (34). We will make the comparison in well-distributed points from a circle with the center and the radius . Numerical results are given in Tables 1 and 2.
Here are the average running time and the relative error of PolyLog[n,p,N[z,Prec]], while are the average running time and the relative error of (59), (4), (28), (30), (33), and (34) in precision , respectively. We always use second as the unit of time.
Seen from Table 1, when the precision , (59), (4), (28), (30), (33), and (34) are faster than the PolyLog. Seen from Table 2, in addition to (33), with improved precision, it is more obvious that the calculation speed of (59), (4), (28), (30), and (34) is faster than PolyLog.
Remark 8. When or , since , is calculated as a value, (59), (4), (28), (30), (33), and (34) cannot be used. But so far we did not find such a transformation . Although in the literature [6] the author gives some transformation , it essentially still belongs to the above type of transformation. For example, in [6], the author gives the following results: where . If the into , (62) becomes this is clearly the type of (4).
6. Conclusions
In this paper, for the Nielsen generalized polylogarithm , the formulas for the transformations , and have been established, and the last three transformation formulas are new. It is worthy to note that the transformation formula (28) is faster than (1.5) in [5] after taking into account numerical calculation. By use of these transformation formulas established above, new fast algorithms for Nielsen generalized polylogarithm can be designed. Numerical results show that when , where , the new algorithm is superior to Mathematica's internal functions PolyLog tens of times faster or even a hundred times. For , we have also given a new recursive formula (36). For and , we have got some new identities, some of which are different from those in [5].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work is supported by National Natural Science Foundation of China (61379009).