Introduction and preliminaries

The polylogarithm is a function in mathematics which was investigated intensively by many mathematicians. Many of them used different definitions but the one we use is the standard modern definition. For more information about the polylogarithm as a function consult the following book (Lewin, 1981). Questions about sums and their evaluations trace back to ancient times. Even the great Euler concerned himself with evaluating the ζ(2) known as the Basel problem, which was later generalized by him in view of finding a formula for even zeta values. More on various sums and evaluations can be found here (Hirschman, 2014; Knopp, 1990; Stojiljković, 2021; Davis, 2015). We will use the following notation throughout the paper. The first known definition is as follows.

Defınıtıon 1. The polylogarithm, see (Lewin, 1981), is defined by a power series in z, given by

This definition is valid for the arbitrary complex order s and for all complex arguments z with |z| < 1. We will also need the definition given by

Also, the special case we will use frequently is

For z = 1 we get the Riemann zeta function ζ which is also a function of the complex variable s. For more information see (Edwards, 1974), (Fabiano, 2020).

The second definition is as follows.

Defınıtıon 2. The harmonic numbers, see (Olaikhan, 2021), are defined as follows

for n > 1 and by definition H₀ = 0.

The main results of this paper are the following.

Theorem 1. Let Lis(z) denote the polylogarithmic function. Then the following equality holds for |z| < 1

Theorem 2. Let Li_s(z) denote the polylogarithmic function. Then the following equality holds for |z| < 1

Theorem 3. Let Li_s(z) denote the polylogarithmic function. Then the following equality holds for |z| < 1

The corollaries of the results are given as follows.

Corollary 1. The following equalities come from theorem 1.

Setting we get

We can also derive

By setting we get

Corollary 2. The following equalities come from Theorem 2

By setting we get

Main results

We will need some lemmas in order to proceed further. The following lemma will be extensively used throughout the paper.

Lemma 1. The following equality holds for |z| < 1.

Proof. Follows from the definition of the polylogarithm.

We will need the following in order to proceed further.

We will need in our analysis Abel’s summation formula (Bonar & Koury, 2006, p.55), (Lewin, 1981, p.258), which states that if (a_n)_n $\ge$ 1 and (b_n)_n $\ge$ 1 are two sequences of real numbers and , then

The second lemma will be given.

Lemma 2. The following identity holds:

Proof. We will prove it using the Abel’s summation (finite version). By choosing a_k = 1, b_k = H_k we get

and the proof is done.

The third lemma that we will need.

Lemma 3. The following equality holds.

Proof. We will prove it using the Abel’s summation (finite version). By choosing a_k = k, b_k = H_k we get

and the proof is complete.

Lemma 4. The following equality holds for any q and for |z| < 1

Proof. Let us observe the expression inside the brackets

What we can realise is that every term is less than ; therefore, by

multiplying both sides by k^q and letting the limit go to infinity, we get

and this will go to zero independently of s − q − 1 because |z| < 1 and z^k goes faster to zero than any power of the form k^s−q−1.

The proof is complete.

We give our first generalization of the zeta function series.

Lemma 5. Let Li_s(z) denote the polylogarithmic function. Then the following equality holds for |z| < 1

Proof. We apply the Abel’s summation formula with a_k = 1 and b_k = Li_s(z) − z − from which we get

Since the first term goes to zero when k → +∞, Lemma 4 (q = 1), we get

Adding and subtracting 1 in the numerator leaves us with two sums

because of Lemma 1. The proof is complete.

In the following we give a proof of Theorem 1.

Theorem 1. Let Li_s(z) denote the polylogarithmic function. Then the following equality holds for |z| < 1

Proof. By using Abel’s theorem (infinite version) and choosing

and Lemma 2 we get

Since the first term goes to zero when k → +∞, Lemma 4 (q = 2), the above equals to:

The second sum follows from Lemma 1. In the first sum we will rewrite the harmonic number as an integral and interchange the sum and the integral thanks to Fubini’s theorem:

By rewriting it as two sums, we get

Using the results from the Lemma 1 leaves us with

Which, when substituted above, gives us:

Now we prove Corollary 1, part a).

When s = 1 it can be shown, after a long and tedious calculation, that the following holds

By setting we get b)

When s = 2 it can be shown, similarly to the case s = 1, that c) part holds

By setting we arrive at d)

In the following we give proof of Theorem 2.

Theorem 2. Let Li_s(z) denote the polylogarithmic function. Then the following equality holds for |z| < 1

Proof. Using the Abel’s summation with a_k = kH_k and b_k = Li_s(z) − − ... − and Lemma 3 for the ak part gives

The expression in the brackets goes to zero by Lemma 4, so we are left with:

We will use Lemma 1:

As we can see, the second sum is the expression above with s shifted by −1 and multiplied by $\frac{\text{1}}{\text{4}}$. For the first sum, we will rewrite the harmonic number into its integral form.

Both sums are of the form given above. Therefore, we get

By incorporating this into the original equality, we get

By setting s = 2 it can be shown that Corollary 2 part a) holds

By setting we arrive at part b)

Our significant result in this paper is given in the following theorem. The next theorem will use all the previous results.

Theorem 3. Let Li_s(z) denote the polylogarithmic function. Then the following equality holds for |z| < 1

Proof. We will use Abel’s summation method, choosing

with Lemma 2, we will use the following notation to minimize the clutter in the formulas, let us call

. By evaluating b_k − b_k+1 we get

By using Abel’s summation we get

The expression in the limit goes to zero by Lemma 4. We are left with the sum

We know the third term from the proof of Theorem 1, the fifth term from Lemma 5 and the sixth term from Lemma 1. Let us focus on the second one,. This is a separate problem we must deal with. So let us write

The first term is known from Theorem 1, but the second one is not, so we will use again Abel’s summation method choosing

We get

The first sum is from the proof of Theorem 1 while the second one is from Lemma 1; therefore, the original second sum is done. Let us deal with the fourth sum:

The first sum is from the proof of Theorem 1 while the second one is from Lemma 1. Therefore, the fourth sum is done. Let us focus on the first one.

The second one is from the proof of Theorem 1, but we need to dig further for the first one

The second term is the same as in Theorem 1 when taking z as zm; therefore, the result follows.

While the first one we have directly from the proof of Theorem 1

Therefore, by putting all together, we obtain

We can see that four of the terms will cancel themselves; then we plug the polylogarithm expressions we have got and establish the equality

And the proof is done.

By setting s=1 in theorem 3 we arrive at Corollary 3 part a)

By setting we arrive at part b) of Corollary 3

Some examples of series

The usage of the previously derived theorems will be displayed in the following examples. Equipped with the series in a closed form we have derived, we can get many series via incorporating the values from the domain which is |z| < 1. By letting in Corollary 1 part c), Corollary 2 part a) and Corollary 3 part a) we get, respectively

More interesting sums can be obtained incorporating in the value z = .

By setting z = in Corollary 1 part c), Corollary 2 part a) and Corollary 3 part a), we get, respectively

The numerical values of Li₂ at the points can be found here (Lewin, 1981). Many more series can be obtained by substituting different values.

Conclusions

1. To assure the accuracy of the results, we verified all the numerical series identities through Wolfram Alpha.

2. Further questions can be asked regarding the sums with harmonic numbers of an arbitrary order as to, whether it is possible to find more of them of the form ${\text{H}}_{\frac{\text{n}}{\text{k}}}$ for some fixed k.

3. In this paper, we generalized the results given in (Furdui, 2016) as the polylogarithm is a generalization of the zeta function since Li_s(1) = ζ(s). We can obtain many more series by varying the two parameters z and s.

References

Bonar, D.D. and Khoury, M.J. 2006. Real Infinite Series. Washington D.C., American Mathematical Society: MAA Press. ISBN: 978-1-4704-4782-3.

Davis, H.T. 2015. The Summation of Series (Dover Books on Mathematics). Mineola, New York: Dover Publications. ISBN-13: 978-0486789682.

Edwards, M.H. 1974. Riemann’s Zeta Function. Mineola, New York: Dover Publications. ISBN-13: 978-0486417400.

Fabiano, N. 2020. Zeta function and some of its properties. Vojnotehnički glasnik/Military Technical Courier, 68(4), pp.895-906. Available at: https://doi.org/10.5937/vojtehg68-28535.

Furdui, O. 2016. Harmonic series with polygamma functions. Journal of Classical Analysis, 8(2), pp.123-130. Available at: https://doi.org/10.7153/jca-08-11.

Hirschman, I.I. 2014. Infinite series (Dover Books on Mathematics). Mineola, New York: Dover Publications. ISBN-13: 978-0-486-78975-0.

Knopp, K. 1990. Theory and Applications of Infinite Series. Mineola, New York: Dover Publications. ISBN-13: 978-0-486-66165-2.

Lewin, L. 1981. Polylogarithms and associated functions. Elsevier Science Ltd. ISBN-13: 978-0444005502.

Olaikhan, A.S. 2021. An Introduction To The Harmonic Series And Logarithmic Integrals: For High School Students Up To Researcher. Ali Shadhar Olaikhan (private edition). ISBN-13: 978-1-7367360-0-5.

Stojiljković, V. 2021. Some Series Associated with Central Binomial Coefficients and Harmonic Numbers. Octogon Mathematical Magazine, 29(2).

Author notes

nicola.fabiano@gmail.com

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