# Abstract

We provide an explicit analytical representation for a number of logarithmic integrals in terms of the Lerch transcendent function and other special functions. The integrals in question will be associated with both alternating harmonic numbers and harmonic numbers with positive terms. A few examples of integrals will be given an identity in terms of some special functions including the Riemann zeta function. In general none of these integrals can be solved by any currently available mathematical package.

## 1 Introduction and Preliminaries

In this paper we will develop explicit analytical representations, identities, new families of integral representations, of the form:

for (*k*, *p*) being the set of positive integers and where

where the Lerch transcendent function *Φ* is defined as the analytic continuation of the series

which converges for any real number *a* > 0 if *z* and *t* are any complex numbers with either |*z*| < 1 or |*z*| = 1 and ℜ (*t*) > 1. It is known that the Lerch transcendent extends by analytic continuation to a function *Φ* (*z*, *t*, *a*) which is defined for all complex *t*, *z* ∈ ℂ − [1, ∞) and *a* > 0, which can be represented, [3], by the integral formula

for ℜ (*t*) > 0. For a fuller account of the Lerch function see the excellent papers, [6], [7] and [8]. The Lerch transcendent generalizes the Hurwitz zeta function at *z* = 1,

and the Polylogarithm, or de Jonquière’s function, when *a* = 1,

Moreover,

Let ℝ and ℂ denote, respectively, the sets of real and complex numbers and let ℕ := {1, 2, 3, …} be the set of positive integers, and ℕ_{0}:= ℕ ∪ {0} . A generalized binomial coefficient *λ*, *μ* ∈ ℂ) is defined, in terms of the familiar gamma function, by

which, in the special case when *μ* = *n*, *n* ∈ ℕ_{0}, yields

where (*λ*)_{ν} (*λ*, *ν* ∈ ℂ) is the Pochhammer symbol. Let

be the *nth* harmonic number. Here, as usual, *γ* denotes the Euler-Mascheroni constant and *ψ* (*z*) is the Psi (or Digamma) function defined by

A generalized harmonic number *n* and *m*, as follows:

In the case of non-integer values of *n* such as (for example) a value *ρ* ∈ ℝ, the generalized harmonic numbers

by

where *ζ* (*z*) is the Riemann zeta function. Whenever we encounter harmonic numbers of the form *ρ*, they may be evaluated by means of this known relation (3). In the exceptional case of (3) when *m* = 0, we may define

We assume (as above) that

In the case of non integer values of the argument

where *ζ* (*z*) is the zeta function. When we encounter harmonic numbers at possible rational values of the argument, of the form
*ψ*^{(α)}(*z*) or, for rational arguments

The evaluation of the polygamma function

Some specific values are listed in the books [13] and [14]. Some results for sums of harmonic numbers may be seen in the works of [2], [15] and references therein.

The following lemma will be useful in the development of the main theorems.

## Lemma 1.1

*Let k be a positive integer*. *Then*:

## Proof

Consider, for *t* ∈ [−1, 1) and *j* ∈ ℝ^{+} ∪ {0}

where the beta function

for ℜ (*s*) > 0 and ℜ (*x*) > 0. We have

now

and with *t* = −1 we obtain the result (4). To prove (5), we note, from the properties of the polygamma function with multiple argument, that

where *δ*_{n,0} is the Kronecker delta. By the use of the digamma function in terms of harmonic numbers, we have

where

the first sum is obtained from [11] and the second sum is deduced from [9].Since

## Lemma 1.2

*Let k be a positive integer*. *Then*:

*where X*(*k*, 0) *is given in*(4).

## Proof

The proof of (7) is concluded in the same manner as used in Lemma 1.1. Consider

and by a change of summation index

The integral identity following (7) is obtained by the Beta method as described in Lemma 1.1 and therefore the details will not be outlined. It is of some interest to note that from (4) and (7)

□

## Lemma 1.3

*Let k and r be positive integers*. *Then*:

and

*with X*(*k*, 1) *given by*(8).

## Proof

By a change of summation index

Since the Lerch transcendent

so that

From (11) we have the recurrence relation

for *r* ≥ 2, and with *X*(*k*, 1) given by (8). The recurrence relation is solved by the subsequent reduction of the

terms, finally arriving at the relation (10). The integral identity (9) is obtained by the Beta method as described in Lemma 1.1 and details will not be outlined. □

A slightly different re-arrangement of the terms in *X*(*k*, *r*) leads to the following Lemma.

## Lemma 1.4

*Let k and r* ≥ 2 *be a positive integers*. *Then*:

*with X*(*k*, *r*) *given by*(10).

## Proof

By expansion,

by re arrangement

The integral (12) is obtained by considering for *t* ∈ [−1, 1) and *j* ∈ ℝ^{+}∪ {0}

Now differentiating with respect to jand replacing the limit as japproaches zero, with *t* = −1, we obtain the result (12). Two special cases, furnish the following. For *r* = 0,

For *r* = 1,

from which we deduce the integral identity,

and for *k* = 4,

□

The next few theorems relate the main results of this investigation, namely the closed form representation of integrals of the type (1).

## 2 Integral and Closed form identities

In this section we investigate integral identities in terms of closed form representations of infinite series of harmonic numbers and inverse binomial coefficients. First we indicate the closed form representation of

for *q* = 0, 1, and *k*, *p* ≥ 1 are positive integers.

## Theorem 2.1

*Let k* ≥ 1 *be real positive integer*, *then from*(15)*with q* = 0 *and p be real positive integer*:

*where X*(*k*, *r*) *is given by*(10).

## Proof

Consider the expansion

where

We can now express

From (10) we have *X*(*k*, *r*), hence substituting into (19), (17) follows. The integral (16) is evaluated as in Lemma 1.4. □

The other case of *q* = 1 can be evaluated in a similar fashion. We list the result in the next Theorem.

## Theorem 2.2

*Under the assumptions of Theorem 2.1*, *with q* = 1, *we have*,

*and where X*(*k*, *r*) *is given by*(10).

## Proof

The proof of (20) follows using the same technique as used in Theorem 2.1 and also using (18). □

It is possible to gain some further integral identities from Theorems 2.1 and 2.2 regarding the representation of a sequence of alternating shifted harmonic numbers as follows.

## Theorem 2.3

*For p* ∈ ℕ ∪ {0} *and k* ∈ ℕ:

*where M*(*k*, *p*) *is given by*(20)*and*

*where* [*x*] *is the integer part of x*.

## Proof

From the properties of harmonic numbers,

the details for the calculation of (23) may be seen in [11]. The integral representation (21) is obtained in the same manner as in Lemma 1.4.

For the simple case of *p* = 0, we have

and when *k* = 6,

It is also possible to represent, individually, some results of shifted harmonic numbers of (22), see for example, [9] and [10]. □

The following integral identities can be exactly evaluated by using the alternating harmonic number sums in Theorems 2.1 and 2.2.

## Theorem 2.4

*Let k and p be real positive integers*, *then*:

*where X* (*k*, *r*) *is given by*(10).

## Proof

From

Re-arranging

and (25) follows. The integral (24) is evaluated as in Lemma 1.4. □

## Theorem 2.5

*Let k and p be real positive integers*, *then*:

*where X (k, r) is given by (10)*.

## Proof

Follows the same pattern as used in Theorem 2.4. □

## Theorem 2.6

*Let k and p be real positive integers*, *then*:

## Proof

Follows the same pattern as used in Theorem 2.4. □

A number of special cases follow in the next Corollary.

## Corollary 2.7

*Some examples of integrals are given below*. *For p* = 0, *Theorem 2.5 reduces to (14)*.

For *p* = 1, from Theorem 2.6 we have

for *k* = 3, we have

this integral is highly oscillatory near the origin of *x*. From Theorem 2.5, with *k* = 6.

this integral is highly oscillatory near the origin of *x*.

## Conclusion 2.8

*We have established a number of integral identities in closed form in terms of special functions. A number of oscillatory integrals are also given in closed form. The integral identities established in this paper complement and extend the results in the paper [12]. Some particular identities obtained are*

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**Received:**2016-09-18

**Accepted:**2017-12-22

**Published Online:**2018-02-19

© 2018 Sofo, published by De Gruyter

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.