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 ℕ0:= ℕ ∪ {0} . A generalized binomial coefficient
which, in the special case when μ = n, n ∈ ℕ0, 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
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
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
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 δn,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
References
[1] Choi J., Cvijović D., Values of the polygamma functions at rational arguments, J. Phys. A: Math. Theor. 40 (2007), 15019–15028, Corrigendum, ibidem, 43 (2010), 239801 (1 p).Search in Google Scholar
[2] Choi J., Srivastava H.M., Some summation formulas involving harmonic numbers and generalized harmonic numbers. Math. Comput. Modelling. 54 (2011), 2220-2234.10.1016/j.mcm.2011.05.032Search in Google Scholar
[3] Guillera J., Sondow J., Double integrals and infinite constants via analytic continuations of Lerch’s transcendent. Ramanujan J. 16 (2008), 247-270.10.1007/s11139-007-9102-0Search in Google Scholar
[4] Kölbig K., The polygamma function ψ(x) for x = 1/4 and x = 3/4. J. Comput. Appl. Math. 75 (1996), 43-46.10.1016/S0377-0427(96)00055-6Search in Google Scholar
[5] Kouba O., The sum of certain series related to harmonic numbers. Octogon Math. Mag. 19(1) (2011), 3-18.Search in Google Scholar
[6] Lagarias J.C., Li W.-C., The Lerch zeta function I. Forum Math. 24 (2012), 1-48.10.1515/form.2011.047Search in Google Scholar
[7] Lagarias J.C., Li W.-C., The Lerch zeta function II. Forum Math. 24 (2012), 49-84.10.1515/form.2011.048Search in Google Scholar
[8] Lagarias J.C., Li W.-C., The Lerch function III. Polylogarithms and special values. Res. Math. Sci. 3 (2016). Art 2, 54 pp.10.1186/s40687-015-0049-2Search in Google Scholar
[9] Sofo A., Harmonic numbers at half integer values. Integral Transforms Spec. Funct. 27 (2016), no. 6, 430–442.10.1080/10652469.2016.1153636Search in Google Scholar
[10] Sofo A., Srivastava H.M., A family of shifted harmonic sums. Ramanujan J. 37 (2015), no. 1, 89–108.10.1007/s11139-014-9600-9Search in Google Scholar
[11] Sofo A., New families of alternating harmonic number sums. Tbilisi Math. J. 8 (2015), no. 2, 195–209.10.1515/tmj-2015-0022Search in Google Scholar
[12] Sofo A., A master integral in four parameters. J. Math. Anal. Appl. 448 (2017), no. 1, 81–92.10.1016/j.jmaa.2016.10.073Search in Google Scholar
[13] Srivastava H.M., Choi J., Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers, London, 2001.10.1007/978-94-015-9672-5Search in Google Scholar
[14] Srivastava H.M., Choi J., Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.10.1016/B978-0-12-385218-2.00002-5Search in Google Scholar
[15] Xu C., Yan Y., Shi Z., Euler sums and integrals of polylogarithmic functions. J. Number Theory. 165 (2016), 84-108.10.1016/j.jnt.2016.01.025Search in Google Scholar
© 2018 Sofo, published by De Gruyter
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.