Evaluation of integrals with hypergeometric and logarithmic functions

: 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


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 is the classical generalized hypergeometric function.We also provide analytical solutions for integrals of the form where the Lerch transcendent function Φ is de ned as the analytic continuation of the series (m + a) t , which converges for any real number a > if z and t are any complex numbers with either z < or z = and R (t) > .It is known that the Lerch transcendent extends by analytic continuation to a function Φ (z, t, a) which is de ned for all complex t, z ∈ C − [ , ∞) and a > , which can be represented, [3], by the integral formula x a− ln x − xz dx for R (t) > .For a fuller account of the Lerch function see the excellent papers, [6], [7] and [8].
which, in the special case when µ = n, n ∈ N , yields λ ∶= and be the nth harmonic number.Here, as usual, γ denotes the Euler-Mascheroni constant and ψ(z) is the Psi (or Digamma) function de ned by A generalized harmonic number H (m) n of order m is de ned, for positive integers 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 H (m+ ) ρ may be de ned, in terms of the Polygamma functions where ζ (z) is the Riemann zeta function.Whenever we encounter harmonic numbers of the form H (m) ρ at admissible real values of ρ, they may be evaluated by means of this known relation (3).In the exceptional case of (3) when m = , we may de ne H We assume (as above) that In the case of non integer values of the argument z = r q , we may write the generalized harmonic numbers, H (α+ ) z , in terms of polygamma functions where ζ (z) is the zeta function.When we encounter harmonic numbers at possible rational values of the argument, of the form H (α) r q they maybe evaluated by an available relation in terms of the polygamma function (z) or, for rational arguments z = r q , and we also de ne = .
The evaluation of the polygamma function ψ (α) r a at rational values of the argument can be explicitly done via a formula as given by Kölbig [4], or Choi and Cvijovic [1] in terms of the Polylogarithmic or other special functions.Polygamma functions at negative rational values of the argument can also be explicitly evaluated, for example Some speci c 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: where the beta function and with t = − we obtain the result (4).To prove (5), we note, from the properties of the polygamma function with multiple argument, that where δ n, is the Kronecker delta.By the use of the digamma function in terms of harmonic numbers, we have where H n− r p may be thought of as shifted harmonic numbers.Summing over the integers , the rst sum is obtained from [11] and the second sum is deduced from [9].Since 5) follows.The closed form representation (6) can be evaluated by contour integration, the details are in [5].

Lemma 1.2. Let k be a positive integer. Then
= k where X (k, ) is given in (4).
Proof.The proof of ( 7) is concluded in the same manner as used in Lemma 1. 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) Let k and r be positive integers.Then: and with X (k, ) given by (8).
Proof.By a change of summation index From (11) we have the recurrence relation for r ≥ , and with X (k, ) given by (8).The recurrence relation is solved by the subsequent reduction of the terms, nally 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 di erent re-arrangement of the terms in X (k, r) leads to the following Lemma.
Lemma 1.4.Let k and r ≥ be a positive integers.Then:
, by re arrangement The integral ( 12) is obtained by considering for t ∈ [− , ) and Now di erentiating with respect to j and replacing the limit as j approaches zero, with t = − , we obtain the result (12).Two special cases, furnish the following.For r = , For r = , from which we deduce the integral identity, and for k = , The next few theorems relate the main results of this investigation, namely the closed form representation of integrals of the type (1).

Integral and Closed form identities
In this section we investigate integral identities in terms of closed form representations of in nite series of harmonic numbers and inverse binomial coe cients.First we indicate the closed form representation of for q = , , and k, p ≥ are positive integers.
Theorem 2.1.Let k ≥ be real positive integer, then from ( 15) with q = and p be real positive integer: where X (k, r) is given by (10).
Proof.Consider the expansion 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 = can be evaluated in a similar fashion.We list the result in the next Theorem.
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.
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 = , we have 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).
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.

Conclusion 2 . 8 .
highly oscillatory near the origin of x.From Theorem 2.5, with k = .highly oscillatory near the origin of x.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 Brought to you by | Victoria University Australia Authenticated Download Date | 4/5/19 6:14 AM