Explicit evaluations of log–log integrals

By investigating a family of log-log type integrals on the unit domain and on the positive half line, we produce a substantial number of new identities, representing the value of the integral with the aid of Euler sums. A new family of Euler sum identities will also be given, thereby extending the current knowledge.


Introduction preliminaries and notation
There exists a vast literature in which an exceptionally large number of integral formula have been developed, refer to [2, 4-6, 10, 12, 13, 23, 26] There are also many research papers dealing with specific evaluations and analysis of representations of log-log type integrals, refer to [1,3,7,9,14,25,28] . In this paper the intention is to extend the knowledge and application of these log-log type integrals by examining families of the type, in terms of special functions including the Riemann zeta function, Clausen functions and harmonic numbers. For example we obtain some difficult to evaluate results of the form À Á are the Clausen functions. Some special cases of (1.1) have been considered in [8] and [14] but only for the case m ¼ 0: The m 6 ¼ 0 case adds some unexpected complexities in the analysis and the need to develop some new Euler sum identities of the form for real parameters a; b; p ð Þ such that W þþ 1;p b; a ð Þ converges and the harmonic numbers where c is the familiar Euler Mascheroni constant and for complex values of z; z 2 Cn 0; À1; À2; Á Á ÁÁ f g ; w z ð Þ is the digamma (or psi) function defined by where C z ð Þ is the Gamma function, see [24] . In this paper we denote C; R; R þ ; Z and N as the sets of complex numbers, real numbers, positive real numbers integers and positive integers respectively and let N 0 :¼ N [ 0 f g and Z À :¼ ZnN 0 : In (1.2) we have the appearance of the Clausen function where the generalized Clausen functions are defined for z 2 C with < z ð Þ [ 1 as, and may be extended to all the complex plane through analytic continuation. When z is replaced by a non negative integer n, the standard Clausen functions are defined by the Fourier series sin kx ð Þ k n ; for n even P k ! 1 cos kx ð Þ k n ; for n odd 8 > > > < > > > : : The polylogarithm function Li p ðzÞ is, for z j j 1 and in terms of the Polylogarithm, Li n e Àih À Á À Li n e ih À Á À Á ; for even n 1 2 Li n e Àih À Á þ Li n e ih À Á À Á ; for odd n: The polygamma function has the recurrence and can be connected to the Clausen function in the following way. The Clausen function of rational argument and even integer order Cl 2m where S n þ 1; j þ 1 ð Þare Stirling numbers of the second kind. Equivalently we can write Li Àn ðzÞ ¼ 1 where the Eulerian numbers n j Some other pertinent papers dealing with Euler sums are [15][16][17] and the excellent books [23,26]. We expect that integrals of the type (1.1) may be represented by Euler sums and therefore in terms of special functions such as the Riemann zeta function, the Clausen function and the polygamma functions. A search of the current literature has found some examples for the representation of the log-log integrals in terms of Euler sums, see [27]. The following papers [11,[18][19][20][21] and [22] also examined some integrals in terms of Euler sums. The two examples (1.2) and (1.3) will be considered in detail, moreover, these integrals are not amenable to a computer mathematical package.

Analysis of integrals
Consider the following.
Theorem 1 Let a 2 R ! À 2 and m; p; q ð Þ2N 0 ; the following integral, I a; m; p; q ð Þ¼ where H n are harmonic numbers, s l; l þ 1 À k ð Þare signed Stirling numbers of the first kind and Li kÀl x q ð Þ are the polylogarithms at zero or negative integer.
Proof From the definition of the polyogarithm Li 1 ðx q Þ ¼ À log 1 À x q ð Þ; a Taylor series expansion, for x 2 0; 1 ð Þ produces The m th derivative and integrating both sides for x 2 0; 1 ð Þ, we have ð2:5Þ Here we require the exclusion of all terms of the form qn þ j þ a À m ¼ 0 and for convergence requirements we put p ! m þ 1: From the equivalent expressions (2.4) and (2.5) The next corollary deals with the degenerate case of m ¼ 0 for the representation of the integral (2.1).
Corollary 1 Let a 2 R ! À 2; m ¼ 0 and p; q ð Þ 2 N 0 ; the following integral, for p ! 1: We utilize the following notation The following required Euler sum identity appears in [18]. Let a be a real number a 6 ¼ À1; À2; À1; . . .; and assume that p 2 Nn 1 f g: Then where c is the Euler Mascheroni constant.
Proof From the Taylor series (2.3) I a; 0; p; q ð Þ¼ Two examples are now given. À Á are the Clausen functions.
In this next example we shall require Euler sum identities of the form (1.4), the next proposition will be essential.
Proof In (2.8) we put a ¼ 1 y ; y 6 ¼ 0; differentiate with respect to y and then rename y as a so that (2.9) follows. Similar analysis allows us to evaluate W þþ 1;pþ1 b; a ð Þ for b 2 N: h Example 2 For this example consider the case a ¼ 0; m ¼ 2; p ¼ 3 and q ¼ 3 We have put in the values of the Stirling numbers of the first kind, and evaluating we obtain ð2:12Þ The Euler sums in (2.12) can be evaluated from (2.8) and ( 2.9). The integrals in (2.12) can be evaluated by standard techniques and we evaluate the first integral to give a hint of the method used. Consider ð2:13Þ upon expansion and simplification using (1.8) and (1.7) we obtain Finally we obtain the identity (1.2), 3 The positive half line x ‡ 0 ð3:2Þ where I 0; m; p; q ð Þ and I m À 1; m; p; q ð Þ are given by (2.2), n þ 1 ð Þ m is the Pochhammer symbol and i ¼ ffiffiffiffiffiffi ffi À1 p : Proof Let us put and notice that lim upon making the transformation xt ¼ 1 (and renaming t as x) in the third integral we obtain J m; p; q ð Þ¼ Expanding the last two integrands in a Taylor series form and then integrating in the interval x 2 0; 1 ð Þ; we have J m; p; q ð Þ¼I 0; m; p; q ð Þþ À 1 ð Þ mþpþ1 I m À 1; m; p; q ð Þ þ À1 ð Þ mþ1 ipp! X The Pochhammer symbol k ð Þ x for k; x ð Þ 2 C can be defined in terms of the Gamma function C Á ð Þ; by and conventionally understood that 0 ð Þ 0 ¼ 1: Finally we obtain The particular case of m ¼ 1 has a complete representation in terms of polygamma functions and the details are developed in the following corollary. Corollary 2 Let m ¼ 1; p; q ð Þ 2 N and put p ! 2; then where w p ð Þ jÀ1 q are the polygamma functions (1.6) and S þþ 1;p 0; jÀ1 q are the Sofo-Cvijović-Euler sums (2.8).
Proof Let m ¼ 1; p; q ð Þ 2 N; The required evaluation of I 0; 1; p; q ð Þcan be done by applying (2.2) so that since the Stirling numbers of the first kind s 1; 1 ð Þ ¼ 1 and Li 0 ðx q Þ ¼ x q 1Àx q ; then A similar evaluation of that used for the integrals in (2.12) yields the result Substituting into (3.4) we have which is the result (3.3). It is evident from (3.4) that in the case of odd integer p, say 2p À 1 then Concluding Remarks We have studied of a family of integrals having log À log and polynomial functions in terms of Euler sums, which themselves incorporate special functions such as Beta functions, Clausen functions and Zeta functions. For higher values of the parameters m, p and q our results are new in the literature. We have evaluated four specific examples which are not amenable to a mathematical computer package. Further work examining integral families containing Polylogarithmic functions will be presented in the near future.
Funding Open Access funding enabled and organized by CAUL and its Member Institutions.

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