Bouzeffour and Halouani Advances in Difference Equations (2016) 2016:126 DOI 10.1186/s13662-016-0811-9
R E S E A R C H Open Access On the asymptotic expansion of the q-dilogarithm Fethi Bouzeffour* and Borhen Halouani
*Correspondence: fbouzaff[email protected] Abstract Department of mathematics, College of Sciences, King Saud In this work, we study some asymptotic expansion of the q-dilogarithm at q =1and University, P.O. Box 2455, Riyadh, q = 0 by using the Mellin transform and an adequate decomposition allowed by the 11451, Saudi Arabia Lerch functional equation. MSC: 33D45; 33D80 Keywords: q-special functions; difference-differential equations
1 Introduction Euler’s dilogarithm is defined by []
∞ zn Li (z)= , |z| < . (.) n n=
In [], Kirillov defines the following q-analog of the dilogarithm Li(z):
∞ zn Li (z; q)= , |z| <, and he observes the following remarkable formula ([], Section ., Lemma ): ∞ zn = exp Li (z, q) , |z| <,|q| <, (.) (q, q) n= n where n– k (q, q) =, (q, q)n = –q , n =,,.... (.) k= It seems a precise formulation of (.)goingbacktoRamanujan(see[], Chapter , En- try ) is given an asymptotic series for Li(z; q) and Hardy and Littlewood []provedthat for |q| = , the identity holds inside the radius of convergence of either series. Let ω = ezx+iθ with Re(z)>,x >,and<θ < . The main result of this work is –x the following complete asymptotic expansion of the q-dilogarithm function Li(ω; e )at © 2016 Bouzeffour and Halouani. This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna- tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Bouzeffour and Halouani Advances in Difference Equations (2016) 2016:126 Page 2 of 9 x → : ∞ (–)n+ Li ω, e–x ∼ Ci (θ) + – z Ci (θ)+ x (n +)(n +)! n= iπθ n × Bn+(z)Bn+ , e x as x → (.) and ∞ γ i ψ(n–)(z)B (θ) π n Li ω, e–x ∼ B (θ) + in n+ , x →∞.(.) π x (n +)! x n= In Section ., Corollary of [], Kirillov and Ueno and Nishizawa derived the asymp- totic expansion (.) by using the Euler-Maclaurin summation formula; see also [], an integral representation of Barnes type for the q-dilogarithm. Second, we use the Lerch functional equation to decompose the integrand and to apply the Cauchy theorem. 2 q-Dilogarithm The polylogarithm is defined in the unit disk by the absolutely convergent series [] ∞ zn Li (z)= , |z| <. (.) s ns n= Several functional identities satisfied by the polylogarithm are available in the literature (see []). For n =,...,thefunctionLin(z) can also be represented as z z Lin–(t) dt Lin(z)= dt, n ∈ N, Li(z)=–log( – z)= , (.) t –t which is valid for all z in the cut plane C \ [, ∞). inπθ The notation F(θ, s) is used for the polylogarithm Lis(e )withθ real, called the peri- odic zeta function (see [], Section .) and is given by the Dirichlet series ∞ einπθ F(θ, s)= , θ ∈ R,(.) ns n= it converges for Re s >ifθ ∈ Z,andforRe s >ifθ ∈ R/Z.Thisfunctionmaybeexpressed in terms of the Clausen functions Cis(θ)andSis(θ), and vice versa (see [], Section .): ±iθ Lis e = Cis(θ) ± iSis(θ). (.) In [], Koornwinder defines the q-analog of the logarithm function ∞ zn – log( – z)= , |z| <, n n= as follows: ∞ zn log (z)= , |z| <, Recall that the q-analog of the ordinary integral (called Jackson’s integral) is defined by z ∞ n n f (t) dqt =(–q)z f zq q .(.) n= One can recover the ordinary Riemann integral as the limit of the Jackson integral for q ↑ . Lemma . The function logq(z) has the following q-integral representation: z | | ( – q) logq(z)= dqt, z <. (.) –t –n Moreover, it can be extended to any analytic function on C – {q , n ∈ N}. Proof Assume that |z| <,thenfrom(.)wehave ∞ ∞ n nm ( – q) logq(z)=(–q) z q n= m= ∞ ∞ =(–q)z qm znqnm m= n= ∞ qm =(–q)z . –zqm m= The inversion of the order of summation is permitted, since the double series converges absolutely when |z| <. –n Let K be a compact subset of C – {q , n ∈ N}.ThereexistsN ∈ N such that, for all z ∈ K, |qN z| < q.Thenforn ≥ N we have m m q q ≤ .(.) –zqm –q ∞ qm Hence, the series m=N –zqm converges uniformly in K. The q-dilogarithm (.) is related to Koornwinder’s q-logarithm (.)by z logq(t) Li(z, q)= dt.(.) t It follows that, for n ≥ , we can also define z Lin–(t, q) Lin(z, q)= dt.(.) t This integral formula proves by induction that Lin(z, q) has an analytic continuation on C –[,∞). Moreover, for |z| <,wehave ∞ zk Li (z, q)= . n kn( – qk) k= Bouzeffour and Halouani Advances in Difference Equations (2016) 2016:126 Page 4 of 9 This converges absolutely for |z| < and defines a germ of a holomorphic function in the neighborhood of the origin. Note that lim( – q)Li ( – q)z, q = Li(z), q↑ lim( – q)Li(z, q)=–Log( – z), |z| <. q↓ Let ω = e–zx+iπθ, θ ∈ R,andRe z >,wedefine ∞ e–zx cos(πnθ) Ci (ω, q)= ,(.) n( – qn) n= ∞ e–zx sin(πnθ) Si (ω, q)= .(.) n( – qn) n= Note that these functions can be considered as q-analogs of the Clausen functions (.) and are related to the q-dilogarithm by Li(ω, q)=Ci(ω, q)+iSi(ω, q). (.) Now, we will use the Mellin transform method to obtain the integral representation c+i∞ –x –s Li ω, e = ζ (s, z)F(θ, s)(s)x ds, c >, (.) iπ c–i∞ where ω = e–zx+iπθ, x >, Re z >, <θ <. Recall that the Mellin transform for a locally integrable function f (x)on(,∞)isdefined by ∞ M(f , s)= f (x)xs– dx, (.) which converges absolutely and defines an analytic function in the strip a < Re s < b, where a and b are real constants (with a < b)suchthat,forε >, ⎧ ⎨ O(x–a–ε)asx → +, f (x)=⎩ (.) O(x–b–ε)asx → +∞. The inversion formula reads c+i∞ f (x)= M(f , s)x–sds,(.) iπ c–i∞ Bouzeffour and Halouani Advances in Difference Equations (2016) 2016:126 Page 5 of 9 where c satisfies a < c < b.Equation(.) is valid at all points x ≥ wheref (x) is contin- uous. We first compute the Mellin transform M(ψn(x), s), where e–nzx ψ (x)= , x >,Re z >,n ∈ N.(.) n n( – e–nx) Since ψ (x) ∼ , x → +,(.) n nx ψ (x) ∼ e–nx(z–), x → +∞.(.) n n We concluded that M(ψn(x), s) is defined in the half-plane Re s > . That is, the constants a and b satisfy a =andb =+∞, which values can be used for all n ≥ andRe z >.The Mellin transform of ψn(x) can be obtained from the following integral representation of the Hurwitz zeta function ζ (s, z): ∞ e–zx ζ s z xs– dx Re s arg z π Re s z ( , )= –x >, ( – ) < ; >, = . (.) (s) –e Note that ζ (s, z)isexpressedalsobytheseries ∞ ζ (s, z)= , Re s >,z =–,–,.... (.) (z + k)s k= For the other values of z, ζ (s, z) is defined by analytic continuation. It has a meromorphic continuation in the s-plane, its only singularity in C being a simple pole at s =, ζ (s, z)= – ψ(z)+O(s – ). (.) s – Applying the Mellin inversion theorem to the integral (.), we then find c+i∞ –s ψn(x)= ζ (s, z)(s)(nx) ds. (.) iπ c–i∞ We use the Stirling formula, which shows that, for finite σ , σ π| | (σ +it)=O |t| –e– t |t|→+∞ (.) and the well-known behavior of ζ (s, z)(see[]) ζ (s, z)=O |t|τ(σ) log |t| ,(.) where ⎧ ⎪ ⎪ – σ , σ ≤ , ⎪ ⎨ ≤ ≤ ,σ , τ(σ )=⎪ ⎪ σ ≤ σ ≤ ⎪– , , ⎩⎪ , σ ≥ . Bouzeffour and Halouani Advances in Difference Equations (2016) 2016:126 Page 6 of 9 Then we obtain the following majorization of the modulus of the integrand in (.): O |t|τ(σ)+σ– log |t| .(.) Consequently, the integral (.) converges absolutely in the whole vertical strip of the half-plane Re s >.Thenwereplacex by nx,wheren is a positive integer, and sum over n, and we then obtain c+i∞ –x –s Li ω, e = ζ (s, z)F(θ, s +)(s)x ds, c >, (.) iπ c–i∞ where ω = e–zx+iπθ, x >, Re z >, <θ <. 3 Asymptotic at q =1 The integral (.) will be used to derive asymptotic expansions of the q-dilogarithm. The contour of integration is moved at first to the left to obtain an asymptotic expansion at q = and then to the right to get an asymptotic expansion at q =. Let us consider the function g(s)=ζ (s, z)F(θ, s +)(s). (.) The periodic function zeta function F(θ, s) has an extension to an entire function in the s-plane (see []). Hence, the function g(s) has a meromorphic continuation in the s-plane, its only singularity in C coincides with the pole of (s)andζ (s, z)beingasimplepoleat s =,,–,–,.... Now we compute the residues of the poles. The special values at s = –,–... ofthe periodic zeta function are reduced to the Apostol-Bernoulli polynomials (see []), B (, eiπθ) F(θ,–n)=– n+ .(.) n + We need also the following asymptotic expansions of (s)andζ (s)ats =: (s)= – γ + O s , (.) s (z) ζ (s)= – z + s log + O s .(.) π Hence, iπθ lim(s –)g(s)=Li e , s→ n (–) iπθ lim (s + n)g(s)= Bn+(z)Bn+ , e . s→–n (n +)(n +)! Here Bn(z) is the Bernoulli polynomial (see []). Let N be an integer and d real number such that –N – the side in Re(s) < parallel to the imaginary axis passes midway between the poles s = ,–,–,...,–N. The contribution from the upper and lower sides s = σ ± iA vanishes as |A|→+∞, since the modulus of the integrand is controlled by τ σ σ π| | O |A| ( )+ –/ log |A|e– A .(.) This follows from Stirling’s formula (.), the behavior ζ (s, z)beinggivenby(.), and the following estimation: F(θ, s +) ≤ ζ (σ +)=O(), |A|→+∞. Displacement of the contour (.)totheleftthenyields Li ω, e–x = Ci (θ) + – z Ci (θ) x N (–)n+ + B (z)B , eiπθ xn + R (x), (.) (n +)(n +)! n+ n+ N n= where the remainder integral RN (z)isgivenby d+i∞ –s RN (x)= ζ (s, z)F(θ, s +)(s)x ds, x >,Re z >. (.) iπ d–i∞ From (.), we find R (x) = O . N xN+ 4 Asymptotic at q =0 Recall that the periodic zeta function satisfies the functional equation (see []) ( – s) πi(–s) πi(s–) F(θ, s)= e ζ ( – s, θ)+e ζ ( – s,–θ) (π)–s (Re s >,<θ < ), (.) first given by Lerch, whose proof follows the lines of the first Riemann proof of the func- tional equation for ζ (x). It is well known that the asymptotic expansion near infinity via the Mellin transform is obtained by displacement of the contour of integration in the Mellin inversion formulas (.) to the right-hand side (see []). However, the integrand (.)hasnopolesinthe half-plane Re s > . The periodic zeta function F(θ, s) has an analytic continuation to the whole s-space for < θ < . Moreover, the poles of ( – s)inequation(.)ats =–,–... are canceled by the zeros of the function πi(–s) πi(s–) e ζ ( – s, θ)+e ζ ( – s,–θ). On the other hand from (.)weeasilyobtain (π)s+ (s) F(θ, s +)+F( – θ, s +) =– πs ζ (–s, θ)+ζ (–s,–θ) ,(.) s sin Bouzeffour and Halouani Advances in Difference Equations (2016) 2016:126 Page 8 of 9 where we are able to simplify (.) by the well-known reflection formulas π sin πs πs π( – s) = (s)( – s), = sin sin . sin πs π π Proceeding similar to above we also obtain (π)s+ θ θ ζ θ ζ θ (s) F( , s)–F( – , s) = π(s) (–s, )– (–s,– ) .(.) s cos Moreover, the integral representation (.)isvalidforall<θ < . So we can replace θ by – θ in its integrand. Using the above decomposition (.)and(.), we then obtain c+i∞ (π)s+ζ (s, z) ds ω –x ζ θ ζ θ Ci , e =– πs (–s, )+ (–s,– ) s (.) iπ c–i∞ s sin x and c+i∞ (π)s+ζ (s, z) ds ω –x ζ θ ζ θ Si , e = πs (–s, )– (–s,– ) s ,(.) iπ c–i∞ s cos x where ω = e–zx+iπθ, Note that the special values ζ (n, z)(n ∈ N) are expressed in terms of the polygamma function ψ(z), (–)n+ ζ (n +,z)= ψ(n)(z), z =,–,–,..., (.) n! and ζ (–n, z)(n ∈ N) is reduced to the Bernoulli polynomial B (z) ζ (–n, z)=– n+ .(.) n + Applying the identities for the Bernoulli polynomial n Bn( – θ)=(–) Bn(θ), we obtain B (θ) ζ (–n, θ)+ζ (–n,–θ)= (–)n+ – n+ ,(.) n + B (θ) ζ (–n, θ)–ζ (–n,–θ)= (–)n – n+ .(.) n + The integrand in (.) has a meromorphic continuation in the s-plane, its only singularity πs in the half-plane Re s > coincides with the pole of / sin being a simple pole at s = ,,....ThenbytheCauchyintegral,wecanshiftthecontourin(.)totheright,picking up the residues at s =,...,N ,withtheresult N ψ(n–)(z)B (θ) π n Ci ω, e–x = (–)n n+ + Q (x), (.) (n +)! x N n= Bouzeffour and Halouani Advances in Difference Equations (2016) 2016:126 Page 9 of 9 where c+N+i∞ (π)s+ζ (s, z) ds ζ θ ζ θ QN (x)=– πs (–s, )+ (–s,– ) s .(.) iπ c+N–i∞ s sin x Using the following estimations in a vertical strip s = σ +it, σ =, ±, ±,..., π | | O | |– – t πs = t e ,(.) sin we obtain Q (x) = O . (.) N xN+ Similarly, γ N ψ(n)(z)B (θ) π n+ Si ω, e–x = B (θ) + (–)n n+ + O .(.) π x (n +)! x xN+ n= Proposition . Let ω = e–zx+iπθ, x >,Re z >and <θ <.Then ∞ γ i ψ(n–)(z)B (θ) π n Li ω, e–x ∼ B (θ) + in n+ , x →∞. (.) π x (n +)! x n= Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Acknowledgements The first author would like to extend his sincere appreciation to the Deanship of Scientific Research at King Saudi University for funding this Research group No. (RG-1437-020). 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