Generalized Log-Sine Integrals

• Generalized log-sine integrals are a family of trigonometric-logarithmic integrals defined via the kernel log|2 sin(θ/2)| that connect classical functions like Clausen functions and zeta values.
• They include various forms—such as generalized moments, shifted, normalized, and iterated integrals—that facilitate conversion into polylogarithmic and Nielsen polylogarithmic expressions.
• These integrals have practical applications in evaluating Mahler measures, central binomial sums, and poly-Bernoulli numbers, thereby advancing analytic and number theory research.

Generalized log-sine integrals are a family of trigonometric-logarithmic integrals built from the kernel $\log\!\left|2\sin \frac{\theta}{2}\right|$ and its moments, shifted variants, normalized forms, and iterated analogues. In the modern literature they include the classical quantities $Ls_n(\sigma)$, the moment integrals $Ls_n^{(k)}(\sigma)$, log-sine-cosine integrals, shifted order-3 forms, normalized integrals $\ls_m^{(k)}(z)$, zeta-type shifted log-sine integrals $\mathrm{SLs}(s;\sigma)$, and multi-indexed iterated log-sine integrals. Across these formulations, generalized log-sine integrals function as an analytic bridge between Clausen functions, zeta values, multiple polylogarithms, multiple zeta values, Mahler measures, central binomial series, and, in recent work, poly-Bernoulli numbers with negative index (Borwein et al., 2011, Matsusaka, 26 Mar 2026).

1. Classical kernel and standard generalizations

The basic log-sine integrals are

$$Ls_n(\sigma):=-\int_0^\sigma \log^{\,n-1}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad n=1,2,\dots,$$

and their generalized moments are

$$Ls_n^{(k)}(\sigma):=-\int_0^\sigma \theta^k\,\log^{\,n-1-k}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad k\ge 0.$$

They satisfy $Ls_1(\sigma)=-\sigma$, $Ls_n^{(0)}(\sigma)=Ls_n(\sigma)$, and

$$Ls_2(\sigma)=Cl_2(\sigma)=\sum_{m=1}^\infty \frac{\sin(m\sigma)}{m^2},$$

so the lowest nontrivial log-sine integral is already the Clausen function (Borwein et al., 2011, Borwein et al., 2011).

Two common extensions appear early in the literature. The first is the log-sine-cosine family

$Ls_n(\sigma)$0

with $Ls_n(\sigma)$1. The second is the shifted order-3 integral

$Ls_n(\sigma)$2

for which $Ls_n(\sigma)$3 (Borwein et al., 2011).

Classical identities already exhibit the arithmetic content of the kernel. Matsusaka recalls Euler’s identity

$Ls_n(\sigma)$4

and Clausen’s identity

$Ls_n(\sigma)$5

These formulas show that log-sine integrals encode special values of zeta-related functions and motivate their later generalizations (Matsusaka, 26 Mar 2026).

A related antiderivative viewpoint is supplied by Dowker’s

$Ls_n(\sigma)$6

whose repeated integration-by-parts reduction and Newman’s 1847 formula recover the standard closed forms in terms of odd zeta values and Clausen functions (Dowker, 2024).

2. Normalized, shifted, and deformed families

One widely used normalization factors out the endpoint value of the sine function: $Ls_n(\sigma)$7 with

$Ls_n(\sigma)$8

A second variant is the shifted normalized integral

$Ls_n(\sigma)$9

which appears naturally when powers of the logarithm are expanded in generating-function arguments (Shalev, 7 Aug 2025).

A more recent zeta-type generalization is Matsusaka’s shifted log-sine integral

$Ls_n^{(k)}(\sigma)$0

defined for $Ls_n^{(k)}(\sigma)$1 and $Ls_n^{(k)}(\sigma)$2. Its defining features are the shift by the ratio $Ls_n^{(k)}(\sigma)$3, the linear factor $Ls_n^{(k)}(\sigma)$4, and the normalization by $Ls_n^{(k)}(\sigma)$5, which makes it behave like a zeta-type function in the parameter $Ls_n^{(k)}(\sigma)$6 (Matsusaka, 26 Mar 2026).

The literature also contains affine and Mellin-type deformations. In the study of the incomplete Mellin transformation of the fractional part, the generalized log-sine integral

$Ls_n^{(k)}(\sigma)$7

appears as one side of an affine complex deformation of the log-sine/fractional-part kernel. In that setting, equalities between hypergeometric series and zeta/log-sine expressions capture meromorphic extension and the functional equation of the Hurwitz zeta function (Adam, 2020).

These variants show that “generalized log-sine integral” is not a single rigid notation but a family of constructions sharing the same logarithmic sine kernel. The common pattern is that normalization, shifting, or deformation is introduced to expose additional analytic structure: endpoint regularization, polylogarithmic generating functions, Mellin transforms, or zeta-type continuation.

3. Special values, generating functions, and explicit evaluation

At $Ls_n^{(k)}(\sigma)$8, the standard log-sine values are controlled by the exponential generating function

$Ls_n^{(k)}(\sigma)$9

This implies that all $\ls_m^{(k)}(z)$0 reduce to zeta values. Explicit examples include

$\ls_m^{(k)}(z)$1

$\ls_m^{(k)}(z)$2

The corresponding generalized values $\ls_m^{(k)}(z)$3 are extracted from a double generating function and lie in the algebra generated by $\ls_m^{(k)}(z)$4 and Nielsen polylogarithms at $\ls_m^{(k)}(z)$5 (Borwein et al., 2011, Borwein et al., 2011).

At $\ls_m^{(k)}(z)$6 and multiples of $\ls_m^{(k)}(z)$7, quasiperiodicity simplifies the picture further. The generating function

$\ls_m^{(k)}(z)$8

shows that $\ls_m^{(k)}(z)$9 reduce to zeta values only, and a harmonic-sum formula reduces all multiples of $\mathrm{SLs}(s;\sigma)$0 to the $\mathrm{SLs}(s;\sigma)$1 case (Borwein et al., 2011).

At $\mathrm{SLs}(s;\sigma)$2, the sixth-root-of-unity structure yields a different type of reduction. The values are expressed in terms of zeta values together with multiple Clausen and Glaisher values at $\mathrm{SLs}(s;\sigma)$3. Representative evaluations are

$\mathrm{SLs}(s;\sigma)$4

$\mathrm{SLs}(s;\sigma)$5

and

$\mathrm{SLs}(s;\sigma)$6

Algorithmic evaluations for general arguments are also available, and an implementation in Mathematica and SAGE was provided (Borwein et al., 2011).

Two complementary closed-form techniques are notable. Dowker recovers standard formulas for $\mathrm{SLs}(s;\sigma)$7 by repeated integration by parts plus Newman’s 1847 antiderivative formula, avoiding Hurwitz-zeta derivatives (Dowker, 2024). Separately, the differentiation method based on

$\mathrm{SLs}(s;\sigma)$8

identifies generalized log-sine integrals with parameter derivatives at $\mathrm{SLs}(s;\sigma)$9, while complete Bell polynomials encode derivatives of central and shifted central binomial coefficients in terms of polygamma functions and harmonic numbers (Orr, 2017).

4. Polylogarithmic, Nielsen, and multiple-zeta structures

A central modern theme is the conversion of generalized log-sine integrals into polylogarithmic objects. For normalized integrals, the generating identity

$$Ls_n(\sigma):=-\int_0^\sigma \log^{\,n-1}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad n=1,2,\dots,$$0

yields, after expansion in $$Ls_n(\sigma):=-\int_0^\sigma \log^{\,n-1}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad n=1,2,\dots,$$1, explicit formulas for polylogarithms and log-sine integrals. In particular,

$$Ls_n(\sigma):=-\int_0^\sigma \log^{\,n-1}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad n=1,2,\dots,$$2

This framework proves that all generalized log-sine integrals of weight $$Ls_n(\sigma):=-\int_0^\sigma \log^{\,n-1}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad n=1,2,\dots,$$3 can be written using only classical polylogarithms, while higher weights require Nielsen polylogarithms (Shalev, 7 Aug 2025).

The higher-weight transition is explicit. Nielsen polylogarithms are defined by

$$Ls_n(\sigma):=-\int_0^\sigma \log^{\,n-1}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad n=1,2,\dots,$$4

with $$Ls_n(\sigma):=-\int_0^\sigma \log^{\,n-1}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad n=1,2,\dots,$$5. The identity

$$Ls_n(\sigma):=-\int_0^\sigma \log^{\,n-1}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad n=1,2,\dots,$$6

implies that every generalized log-sine integral $$Ls_n(\sigma):=-\int_0^\sigma \log^{\,n-1}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad n=1,2,\dots,$$7 can be expressed in terms of Nielsen polylogarithms. The same method also produces functional equations for $$Ls_n(\sigma):=-\int_0^\sigma \log^{\,n-1}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad n=1,2,\dots,$$8 and $$Ls_n(\sigma):=-\int_0^\sigma \log^{\,n-1}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad n=1,2,\dots,$$9 and a hyperbolic analogue obtained by replacing $$Ls_n^{(k)}(\sigma):=-\int_0^\sigma \theta^k\,\log^{\,n-1-k}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad k\ge 0.$$0 with $$Ls_n^{(k)}(\sigma):=-\int_0^\sigma \theta^k\,\log^{\,n-1-k}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad k\ge 0.$$1 (Shalev, 7 Aug 2025).

The iterated theory enlarges the scope from single integrals to multi-indexed simplicial integrals. For

$$Ls_n^{(k)}(\sigma):=-\int_0^\sigma \theta^k\,\log^{\,n-1-k}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad k\ge 0.$$2

Umezawa defines

$$Ls_n^{(k)}(\sigma):=-\int_0^\sigma \theta^k\,\log^{\,n-1-k}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad k\ge 0.$$3

with the convergence condition $$Ls_n^{(k)}(\sigma):=-\int_0^\sigma \theta^k\,\log^{\,n-1-k}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad k\ge 0.$$4. When $$Ls_n^{(k)}(\sigma):=-\int_0^\sigma \theta^k\,\log^{\,n-1-k}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad k\ge 0.$$5, these reduce to the generalized log-sine integrals. The main evaluation theorem states that for $$Ls_n^{(k)}(\sigma):=-\int_0^\sigma \theta^k\,\log^{\,n-1-k}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad k\ge 0.$$6 every such iterated log-sine integral is a $$Ls_n^{(k)}(\sigma):=-\int_0^\sigma \theta^k\,\log^{\,n-1-k}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad k\ge 0.$$7-linear combination of powers of $$Ls_n^{(k)}(\sigma):=-\int_0^\sigma \theta^k\,\log^{\,n-1-k}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad k\ge 0.$$8, powers of $$Ls_n^{(k)}(\sigma):=-\int_0^\sigma \theta^k\,\log^{\,n-1-k}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad k\ge 0.$$9, multiple zeta values, and multiple polylogarithms $Ls_1(\sigma)=-\sigma$0 with admissible indices (Umezawa, 2019).

The algebraic mechanism behind this reduction already appears in Umezawa’s earlier iterated-integral formulation. There the iterated log-sine integrals satisfy a shuffle product, a reduction identity when some exponent $Ls_1(\sigma)=-\sigma$1 vanishes, and a conversion formula expressing $Ls_1(\sigma)=-\sigma$2 in terms of iterated log-sine integrals. Combined with the Borwein–Broadhurst–Kamnitzer identity, this yields a method for deriving relations among multiple zeta values; the paper records, for example, $Ls_1(\sigma)=-\sigma$3 and recovers classical relations in weights $Ls_1(\sigma)=-\sigma$4 and $Ls_1(\sigma)=-\sigma$5 (Umezawa, 2019).

5. Mahler measures, central binomial sums, and related series

Generalized log-sine integrals became especially prominent through their role in higher and multiple Mahler measures. For the family

$Ls_1(\sigma)=-\sigma$6

one has

$Ls_1(\sigma)=-\sigma$7

Consequently,

$Ls_1(\sigma)=-\sigma$8

and higher values involve Clausen or Glaisher constants at $Ls_1(\sigma)=-\sigma$9, zeta values, and sometimes central-binomial-sum constants (Borwein et al., 2011).

Multiple Mahler measures of simpler polynomials also reduce directly to generalized log-sine-cosine integrals. For $Ls_n^{(0)}(\sigma)=Ls_n(\sigma)$0,

$Ls_n^{(0)}(\sigma)=Ls_n(\sigma)$1

For $Ls_n^{(0)}(\sigma)=Ls_n(\sigma)$2, one has

$Ls_n^{(0)}(\sigma)=Ls_n(\sigma)$3

while $Ls_n^{(0)}(\sigma)=Ls_n(\sigma)$4 is expressed in terms of $Ls_n^{(0)}(\sigma)=Ls_n(\sigma)$5, $Ls_n^{(0)}(\sigma)=Ls_n(\sigma)$6, Clausen values, and $Ls_n^{(0)}(\sigma)=Ls_n(\sigma)$7. The auxiliary quantity

$Ls_n^{(0)}(\sigma)=Ls_n(\sigma)$8

is shown to be a finite linear combination of multiple polylogarithms at $Ls_n^{(0)}(\sigma)=Ls_n(\sigma)$9; in low weight,

$$Ls_2(\sigma)=Cl_2(\sigma)=\sum_{m=1}^\infty \frac{\sin(m\sigma)}{m^2},$$0

(Borwein et al., 2011).

At $$Ls_2(\sigma)=Cl_2(\sigma)=\sum_{m=1}^\infty \frac{\sin(m\sigma)}{m^2},$$1, generalized log-sine moments are tied to central binomial sums. The relation

$$Ls_2(\sigma)=Cl_2(\sigma)=\sum_{m=1}^\infty \frac{\sin(m\sigma)}{m^2},$$2

links the log-sine values to inverse central binomial series, and specific evaluations such as

$$Ls_2(\sigma)=Cl_2(\sigma)=\sum_{m=1}^\infty \frac{\sin(m\sigma)}{m^2},$$3

follow from this framework (Borwein et al., 2011).

A complementary development derives Apéry-like and Leshchiner-type identities from the same generating-function mechanism that evaluates normalized log-sine integrals. In that setting, specializations produce central binomial coefficient series, parametric Apéry-like formulas for $$Ls_2(\sigma)=Cl_2(\sigma)=\sum_{m=1}^\infty \frac{\sin(m\sigma)}{m^2},$$4 and $$Ls_2(\sigma)=Cl_2(\sigma)=\sum_{m=1}^\infty \frac{\sin(m\sigma)}{m^2},$$5, and identities of Koecher–Leshchiner type; these are not ancillary results but arise from the same polylogarithmic-sine integral correspondence (Shalev, 7 Aug 2025).

6. Shifted zeta-type formulations, poly-Bernoulli numbers, and conjectural directions

Matsusaka’s shifted log-sine integral provides a zeta-type object whose negative integer values are governed by central binomial series and Lehmer polynomials. A key structural theorem expresses

$$Ls_2(\sigma)=Cl_2(\sigma)=\sum_{m=1}^\infty \frac{\sin(m\sigma)}{m^2},$$6

where $$Ls_2(\sigma)=Cl_2(\sigma)=\sum_{m=1}^\infty \frac{\sin(m\sigma)}{m^2},$$7 is a central binomial series and $$Ls_2(\sigma)=Cl_2(\sigma)=\sum_{m=1}^\infty \frac{\sin(m\sigma)}{m^2},$$8 its half-integer-shift analogue. This yields analytic continuation of $$Ls_2(\sigma)=Cl_2(\sigma)=\sum_{m=1}^\infty \frac{\sin(m\sigma)}{m^2},$$9 to all $Ls_n(\sigma)$00. For integers $Ls_n(\sigma)$01 and $Ls_n(\sigma)$02,

$Ls_n(\sigma)$03

with $Ls_n(\sigma)$04 the Lehmer polynomials determined recursively by

$Ls_n(\sigma)$05

$Ls_n(\sigma)$06

The cancellation of the $Ls_n(\sigma)$07-term is the mechanism behind the closed formula (Matsusaka, 26 Mar 2026).

The special point $Ls_n(\sigma)$08 produces the poly-Bernoulli connection. Since $Ls_n(\sigma)$09,

$Ls_n(\sigma)$10

and for $Ls_n(\sigma)$11 this becomes

$Ls_n(\sigma)$12

This is the anti-diagonal sum of poly-Bernoulli numbers with negative index highlighted in the abstract. The identity was observed by Stephan and later proved in the cited work of Bényi–Matsusaka; within Matsusaka’s framework it arises naturally from analytic continuation of the shifted log-sine integral (Matsusaka, 26 Mar 2026).

This formulation is explicitly compared with the Arakawa–Kaneko zeta function. Arakawa and Kaneko introduced a zeta function whose special values at negative integers yield the poly-Bernoulli numbers “by design,” whereas the shifted log-sine integral is presented as a natural analytic object coming from classical log-sine geometry whose negative values automatically encode poly-Bernoulli numbers (Matsusaka, 26 Mar 2026).

Conjectural directions remain active in the iterated setting. Umezawa proposes that suitable families of shifted log-sine values at $Ls_n(\sigma)$13 should span, and in stronger form should form bases of, the spaces of multiple zeta values, multiple Clausen values, multiple Glaisher values, and iterated log-sine integrals, with predicted dimensions governed by Fibonacci-type recurrences. Proven depth-one results show that every Riemann zeta value, every Clausen value $Ls_n(\sigma)$14, and every Glaisher value $Ls_n(\sigma)$15 can be written as a $Ls_n(\sigma)$16-linear combination of appropriate shifted log-sine generators (Umezawa, 2019).

In aggregate, generalized log-sine integrals constitute a broad analytic framework rather than a single special function. Their unifying feature is the logarithmic sine kernel; their distinguishing feature is the range of arithmetic and polylogarithmic structures exposed by moments, shifts, iterated integration, endpoint normalization, Mellin deformation, and analytic continuation.