On multiple L-values

Evaluations of a class of double 𝐿-values

“…respectively. It has been shown previously that the last sum converges if ψ = χ 0 , and converges absolutely if b > 1 [1,8]. It turns out that rearrangement of the limit symbols leaves the sum invariant.…”

“…and making the change of variables z = e(τ ), we see J(−1) equals J(1) + h (1). By Lemma 3, (1) and (2), h(1) is (2πi) 2 times a K-linear finite combination of products of powers of 2πi and positive-integer values of L-series of characters with conductors dividing F .…”

“…In this paper, the term double L-value will refer to a sum of the second type. Multiple L-values have previously been defined more generally [1], and have been shown to have an extensive and interesting algebraic structure [4,7]. In the special case χ = ψ = 1, L(χ, ψ; a, b) equals the double zeta value ζ(a, b).…”

On evaluation formulas for double L-values

“…Independently, Mordell also considered these series in [5]. Recently Huard, Williams and Zhang Nan-Yue gave an explicit formula for T(k, I, N -k -I) as a rational linear combination of the products C(2j)<(JV -2j) {0 ^ j ^ (N -3)/2) when N is odd, N > 3, and k,l € N o satisfying l^k + l^N-l, k^N-2&ndl^N-2 (see [3] [1,6,7]). Recently Terhune proved that if x ( -l ) t / ' ( -l ) = ( -l ) formulas for double L-series given in [6,7].…”

Generalized multiple Dirichlet series and generalized multiple polylogarithms

“…The special case u = 1 and a j (n) = 1 (1 ≤ j ≤ r) has been studied by the first author in [12,13]; it can be regarded as a generalization of both the Euler-Zagier multiple zeta function and the Barnes multiple zeta function. On the other hand, the special case u = 1, α j = j and w j = 1 (1 ≤ j ≤ r) has also been studied before: see Arakawa-Kaneko [2] when a j s are periodic functions on Z, and MatsumotoTanigawa [14] for more general a j s.…”