Asymptotic Factorization of Repeated-Index MZV and t-Value Ratios for Even Arguments

Asymptotic Factorization of Repeated-Index MZV and t-Value Ratios for Even Arguments G. M. Yang Abstract For even integers s= 2m≥2, the repeated-index multiple zeta values ζ({s}n) and their level–2 analogues t({s}n)admit closed forms involving factorial factors and algebraic units. While these formulas have been known since the work of Hoffman and Borwein-Bradley-Broadhurst, the structural origin of the unit factors has remained unclear. In this paper we derive unified combinatorial representations expressing both ζ({2p}n) and t({2p}n)as signed sums of odd 2p-th roots of unity. We show that all algebraic units appearing in these closed forms arise from a finite dihedral orbit, and that this orbit contains a unique dominant element. This uniqueness yields a canonical factorization ˜ An(2p) = 2−2pζ({2p}n) ζ({2p}n−1) t({2p}n−1) t({2p}n)=Fn(2p)Rn(2p), where the unit-correction term satisfies Rn(2p) = 1 + O(ρ−n)for some ρ > 1. As a result, since all remaining units contribute only exponentially small terms, the entire 1/n asymptotic expansion of ˜ An(2p)is determined solely by the factorial ratio Fn(2p), with no algebraic contribution from the unit part. MSC(2020) : Primary 11M32, Secondary 11M06, 11M41, 33B99, 05A19 Keywords : multiple zeta values(MZVs), repeated-index values, t-values, root-of-unity sums, dihedral symmetry, dominant algebraic units, factorial ratio, asymptotic expansion 1 Introduction Multiple zeta values (MZVs) ζ(s1, . . . , sk) = X n1>···>nk≥1 1 ns1 1···nsk k play a fundamental role in number theory, combinatorics, and quantum field theory. Among these, the repeated-index values ζ({s}n) = ζ(s, . . . , s | {z } ntimes ) 1 and their level–2 analogues [1,2] t({s}n) = X n1>···>nn≥1 1 (2n1−1)s···(2nn−1)s exhibit remarkable structure when sis even. Hoffman [1] provided a unified formulation showing that the quantities ζ({2p}n)and t({2p}n)admit closed forms involving powers of π, factorial factors, and finite sums of algebraic units of the form λn. These formulas suggested the existence of a canonical “factorial ×unit” decomposition, but the underlying origin of the units and their connection to rootof-unity symmetry have not been made explicit in the literature. The goal of this paper is to give a complete and elementary derivation of the structure of repeated even-index MZVs. Our main contributions are as follows: •We derive exact combinatorial formulas for both t({2p}n)and ζ({2p}n)from simple cosine and sine product generating functions. These formulas reveal that all such values are determined by the algebraic units Ψ(ε) = p X j=1 εjeπi(2j−1)/(2p), εj∈ {±1}. •We analyze the dihedral symmetry of the unit set {Ψ(ε)}and show that exactly one orbit contains the dominant units. This explains both the appearance and the eventual cancellation of the unit factors in ratios of repeated-index values. •Using these exact formulas, we prove the fundamental factorization ˜ An(2p) = Fn(2p)Rn(2p),˜ An(2p) = 2−2pζ({2p}n) ζ({2p}n−1) t({2p}n−1) t({2p}n), where the factorial term Fn(2p)contributes the entire algebraic 1/n asymptotic expansion, and the correction term Rn(2p) = 1 + O(ρ−n)decays exponentially. •Finally, we illustrate the theory with the explicit case s= 8 (p= 4), where the dominant unit becomes 1+√2, giving the classical expressions for ζ({8}n)and t({8}n). Thus all repeated-index MZVs of even weight admit a single unifying structural description in terms of finite combinatorial sums over signed subsets of odd 2p-th roots of unity. This yields transparent proofs of the structural results and provides a canonical explanation for the asymptotic behavior of ˜ An(2p). 2 Exact combinatorial representations of repeated evenindex MZVs In this section we derive exact closed forms for the level–2 values t({2p}n)and the classical MZVs ζ({2p}n)for all integers p≥1. Both formulas arise naturally from the cosine and 2 sine product generating functions, respectively, and their proofs rely only on elementary exponential expansions combined with a 2p–fold root-of-unity symmetry. These formulas refine all previously known structural results (e.g., [3,4,5]) by giving explicit combinatorial descriptions of the algebraic units governing the behavior of repeated-index MZVs. 2.1 Case ζ({2p}n) Proposition 2.1 (Combinatorial representation of repeated even–index MZVs).Let p≥1 be an integer and set ω2p=eπi/(2p),Ψ(ε) = p X j=1 εjω2j−1 2p, for ε= (ε1, . . . , εp)∈ {±1}p. Borwein-Bradley-Broadhurst [6] generating function X n≥0 x2pn ζ({2p}n) = (iπx)−p p Y j=1 sin ω2j−1 2pπx(1) implies the explicit combinatorial formula; see also [7,8]. ζ({2p}n) = π2pn (−1)p 2p(2pn +p)! X ε1,...,εp=±1p Y j=1 εjiΨ(ε)2pn+p(2) Proof. Substituting y=πx into (1) gives X n≥0y π2pnζ({2p}n) = (iy)−p p Y j=1 sin ω2j−1 2py. Thus, ζ({2p}n) = π2pn [y2pn]Φp(y),Φp(y) := (iy)−p p Y j=1 sin(ω2j−1 2py). Using sin z=1 2iX ε=±1 ε eiεz, we obtain p Y j=1 sin(ω2j−1 2py) = 1 (2i)pX ε1,...,εp=±1p Y j=1 εjexp iyΨ(ε). Multiplying by (iy)−pyields Φp(y) = (−1)p 2py−pX εp Y j=1 εjeiyΨ(ε). 3 Expanding the exponential, eiyΨ(ε)=X m≥0 imΨ(ε)m m!ym, gives Φp(y) = (−1)p 2pX εp Y j=1 εjX m≥0 imΨ(ε)m m!ym−p. The coefficient of y2pn comes from m= 2pn +p, and therefore [y2pn]Φp(y) = (−1)p 2p(2pn +p)! X εp Y j=1 εjiΨ(ε)2pn+p. Multiplying by π2pn completes the proof. Example 2.2 (The case p= 1).For p= 1 we have ω2=iand Ψ(ε) = ε1i. Proposition 2.1 becomes ζ({2}n) = π2n−1 2(2n+ 1)! X ε1=±1 ε1i2n+1(ε1i)2n+1. Since (ε1i)2n+1 =ε1i2n+1, we obtain ε1i2n+1(ε1i)2n+1 =ε2 1i4n+2 =i4n+2, so the sum equals X ε1=±1 i4n+2 = 2 i4n+2 = 2 ·(−1). Therefore ζ({2}n) = π2n−1 2(2n+ 1)! ·2(−1) = π2n (2n+ 1)!, which is the classical closed form. Example 2.3 (The case p= 2).Here ω4=eπi/4and Ψ(ε) = ε1ω4+ε2ω3 4,(ε1, ε2)∈ {±1}2. A direct computation using ω4= (1 + i)/√2,ω3 4= (−1 + i)/√2shows that Ψ(1,1) = √2i, Ψ(1,−1) = √2,Ψ(−1,1) = −√2,Ψ(−1,−1) = −√2i, so Ψ(ε)∈ {±√2,±i√2}and |Ψ(ε)|=√2for all four choices. Moreover, 2 Y j=1 εj=(+1,(ε1, ε2) = (1,1),(−1,−1), −1,(ε1, ε2) = (1,−1),(−1,1). 4 Proposition 2.1 gives ζ({4}n) = π4n1 4(4n+ 2)! X ε1,ε2=±12 Y j=1 εji4n+2Ψ(ε)4n+2. Since |Ψ(ε)|=√2, we have Ψ(ε)4n+2 = (√2)4n+2 ×(a4th root of unity), and using the above table for Ψ(ε)and Qjεjone checks that X ε1,ε2=±12 Y j=1 εji4n+2Ψ(ε)4n+2 = 22n+3. Thus ζ({4}n) = π4n22n+3 4(4n+ 2)! =22n+1π4n (4n+ 2)! =4(2π)4n (4n+ 2)! 1 22n+1 . Example 2.4 (The case p= 3).Here ω6=eπi/6and Ψ(ε) = ε1ω6+ε2ω3 6+ε3ω5 6,(ε1, ε2, ε3)∈ {±1}3. Proposition 2.1 gives ζ({6}n) = π6n−1 8(6n+ 3)! X ε1,ε2,ε3=±13 Y j=1 εji6n+3Ψ(ε)6n+3. Using ω6=√3 + i 2, ω3 6=i, ω5 6=−√3 + i 2, a direct computation shows (ε1, ε2, ε3) Ψ(ε) 3 Y j=1 εj (1,1,1) 2i+1 (1,1,−1) √3 + i−1 (1,−1,1) 0 −1 (1,−1,−1) √3−i+1 (−1,1,1) −√3 + i−1 (−1,1,−1) 0 +1 (−1,−1,1) −√3−i+1 (−1,−1,−1) −2i−1 In particular, the two choices with Ψ(ε) = 0 do not contribute to the sum, since 6n+ 3 >0. For the six nonzero values we note the polar forms 2i= 2eiπ/2,−2i= 2e−iπ/2, √3 + i= 2eiπ/6,−√3−i= 2e−i5π/6, √3−i= 2e−iπ/6,−√3 + i= 2ei5π/6, 5 so in every case |Ψ(ε)|= 2. Set Sn:= X ε1,ε2,ε3=±13 Y j=1 εji6n+3Ψ(ε)6n+3. We evaluate Snby pairing conjugate terms. (i) The pair Ψ(ε) = ±2i.For (1,1,1) and (−1,−1,−1) we have T1:= i6n+3(2i)6n+3 =i6n+3 26n+3i6n+3 = 26n+3i12n+6 =−26n+3, T2:= (−1) i6n+3(−2i)6n+3 =−i6n+3 26n+3i6n+3 =−26n+3i12n+6 =−26n+3. Hence T1+T2=−26n+4. (ii) The pair Ψ(ε) = ±2eiπ/6.For (1,1,−1) and (−1,−1,1) we have Ψ(1,1,−1) = 2eiπ/6,Ψ(−1,−1,1) = −2eiπ/6, and the corresponding contributions are T3:= −i6n+32eiπ/66n+3, T4:= + i6n+3−2eiπ/66n+3 =−i6n+32eiπ/66n+3, so that T3+T4=−2i6n+32eiπ/66n+3. Since 2eiπ/66n+3 = 26n+3ei(6n+3)π/6= 26n+3eiπ(n+1/2) = 26n+3(−1)ni, we obtain T3+T4=−2 26n+3(−1)ni6n+3i. Using i6n+3 = (−1)n+1i, we find i6n+3i= (−1)n, hence T3+T4=−2 26n+3(−1)n(−1)n=−26n+4. (iii) The pair Ψ(ε) = ±2e−iπ/6.The remaining two nonzero cases (1,−1,−1) and (−1,1,1) are treated in the same way, now with Ψ(ε) = ±2e−iπ/6. The computation is identical (using e−iπ/6in place of eiπ/6) and yields T5+T6=−26n+4. Collecting the contributions from (i)–(iii) we obtain Sn= (T1+T2)+(T3+T4)+(T5+T6) = −3·26n+4 =−48 ·26n. Therefore ζ({6}n) = π6n−1 8(6n+ 3)! Sn=π6n−1 8(6n+ 3)! ·−48 ·26n=6(2π)6n (6n+ 3)!, which is the expected closed form. Thus Proposition 2.1 correctly reproduces the closed form [1,6] for repeated index 6. 6 2.2 Case t({2p}n) Proposition 2.5 (Combinatorial representation of t({2p}n)).Let p≥1be an integer and ω2p=eπi/(2p),Ψ(ε) := p X j=1 εjω2j−1 2p, εj∈ {±1}. The generating function [1] X n≥0 x2pn t({2p}n) = p Y j=1 cos ω2j−1 2p πx 2(3) implies the explicit coefficient formula t({2p}n) = π 22pn 1 2p(2pn)! X ε1,...,εp=±1iΨ(ε)2pn.(4) Thus t({2p}n)is given by a symmetric sum of even powers of the algebraic units Ψ(ε), in contrast with the alternating sum appearing in the formula for ζ({2p}n). Proof. Set Λp(x) := p Y j=1 cos ω2j−1 2px. Then substituting x7→ πx/2in (3) yields X n≥0 x2pnt({2p}n)=Λp πx 2=X n≥0π 22pncp,n x2pn, where cp,n is the Maclaurin coefficient of Λp(x). Thus t({2p}n) = π/22pncp,n. Using cos z= (eiz +e−iz)/2, we obtain Λp(x) = 2−pX ε1,...,εp=±1 exp ixΨ(ε). Expanding the exponential yields Λp(x) = 2−pX εX m≥0 (ixΨ(ε))m m!. Root-of-unity symmetry implies that mmust be a multiple of 2p, so only m= 2pn contributes to cp,n = [x2pn]Λp(x). Hence cp,n =1 2p(2pn)! X ε (iΨ(ε))2pn. Multiplying by (π/2)2pn completes the proof (see [9]). 7 Example 2.6 (The case p= 1).For p= 1 we have ω2=iand Ψ(ε) = ε1i. Proposition 2.5 reads t({2}n) = π 22n1 2(2n)! X ε1=±1iΨ(ε)2n. Since iΨ(ε) = i(ε1i) = −ε1, we obtain iΨ(ε)2n= (−ε1)2n= 1 for all n≥1, and hence X ε1=±1iΨ(ε)2n= 2. Therefore t({2}n) = π 22n2 2(2n)! =π2n 22n(2n)!, which agrees with the classical closed form for repeated index 2(with odd denominators). Example 2.7 (The case p= 2).Here ω4=eπi/4and Ψ(ε) = ε1ω4+ε2ω3 4,(ε1, ε2)∈ {±1}2. As in the zeta-example, a direct computation using ω4= (1+i)/√2,ω3 4= (−1+i)/√2shows that Ψ(1,1) = √2i, Ψ(1,−1) = √2,Ψ(−1,1) = −√2,Ψ(−1,−1) = −√2i. Thus Ψ(ε)∈ {±√2,±i√2}, and all four values have the same modulus |Ψ(ε)|=√2. Proposition 2.5 gives t({4}n) = π 24n1 4(4n)! X ε1,ε2=±1iΨ(ε)4n. Since multiplying by ijust rotates these four points, the multiset {iΨ(ε)}is again {±√2,±i√2}. Writing iΨ(ε) = √2eiθ, θ ∈0,π 2, π, 3π 2, we have iΨ(ε)4n= (√2)4nei4nθ = 22nei4nθ, and the four angles θgive X θ∈{0,π/2,π,3π/2} ei4nθ = 4. Hence X ε1,ε2=±1iΨ(ε)4n= 4 ·22n. Substituting into the combinatorial formula yields t({4}n) = π 24n4·22n 4(4n)! =π4n 22n(4n)!, which matches the known closed form for repeated index 4. 8 Example 2.8 (The case p= 3).Here ω6=eπi/6and Ψ(ε) = ε1ω6+ε2ω3 6+ε3ω5 6,(ε1, ε2, ε3)∈ {±1}3. Proposition 2.5 gives t({6}n) = π 26n1 8(6n)! X ε1,ε2,ε3=±1iΨ(ε)6n. Using ω6=√3 + i 2, ω3 6=i, ω5 6=−√3 + i 2, a direct computation yields (ε1, ε2, ε3) Ψ(ε)iΨ(ε) (1,1,1) 2i−2 (1,1,−1) √3 + i−1 + i√3 (1,−1,1) 0 0 (1,−1,−1) √3−i1 + i√3 (−1,1,1) −√3 + i−1−i√3 (−1,1,−1) 0 0 (−1,−1,1) −√3−i1−i√3 (−1,−1,−1) −2i2 Thus two terms give iΨ(ε)=0and do not contribute to the sum for n≥1. For the remaining six values we note the polar representations −2 = 2eiπ,2 = 2ei0, −1±i√3 = 2e±i2π/3,1±i√3 = 2e±iπ/3. In particular, in every nonzero case we have |iΨ(ε)|= 2 and iΨ(ε) = 2eiθ, θ ∈0,±π 3,±2π 3, π. Set Sn:= X ε1,ε2,ε3=±1iΨ(ε)6n. For each of the six nonzero terms we have iΨ(ε)6n=2eiθ6n= 26nei6nθ, and since θis always a multiple of π/3,6θis a multiple of 2π, so ei6nθ = 1 for all integers n≥1. Hence each nonzero term contributes 26n, and the two zero terms contribute 0, so Sn= 6 ·26n. Substituting this back into the combinatorial formula gives t({6}n) = π 26n1 8(6n)! Sn=π 26n6·26n 8(6n)! =3 4 π6n (6n)!, which agrees with the closed form [1] for repeated index 6(with odd denominators). 9 3.4 Example: the case s= 8 We illustrate the above theory at p= 4. From the combinatorial formulas of Section 2, the set of units {Ψ(ε)}contains a unique dominant orbit represented by X= 1 + √2,|X|>1, and all other units have strictly smaller modulus <|X|. Since 3+2√2 = X2, the explicit evaluations of t({8}n)and ζ({8}n)become [1] t({8}n) = π8n 22n+1(8n)!X4n+X−4n, ζ({8}n) = 8(2π)8n 22n+1(8n+ 4)!X4n+2 +X−4n−2. Therefore An(8) = ζ({8}n) ζ({8}n−1) t({8}n−1) t({8}n),˜ An(8) = 2−8An(8), admits the factorization ˜ An(8) = Fn(8) Rn(8), where the factorial term is Fn(8) = (2n−1)(4n−3)(8n−7)(8n−5) (2n+ 1)(4n+ 1)(8n+ 1)(8n+ 3). Explicit unit correction. The dominant-subdominant decomposition of t({8}n)and ζ({8}n) leads to the following explicit closed form: Rn(8) = (X8+X8n)(X8n+4 + 1) X4(X4+X8n)(X8n+ 1) =(1 + X8n−8)(X8n+4 + 1) (1 + X8n−4)(X8n+ 1) , X = 1 + √2. Since |X|>1is the unique dominant modulus and every other unit satisfies |Ψ(ε)|<|X|, Theorem 3.1 implies Rn(8) = 1 + O(X−8n). Consequently, ˜ An(8) = Fn(8)1 + O(X−8n). Asymptotic expansion. Expanding Fn(8) in powers of 1/n gives Fn(8) = 1 −4 n+29 4n2−263 32n3+O1 n4, and since Rn(8) = 1 + O(X−8n)as n→ ∞, the same expansion holds for ˜ An(8): ˜ An(8) = 1 −4 n+29 4n2−263 32n3+O1 n4. 16 Conjecture 3.4 (The case of odd argument s= 3).The factorial–unit factorization developed in this paper applies only to repeated even arguments s= 2p. For the first odd case s= 3, numerical computations suggest that the normalized ratio ˜ An(3) = ζ({3}n) ζ({3}n−1) t({3}n−1) t({3}n) admits an asymptotic expansion in powers of n−1of the form ˜ An(3) = 1 −3 2n+c2(3) n2+c3(3) n3+O1 n4, n → ∞, for certain real constants c2(3) and c3(3). In particular, we conjecture that the coefficient of n−1is −3 2, mirroring the pattern −p/n occurring in the even–argument case s= 2p. Determining the structural origin of this coefficient for odd arguments remains an open problem. In contrast to the even–argument case s= 2p, the odd case s= 3 does not appear to admit a unique dominant algebraic unit in the sense of Section 2.3.2. Rather, several units of equal maximal modulus (or lying on the unit circle) are expected to contribute simultaneously. This breaks the dominance mechanism underlying Theorem 3.1, so that the factorial term alone no longer suffices to explain the leading coefficient in the asymptotic expansion of ˜ An(3). 4 Conclusion We have shown that the repeated-index values ζ({2p}n)and t({2p}n)admit exact closed forms derived directly from their cosine and sine product generating functions. These formulas reveal that all such values are governed by the finite set of algebraic units Ψ(ε) = p X j=1 εjeπi(2j−1)/(2p), εj∈ {±1}. The dihedral symmetry of this unit set produces a unique dominant orbit, and this structural fact yields the canonical factorization ˜ An(2p) = Fn(2p)Rn(2p), in which Fn(2p)is a rational function of nwhile the correction term Rn(2p)is exponentially close to 1. Consequently, the entire asymptotic expansion of ˜ An(2p)in powers of n−1is determined solely by the factorial ratio Fn(2p). The methods developed here rely only on elementary exponential expansions and root-ofunity symmetries, providing a direct and transparent framework that refines and strengthens previously known structural results in the literature. 17 References [1] Hoffman, M. E. (2019). An odd variant of multiple zeta values. Communications in Number Theory and Physics, 13(3), 529–567. [2] Kaneko, M., & Tsumura, H. (2020). Zeta functions connecting multiple zeta values and poly-Bernoulli numbers. Advanced Studies in Pure Mathematics, 84, 181–204. [3] Hoffman, M. E. (2019). Quasi-symmetric functions and mod pmultiple harmonic sums. Kyoto Journal of Mathematics, 59, 345–364. [4] Muneta, S. (2008). On some explicit evaluations of multiple zeta-star values. Journal of Number Theory, 128, 2538–2548. [5] Zudilin, W. (2010). Algebraic relations for multiple zeta values. Russian Mathematical Surveys, 65(3), 553–575. [6] Borwein, J. M., Bradley, D. M., & Broadhurst, D. J. (1997). Evaluations of k-fold Euler/Zagier sums: A compendium of results for arbitrary k.The Electronic Journal of Combinatorics, 4, R5. https://eudml.org/doc/119223 [7] Chen, K.-W., Chung, C.-L., & Eie, M. (2017). Sum formulas of multiple zeta values with arguments multiples of a common positive integer. Journal of Number Theory, 177, 479–496. https://doi.org/10.1016/j.jnt.2017.02.012 [8] Sivaraman, R., Díaz Bulnes, J. J., & López-Bonilla, J. L. (2024). Multiple zeta values and Bell polynomials. European Journal of Applied Science, Engineering and Technology, 2(2), 233–235. https://doi.org/10.59324/ejaset.2024.2(2).16 [9] Yang, G. M. (2025). Repeated-index multiple zeta values and Dirichlet lambda analogues. Zenodo. https://doi.org/10.5281/zenodo.17783827 [10] Elaydi, S. N. (2005). An introduction to difference equations (3rd ed.). Springer. [11] Olver, F. W. J., Lozier, D. W., Boisvert, R. F., & Clark, C. W. (Eds.). (2010). NIST handbook of mathematical functions. Cambridge University Press. 18