Wen-Chin Liaw

On Russell-Type Modular Equations

In this paper, we revisit Russell-type modular equations, a collection of modular equations first studied systematically by R. Russell in 1887. We give a proof of Russells main theorem and indicate the relations between such equations and the constructions of Hilbert class fields of imaginary quadratic fields. MotivatedbyRussellstheorem,westateandproveitscubicanaloguewhichallowsustoconstructRussell-type modular equations in the theory of signature 3.

ON GENERALIZATIONS OF WEIGHTED SUM FORMULAS OF MULTIPLE ZETA VALUES

In this paper, we compute shuffle relations from multiple zeta values of the form ζ({1}m-1, n+1) or sums of multiple zeta values of fixed weight and depth. Some interesting weighted sum formulas are obtained, such as where m and k are positive integers with m ≥ 2k. For k = 1, this gives Ohno–Zudilins weighted sum formula.

ON η 3 ( aτ ) η 3 ( bτ ) WITH a + b =8

We prove an observation associated with 3 () 3 (7) which is found on page 54 of Ramanujan’s Lost Notebook (S. Ramanujan, The Lost Notebook and Other Unpublished Papers(Narosa, New Delhi, 1988)). We then study functions of the type 3 (a) 3 (b) with a + b = 8. 2000 Mathematics subject classification: 11F03, 11F11, 11F20.

EXPLICIT EVALUATION OF TRIPLE EULER SUMS

The triple Euler sum defined by \[ \begin{array}{rcl} \zeta(p, q, r) & = & \sum\limits_{\ell=3}^{\infty} \frac{1}{\ell^{p}} \sum\limits_{k=2}^{\ell-1} \frac{1}{k^{q}} \sum\limits_{j=1}^{k-1} \frac{1}{j^{r}} \\ [10pt] & = & \sum\limits_{l=1}^{\infty} \sum\limits_{k=1}^{\infty} \sum\limits_{j=1}^{\infty} \frac{1}{(j+k+\ell)^{p}(j+k)^{q}j^{r}} \end{array} \] with positive integers p, q, r, p ≥ 2, has not been fully evaluated yet except for the case of q = r = 1 and some particular cases such as p + q + r ≤ 10 or p = q = r. With the general theory developed for double Euler sums, we are able to produce identities among triple Euler sums with a variable or two when q = r =. Performing differentiations with respect to the specified variable gives the explicit evaluations of ζ(p, q, r) for general p, q, r. In particular, the values of ζ(n, 1, 1), ζ(2n + 1, 1, 2), ζ(2n + 1, 2, 1) and ζ(2n, 2, 2) are given explicitly in terms of classical double Euler sums. Also ζ(2n, 1, 2m + 1) and ζ(2n + 1, 1, 2m) are obtained f...

The Rogers-Ramanujan continued fraction and a quintic iteration for 1/

Properties of the Rogers-Ramanujan continued fraction are used to obtain a formula for calculating 1/π with quintic convergence.

ON EVALUATION OF GENERALIZED EULER SUMS OF EVEN WEIGHT

The classical Euler sum cannot be evaluated when the weight p + q is even unless p = 1 or p = q or (p, q) = (2, 4) or (p, q) = (4, 2) [7]. However it is a different story if instead we consider the alternating sums and They can be evaluated for even weight p + q. In this paper, we shall evaluate a family of generalized Euler sums containing when the weight p + q is even via integral transforms of Bernoulli identities.

Several weighted sum formulas of multiple zeta values

For a real number β≠0 and positive integers m and n with m ≥ 2, we evaluate the sum of multiple zeta values ∑k=1n∑ |α|=n βα 1 βα2 ⋯ βαk ζ(mα1,mα2,…,mαk) explicitly in terms of ζ(m),ζ(2m),…, and ζ(nm). The special case β = 1 gives an evaluation of ζ({m}n). An explicit evaluation of the multiple zeta-star value ζ⋆({m}n) is also obtained, as well as some applications to evaluation of multiple zeta values with even arguments.

The decomposition theorem of products of multiple zeta values of height one

In this paper, we extend the Euler decomposition theorem to a much more general form of the decomposition of the product of n multiple zeta values of height one ∏j=1nζ({1}cj,dj+2) = ∑ |a|=k|b|=r+n∑ |αj|=aj+bj+11≤j≤n−1ζ({1}a0,α1,…,αn−1,bn+1) ×∑σ∈Snσc ∏j=1n ḡj cj σd ∏j=1n hjdj + 1 , where αj = (αj,0,…,αj,aj), ḡj =∑0≤l<j(al − cl), hj = dj + 1 +∑l≥j(bl − dl − 1), Sn is the symmetric group of n objects, and σc, σd are induced permutations of σ on the sets {c1,c2,…,cn} and {d1,d2,…,dn}. The case c = 0 and n = 2 gives the classical Euler decomposition theorem.