Seminar in Number Theory

主講者: (取消)1.鍾展良 (福州大學) 2.余文卿 (國立中正大學)
講題: 1.Some polynomial sequence relations 2.Multiple zeta values with extra insertions
時間: 2020-03-20 (Fri.)  14:00 - ,15:00 -
地點: 數學所 638 研討室 (台大院區)
Abstract: 1.A Lucas polynomial sequence is a pair of generalized polynomial sequences which satisfies the Lucas recursive relation and involves Fibonacci polynomials, Lucas polynomials, Chebyshev polynomials, and Balancing polynomials as its special cases. We give some polynomial sequence relations that are generalizations of the Sury-type identities. Two proofs are provided. One is based on an elementary identity and the other uses the method of generating functions. Secondly, we focus on the $(a,b)$-type Lucas polynomial sequence and then derive a generalized polynomial version of Melham's sums. As an application, we discuss some divisible properties of such power sums.

2.Multiple zeta values and Multiple zeta-star values are natural generalizations of the classical Euler double sums, which are nothing but multiple series with increasing dummy varies.
Here we are going to introduce three kinds of multiple zeta values with extra insertions. The first kind is obtained from a multiple zeta-star values attached to a multiple zeta value in the head to form a double string with the same head. The second kind is obtained from a multiple zeta-star value attached to a multiple zeta value in the tail to form a double string with the same tail. The third kind is obtained from a multiple zeta-star value attached to a multiple zeta value in both the head and tail to form a double string with the same head and tail. Then we perform the operation which is like the stuffle product and solve the following problems.
(1)The evaluations of two families of multiple zeta-star values with arguments consisting of only 1 and 2.
(2) The sum formula due to Granville in 1997.
(3) The evaluation of Riemann zeta value at even integers due to Euler in 1735.
(4) Reflection formulas of some particular multiples zeta values with arguments consisting of one r+2 and the rest 2.

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