140 research outputs found
Convexity of Quotients of Theta Functions
For fixed and such that , the monotonicity of the
quotients of Jacobi theta functions, namely, , , on has been established
in the previous works of A.Yu. Solynin, K. Schiefermayr, and Solynin and the
first author. In the present paper, we show that the quotients
and are convex on .Comment: 17 pages, 6 figure
The Zagier modification of Bernoulli numbers and a polynomial extension. Part I
The modified B_{n}^{*} = \sum_{r=0}^{n} \binom{n+r}{2r} \frac{B_{r}}{n+r},
\quad n > 0 introduced by D. Zagier in 1998 are extended to the polynomial case
by replacing by the Bernoulli polynomials . Properties of
these new polynomials are established using the umbral method as well as
classical techniques. The values of that yield periodic subsequences
are classified. The strange 6-periodicity of ,
established by Zagier, is explained by exhibiting a decomposition of this
sequence as the sum of two parts with periods 2 and 3, respectively. Similar
results for modifications of Euler numbers are stated.Comment: 35 pages, Submitted for publicatio
- …