Convexity of Quotients of Theta Functions

For fixed $u$ and $v$ such that $0\leq u<v<1/2$, the monotonicity of the quotients of Jacobi theta functions, namely, $\theta_{j}(u|i\pi t)/\theta_{j}(v|i\pi t)$, $j=1, 2, 3, 4$, on $0<t<\infty$ 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 $\theta_{2}(u|i\pi t)/\theta_{2}(v|i\pi t)$ and $\theta_{3}(u|i\pi t)/\theta_{3}(v|i\pi t)$ are convex on $0<t<\infty$.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 $B_{r}$ by the Bernoulli polynomials $B_{r}(x)$. Properties of these new polynomials are established using the umbral method as well as classical techniques. The values of $x$ that yield periodic subsequences $B_{2n+1}^{*}(x)$ are classified. The strange 6-periodicity of $B_{2n+1}^{*}$, 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