Incomplete higher-order Gauss sums

We consider the classical incomplete higher-order Gauss sums

     

The structure of the spectra of Pisot numbers

For q∈(1,2), Erdös, Joó and Komornik studied the set:

     

New identities in the Catalan triangle

In this paper we prove new identities in the Catalan triangle whose (n,p) entry is defined by

     

Moments of combinatorial and Catalan numbers

In this paper we obtain the moments {Φm}_m?0 defined by

     

New proofs of identities of Lebesgue and Göllnitz via tilings

In 1840, V.A. Lebesgue proved the following two series-product identities:

     

Hankel determinants of sums of consecutive Motzkin numbers

We use combinatorial methods to evaluate Hankel determinants for the sequence of sums of consecutive t-Motzkin numbers. More specifically, we consider the following determinant:

     

Comments on the spectra of Pisot numbers

For q∈(1,2), Erd?s, Joó and Komornik studied the spectra of q, defined as

     

On p-adic q-l-functions and sums of powers

In this paper, we give an explicit p-adic expansion of

     

The Herglotz-Zagier function, double zeta functions, and values of L-series

In this paper, we define L-series generalizing the Herglotz-Zagier function (Ber. Verhandl. Sächsischen Akad. Wiss. Leipzig 75 (3-14) (1923) 31, Math. Ann. 213 (1975) 153), and the double zeta function (First European Congress of Mathematics, Vol. II, Paris, 1992, pp. 497-512; Progress in Mathematics, Vol. 120, Birkhäuser, Basel, 1994) and evaluate them after meromorphic continuation at integer points in their extended domains. This is accomplished in three steps. First, when is a periodic function and are the harmonic numbers, we establish identities relating these series to the L-series

     

On a max-type and a min-type difference equation

This note shows that every positive solution to the following third order non–autonomous max-type difference equation

when is a three-periodic sequence of positive numbers, is periodic with period three. The same result was proved for the following min-type difference equation

     

Weighted estimates for a class of multilinear fractional type operators

In this paper, a class of multiple fractional type weights is defined as

     

GLOBAL WELL-POSEDNESS FOR A FIFTH-ORDER SHALLOW WATER EQUATION ON THE CIRCLE

The periodic initial value problem of a fifth-order shallow water equation t u 2 x t u + 3 x u 5 x u + 3u x u 2 x u 2 x u u 3 x u = 0 is shown to be globally well-posed in Sobolev spaces˙ H s (T) for s > 2/3 by I-method. For this equation lacks scaling invariance, we first reconsider the local result and pay special attention to the relationship between the lifespan of the local solution and the initial data, and then prove the almost conservation law, and finally obtain the global well-posedness by an iteration process.