Schur type poly-Bernoulli numbers
The poly-Bernoulli numbers and its relative are defined by the generating series using the polylogarithm series, and we call them type $B$ and $C$, respectively. As a generalization of these poly-Bernoulli numbers, we introduce Schur type poly-Bernoulli numbers and investigate their properties. First, we define a generalization of Arakawa-Kaneko multiple zeta functions and obtain their expression in terms of Schur type Bernoulli numbers. Next, under the restriction to the hook type, we define a generalization of Kaneko-Tsumura multiple zeta functions and obtain similar expression in terms of Schur type Bernoulli numbers. Lastly, we study more properties such as a recurrence formula, a relation formula between Bernoulli numbers and a description in terms of the Stirling numbers.
We deal with linear parabolic (in the sense ::: of Petrovskii) systems of order $2b$ with discontinuous principal ::: coefficients. A priori estimates in Sobolev and ::: Sobolev--Morrey spaces are proved for the strong solutions by ::: means of potential analysis and boundedness of certain singular ::: integral operators with kernels of mixed homogeneity. As a ::: byproduct, precise characterization of the Morrey, $BMO$ and ::: Holder regularity is given for the solutions and their ::: derivatives up to order $2b-1.$