4,423 research outputs found
Extension of the Bernoulli and Eulerian Polynomials of Higher Order and Vector Partition Function
Following the ideas of L. Carlitz we introduce a generalization of the
Bernoulli and Eulerian polynomials of higher order to vectorial index and
argument. These polynomials are used for computation of the vector partition
function , i.e., a number of integer solutions to a linear
system . It is shown that can be expressed through the vector Bernoulli polynomials of higher order.Comment: 18 page
An Explicit Formula for Restricted Partition Function through Bernoulli Polynomials
Explicit expressions for restricted partition function and
its quasiperiodic components (called Sylvester waves) for a
set of positive integers are derived. The
formulas are represented in a form of a finite sum over Bernoulli polynomials
of higher order with periodic coefficients.Comment: 8 pages, submitted to The Ramanujan Journa
Restricted Partition Functions as Bernoulli and Euler Polynomials of Higher Order
Explicit expressions for restricted partition function and
its quasiperiodic components (called {\em Sylvester waves})
for a set of positive integers are
derived. The formulas are represented in a form of a finite sum over Bernoulli
and Euler polynomials of higher order with periodic coefficients. A novel
recursive relation for the Sylvester waves is established. Application to
counting algebraically independent homogeneous polynomial invariants of the
finite groups is discussed.Comment: 15 pages, 2 figures, references added, submitted to The Ramanujan
Journa
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