44 research outputs found
General Convolution Identities for Bernoulli and Euler Polynomials
Using general identities for difference operators, as well as a technique of
symbolic computation and tools from probability theory, we derive very general
kth order (k \ge 2) convolution identities for Bernoulli and Euler polynomials.
This is achieved by use of an elementary result on uniformly distributed random
variables. These identities depend on k positive real parameters, and as
special cases we obtain numerous known and new identities for these
polynomials. In particular we show that the well-known identities of Miki and
Matiyasevich for Bernoulli numbers are special cases of the same general
formula.Comment: 20 page
Some new identities on the Apostol-Bernoulli polynomials of higher order derived from Bernoulli basis
In the present paper, we obtain new interesting relations and identities of
the Apostol-Bernoulli polynomials of higher order, which are derived using a
Bernoulli polynomial basis. Finally, by utilizing our method, we also derive
formulas for the convolutions of Bernoulli and Euler polynomials, expressed via
Apostol-Bernoulli polynomials of higher order.Comment: 8 pages, submitte
Identities Involving Some New Special Polynomials Arising from the Applications of Fractional Calculus
Inspired by a number of recent investigations, we introduce the new analogues of the Apostol-Bernoulli polynomials and the
Apostol-Euler polynomials, the Apostol-Genocchi polynomials based on Mittag-Leffler function. Making use of the Caputo-fractional
derivative, we derive some new interesting identities of these polynomials. It turns out that some known results are derived as special
cases
Central factorials under the Kontorovich-Lebedev transform of polynomials
We show that slight modifications of the Kontorovich-Lebedev transform lead
to an automorphism of the vector space of polynomials. This circumstance along
with the Mellin transformation property of the modified Bessel functions
perform the passage of monomials to central factorial polynomials. A special
attention is driven to the polynomial sequences whose KL-transform is the
canonical sequence, which will be fully characterized. Finally, new identities
between the central factorials and the Euler polynomials are found.Comment: also available at http://cmup.fc.up.pt/cmup/ since the 2nd August
201