The Legendre Polynomials Associated with Bernoulli, Euler, Hermite and Bernstein Polynomials

Some symmetric identities for the generalized Bernoulli, Euler and Genocchi polynomials associated with Hermite polynomials

In 2008, Liu and Wang established various symmetric identities for Bernoulli, Euler and Genocchi polynomials. In this paper, we extend these identities in a unified and generalized form to families of Hermite-Bernoulli, Euler and Genocchi polynomials. The procedure followed is that of generating functions. Some relevant connections of the general theory developed here with the results obtained ...

Extended Laguerre Polynomials Associated with Hermite, Bernoulli, and Euler Numbers and Polynomials

and Applied Analysis 3 The Hermite polynomials are given by Hn x H 2x n n ∑ l 0 ( n l ) 2xHn−l, 1.11 see 23, 24 , with the usual convention about replacing H by Hn. In the special case, x 0, Hn 0 Hn are called the nth Hermite numbers. From 1.11 , we note that d dx Hn x 2n H 2x n−1 2nHn−1 x , 1.12 see 23, 24 , and Hn x is a solution of Hermite differential equation which is given by y′′ − 2xy′ n...

On the Fermionic p-adic Integral Representation of Bernstein Polynomials Associated with Euler Numbers and Polynomials

Throughout this paper, let p be an odd prime number. The symbol, p, p , and p denote the ring of p-adic integers, the field of p-adic rational numbers, the complex number field and the completion of algebraic closure of p , respectively. Let be the set of natural numbers and ∪ {0}. Let νp be the normalized exponential valuation of p with |p|p p−νp p 1/p. Note that p {x | |x|p ≤ 1} lim← N /p p. ...

Some Identities of Bernoulli Numbers and Polynomials Associated with Bernstein Polynomials

On Bernoulli Sums and Bernstein Polynomials

In the paper we discuss a technology based on Bernstein polynomials of asymptotic analysis of a class of binomial sums that arise in information theory. Our method gives a quick derivation of required sums and can be generalized to multinomial distributions. As an example we derive a formula for the entropy of multinomial distributions. Our method simplifies previous work of Jacquet, Szpankowsk...