New Formulas of Bernoulli Polynomials of the Second Kind Using Several Approaches

This article presents several new analytical results for a modified class of Bernoulli polynomials, namely, the Bernoulli polynomials of the second kind (BPs2). The paper mainly develops new connection and inverse connection formulas between the first and second kinds of Bernoulli polynomials using two different approaches. One of these approaches uses the generating functions for both polynomial families, whereas the other employs the power series representation, along with its inverse formula and certain closed forms of sums. Another principal contribution of the paper is the derivation of new explicit formulas for moments, derivatives, and higher-order derivatives of the BPs2, together with inverse derivative formulas and mixed linearization formulas involving several polynomial families, including Chebyshev-type and generalized Fibonacci polynomials. Furthermore, a collection of new definite integral formulas associated with the BPs2 is established. The obtained formulas provide new operational representations for the BPs2 and may be useful in spectral methods, basis transformations, and the treatment of differential equations involving polynomial approximations.