Binomial Transform of Products

Abstract

Given the binomial transforms \(\{b_n\}\) and \(\{c_n\}\) of the sequences \(\{a_n\}\) and \(\{d_n\}\) correspondingly, we compute the binomial transform of the sequence \(\{a_nc_n\}\) in terms of \(\{b_n\}\) and \(\{d_n\}\). In particular, we compute the binomial transform of the sequences \(\{n{n-1}\ldots (n-1-m)a_n\}\) and \(\{a_k x^k\}\) in terms of \(\{b_n\}\). Further applications include new binomial identities with the binomial transforms of the products \(H_n B_n\), \(H_n F_n\), \(H_n L_n(X)\), and \(B_n F_n\), where \(H_n\), \(B_n\), \(F_n\), and \(L_n(X)\) are correspondingly the harmonic numbers, the Bernoulli numbers, the Fibonacci numbers, and the Laguerre polynomials.