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′ ny 0, 1.13 see 1–6, 23–32 . Throughout this paper we assume that α ∈ R with α > −1. Let Pn {p x ∈ R x | deg p x ≤ n}. Then Pn is an inner product space with the inner product 〈p x , q x 〉 ∫∞ 0 x αe−xp x q x dx, where p x , q x ∈ Pn. By 1.4 the set of the extended Laguerre polynomials {L0 x , Lα1 x , . . . , Ln x } is an orthogonal basis for Pn. In this paper we study the properties of the extended Laguerre polynomials which are an orthogonal basis for Pn. From those properties, we derive some new and interesting relations and identities of the extended Laguerre polynomials associated with Hermite, Bernoulli and Euler numbers and polynomials. 2. On the Extended Laguerre Polynomials Associated with Hermite, Bernoulli, and Euler Polynomials For p x ∈ Pn, p x is given by p x n ∑ k 0 CkL α k x , for uniquely determined real numbers Ck. 2.1 From 1.3 , 1.4 , and 2.1 , we note that 〈 p x , Lαk x 〉 Ck 〈 Lαk x , L α k x 〉 Ck ∫∞ 0 xeLk x L α k x dx Ck Γ α k 1 k! . 2.2 4 Abstract and Applied Analysis Thus, by 2.2 , we get Ck k! Γ α k 1 〈 p x , Lαk x 〉