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

Let 𝐏𝑛={𝑝(𝑥)∈ℝ[𝑥]∣deg𝑝(𝑥)≤𝑛} be an inner product space with the inner product ∫⟨𝑝(𝑥),𝑞(𝑥)⟩=∞0𝑥𝛼𝑒−𝑥𝑝(𝑥)𝑞(𝑥)𝑑𝑥, where 𝑝(𝑥),𝑞(𝑥)∈𝐏𝑛 and 𝛼∈ℝ with 𝛼g−1. 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 interesting relations and identities of the extended Laguerre polynomials associated with Hermite, Bernoulli, and Euler numbers and polynomials.

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