Research Article | Open Access
Some Identities on Laguerre Polynomials in Connection with Bernoulli and Euler Numbers
We study some interesting identities and properties of Laguerre polynomials in connection with Bernoulli and Euler numbers. These identities are derived from the orthogonality of Laguerre polynomials with respect to inner product .
As is well known, Laguerre polynomials are defined by the generating function as (see [1, 2]). By (1.1), we get Thus, from (1.2), we have By (1.3), we see that is a polynomial of degree with rational coefficients and the leading coefficient . It is well known that Rodrigues' formula is given by (see [1–27]). From (1.1), we can derive the following of Laguerre polynomials: By (1.7), we easily see that is a solution of the following differential equation of order 2: The Bernoulli numbers, , are defined by the generating function as (see [1–28, 28]), with the usual convention about replacing by .
It is well known that Bernoulli polynomials of degree are given by (see [2, 26]). Thus, from (1.10), we have (see [3–12]). From (1.9) and (1.10), we can derive the following recurrence relation: where is Kronecker's symbol.
In this special case, , are called the th Euler numbers. From (1.13), we note that the recurrence formula of is given by (see ). Finally, we introduce Hermite polynomials, which are defined by (see ). In the special case, , is called the -th Hermite number. By (1.15), we get (see ). It is not difficult to show that In the present paper, we investigate some interesting identities and properties of Laguerre polynomials in connection with Bernoulli, Euler, and Hermite polynomials. These identities and properties are derived from (1.17).
2. Some Formulae on Laguerre Polynomials in Connection with Bernoulli, Euler, and Hermite Polynomials
Theorem 2.1. For , one has
Theorem 2.2. For , one has
Let us take . By the same method, we easily see that
Theorem 2.3. For , one has
Let . Then we have where In , it is known that By (2.14) and (2.15), we get From (2.16), we can derive the following equations ((2.17)-(2.18)): For , we have Therefore, by (2.13), (2.17), and (2.18), we obtain the following theorem.
Theorem 2.4. For , one has
Let us take . By (2.3) and (2.4), we get where It is known (see ) that From (2.20), (2.21), and (2.22), we can derive the following equations ((2.23)-(2.24)): For , we have Therefore, by (2.20) and (2.24), we obtain the following theorem.
Theorem 2.5. For , one has
Theorem 2.6. For , one has
Remark 2.7. Laguerre's differential equation is known to possess polynomial solutions when is a nonnegative integer. These solutions are naturally called Laguerre polynomials and are denoted by . That is, are solutions of (2.29) which are given by From (2.30), we note that Laplace transform of is given by It is not difficult to show that Thus, we conclude that
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