Some identities of Laguerre polynomials arising from differential equations

Abstract

In this paper, we derive a family of ordinary differential equations from the generating function of the Laguerre polynomials. Then these differential equations are used in order to obtain some properties and new identities for those polynomials.MSC: 05A19; 33C45; 11B37; 35G35

Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers

“…See [11,Theorem 7]. It is clear that the quantities a i (n) defined by (6.1) play a key role in the above-mentioned conclusions obtained in the paper [11]. However, the quantities a i (n) are expressed complicatedly and can not be computed easily.…”

“…Now we explain the motivation of Theorem 3.1 as follows. In [11], the following results were inductively and recursively obtained.…”

Identities of the degenerate Daehee numbers with the Bernoulli numbers of the second kind arising from nonlinear differential equation

“…New identities were obtained for degenerate Daehee numbers [8] and for Frobenius-Euler polynomials [9] using linear and nonlinear differential equations. Using differential equations, new identities for Bernoulli numbers of the second kind were obtained in [12], for degenerate Euler numbers and polynomials in [14], and for Laguerre polynomials in [18]. Jang and Kim presented some identities of the ordered Bell numbers arising from differential equation [7].…”

“…In [17], Kim et al showed that the following identity holds for the Bernoulli numbers of the second kind…”

Some identities of degenerate Daehee numbers arising from certain differential equations

“…This idea of obtaining some interesting combinatorial identities by using differential equations satisfied by the generating function of special numbers or special polynomials turned out to be very fruitful (see [7,11,13,15]). …”