are called generalized Laguerre polynomials, or associated Laguerre polynomials. One can also define the generalized Laguerre polynomials recursively, defining the first two polynomials as and then using the following recurrence relation for any : The simple Laguerre polynomials are the special case of the generalized Laguerre polynomials: The Rodrigues formula for them is The generating function for them is Explicit examples and properties of the generalized Laguerre polynomials Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation as where is a generalized binomial coefficient. When is an integer the function reduces to a polynomial of degree . It has the alternative expression in terms of Kummer's function of the second kind. The closed form for these generalized Laguerre polynomials of degree is derived by applying Leibniz's theorem for differentiation of a product to Rodrigues' formula. Laguerre polynomials have a differential operator representation, much like the closely related Hermite polynomials. Namely, let and consider the differential operator . Then . The first few generalized Laguerre polynomials are: The coefficient of the leading term is ; The constant term, which is the value at 0, is If is non-negative, then Ln(α) has n real, strictly positive roots (notice that is a Sturm chain), which are all in the interval The polynomials' asymptotic behaviour for large , but fixed and , is given by and summarizing by where is the Bessel function. As a contour integral Given the generating function specified above, the polynomials may be expressed in terms of a contour integral where the contour circles the origin once in a counterclockwise direction without enclosing the essential singularity at 1 Recurrence relations The addition formula for Laguerre polynomials: Laguerre's polynomials satisfy the recurrence relations in particular and or moreover They can be used to derive the four 3-point-rules combined they give this additional, useful recurrence relations Since is a monic polynomial of degree in , there is the partial fraction decomposition The second equality follows by the following identity, valid for integer i and and immediate from the expression of in terms of Charlier polynomials: For the third equality apply the fourth and fifth identities of this section. Derivatives of generalized Laguerre polynomials Differentiating the power series representation of a generalized Laguerre polynomial times leads to This points to a special case () of the formula above: for integer the generalized polynomial may be written the shift by sometimes causing